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Only the declared critical replay is supported by this focused import.", + "prior_cumulative_preserved_locally": true, + "relocated_provenance": [ + { + "original_path": "research/kappa-next-regime/README.md", + "reason": "Canonical main already has a different README. Preserve that file and retain the historical notice separately. This README is not an executed scientific input.", + "retained_path": "research/kappa-0023-productive-jobs/predecessor-notices/kappa-next-regime-README.md", + "sha256": "bc5f95abac1e31288b0bed45fa550a6fbdfbec666525265fe8ed83128d42bf65" + } + ], + "schema": "conditional-kappa-0023-import/1" +} diff --git a/research/kappa-0023-productive-jobs/LITERATURE.md b/research/kappa-0023-productive-jobs/LITERATURE.md new file mode 100644 index 0000000000..c6b6623c52 --- /dev/null +++ b/research/kappa-0023-productive-jobs/LITERATURE.md @@ -0,0 +1,300 @@ +# Literature and its role in the construction + +These sources shaped the research behind the proposed κ = 0.0023. They are +grouped by the part of the construction they informed. Each entry gives a +standard citation, a primary link and a short note on what the source suggested +and what we actually used. The notes describe this research's use of each +source, not claims made by its authors. Sources behind unsuccessful branches +are included because they explain the turns the search took. An analogy +supplies no native instruction, restoration guarantee or unpaid recursive +operation; every borrowed idea was rebuilt and checked in the repository's +native model. + +This consolidates the delivered checkpoint's `LITERATURE-AND-PROBES.md`, +`architecture/LITERATURE.md`, `bit/LITERATURE.md` and the earlier research +handoff. Citation metadata was checked against primary records on 21 October +2026. + +## Productive work and mixed accounting + +1. **Jacob Alan Sussman** (2026). “Theorem 1G: mixed-target recursion.” + [PR #4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268) · + [pinned proof](https://github.com/jacobalansussman/wht-power-saving-lean/blob/563b4e6e913625637f36ba7869bf5fcd0dd4975a/research/record-track/proofs/theorem-1g.md) + + The direct algorithmic predecessor. Its integer Pascal-class lockstep flow + matches unequal target jobs with smaller paid demands under an exact moment + balance, splitting each source as `s + (n − s)` and paying the remainder + recursively. It is the core of our compiler. Our contribution is realizing + that mechanism as a finite integer compiler with native adapters, an explicit + winning rank and the outer transfer. [#4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268) claims a DFT improvement and no κ; its Lean results + do not verify this compiler or the multiplication claim. + +2. **Arnold Schönhage** (1981). “Partial and Total Matrix Multiplication.” + *SIAM Journal on Computing* 10(3), 434–455. + [doi:10.1137/0210032](https://doi.org/10.1137/0210032) + + The classic precedent for assembling useful partial computations into a + larger complete one, which is the shape of the productive-job idea. The + tensor theorem served only as motivation; the native source/exit schedule + and paid complements carry the actual bridge. + +3. **Péter Vrana** (2024). “Partial and weighted matrix multiplication.” + arXiv:2408.15728. [arXiv](https://arxiv.org/abs/2408.15728v1) + + Reinforced that unequal, weighted useful work deserves its own accounting. + It encouraged us to keep mixed ranks rather than round them to a uniform + job. Its support-rank theorem is not applied to transform programs. + +4. **Robert Robere and Jeroen Zuiddam** (2021). “Amortized Circuit Complexity, + Formal Complexity Measures, and Catalytic Algorithms.” ECCC TR21-035. + [ECCC](https://eccc.weizmann.ac.il/report/2021/035/) + + Motivated one production-and-consumption ledger with an explicit price for + every resource. Its repeated-output setting reuses a single input, so it + grants no independent-output credit here. + +## Building the controller over the legal coefficient ring + +5. **J. J. Koliha** (2001). “Block diagonalization.” *Mathematica Bohemica* + 126(1), 237–246. [EMIS](https://emis.de/ft/45438) + + Resolutions of the identity by idempotents pointed us at the operator + coefficients of the whole controller rather than its individual calls. That + view informed the exact coefficient factorizations tested for p3, including + the integral boundary factorization that removes two idle windows. + +6. **Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel** (2023). + “Dyadic linear programming and extensions.” arXiv:2309.04601. + [arXiv](https://arxiv.org/abs/2309.04601) + + Made the denominator restriction a first-class constraint. Power-of-two + balancing and shear factorizations then avoided an unavailable division + by 13. Its general LP algorithm was not implemented. + +7. **Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel** + (2021; rev. 2022). “Total dual dyadicness and dyadic generating sets.” + arXiv:2111.05749. [arXiv](https://arxiv.org/abs/2111.05749v2) + + Separated a rational feasibility witness from a realization over the + allowed dyadic ring. This shaped the search's rejection gates. + +8. **Ingrid Daubechies and Wim Sweldens** (1998). “Factoring Wavelet Transforms + into Lifting Steps.” *Journal of Fourier Analysis and Applications* 4(3), + 247–269. [author manuscript](https://9p.io/who/wim/papers/factor/factor.pdf) + + Lifting suggested reversible triangular factors for the six-bank transfer in + place of generic scalar elimination. Frame contacts, coefficients, + preparation and restoration were still checked natively. + +9. **Christopher M. Brislawn** (2024). “The Causal Complementation Algorithm + for Lifting Factorization of Perfect Reconstruction Multirate Filter Banks.” + arXiv:2408.07970. [arXiv](https://arxiv.org/abs/2408.07970v1) + + Suggested factors constrained by *when* boundary data become available, not + unconstrained matrix elimination. We used exact legal shears and verified + the actual contact times. + +10. **Christopher M. Brislawn** (2019; final rev. 2024). “Factoring Perfect + Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine + Approach.” *Journal of Computational Algebra* 12, 100024. + [arXiv:1902.09040](https://arxiv.org/abs/1902.09040v5) + + Sharpened the same causal-divisibility question in an earlier handoff. It + never licensed an unavailable contact or an uncharged inverse. + +11. **Hyung-Ju Park** (1995). *A Computational Theory of Laurent Polynomial + Rings and Multidimensional FIR Systems.* PhD thesis, UC Berkeley, + UCB/ERL M95/39. + [report](https://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/ERL-95-39.pdf) + + The factor `I − P + zP` (§7.4) guided exact projector/delay + representations, checked with rational oblique projectors and their full + coefficient identities. No orthogonal rotations were imported. + +12. **Zinnur Doğanata, P. P. Vaidyanathan and Truong Q. Nguyen** (1988). + “General synthesis procedures for FIR lossless transfer matrices, for + perfect-reconstruction multirate filter bank applications.” *IEEE Trans. + Acoustics, Speech, and Signal Processing* 36(10). + [doi:10.1109/29.7544](https://doi.org/10.1109/29.7544) · + [Caltech copy](https://authors.library.caltech.edu/records/mv71c-vqj75) + + Inspired a transfer-matrix view of the controller with an explicit boundary + basis. Its lossless lattices do not license irrational rotations or free + frame changes. + +13. **John Urschel** (2023). “Representing the Special Linear Group with Block + Unitriangular Matrices.” arXiv:2305.00485. + [arXiv](https://arxiv.org/abs/2305.00485v2) + + Motivated short block-shear controllers in the earlier redesign. Its + field-level factor counts were leads only; dyadic coefficients, contacts + and the recursive bill were checked separately. + +## Arbitrary dirty state, restoration and causal flow + +14. **Jean-Guillaume Dumas and Bruno Grenet** (2024). “In-place accumulation + of fast multiplication formulae.” ISSAC 2024; arXiv:2307.12712. + [arXiv](https://arxiv.org/abs/2307.12712v3) + + Suggested borrowing completed output arrays as dirty scratch and restoring + them exactly. Every native contact and reverse operation stayed priced. + +15. **Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff and Florian + Speelman** (2014). “Computing with a full memory: Catalytic space.” + ECCC TR14-053. [ECCC](https://eccc.weizmann.ac.il/report/2014/053/) + + Supplied the discipline for borrowed scratch: it starts arbitrary and must + return exactly. Productive jobs meet a different contract, finishing with + the declared transform of their arbitrary input. Keeping the two + requirements apart is what lets the construction credit productive work. + It also motivated all-column dirty-bank tests rather than zero-initialized + examples. + +16. **Antoine Vinciguerra** (2026). “An Operator Approach to Register Programs + for Catalytic Computing.” arXiv:2609.18692. + [arXiv](https://arxiv.org/abs/2609.18692v1) + + Motivated stating restoration as a whole-operator identity and checking + whole arbitrary register columns. Its access counts were not imported. Its + polynomial-degree and passive-output assumptions differ from our linear + jobs, and its matrix lifting and coefficient divisions need not stay + within the permitted dyadic coefficients. + +17. **Marten Folkertsma, Ian Mertz, Sergii Strelchuk and Sathyawageeswar + Subramanian** (2026). “Fast Cliffords When Your Quantum Memory Is Full.” + arXiv:2609.40144. [arXiv](https://arxiv.org/abs/2609.40144v1) + + Prompted a dirty-workspace cycling probe, which was rejected. Its + restored-workspace premise fails for our accumulating dirty logger. + +18. **Joseph Connelly and Kenneth Zeger** (2016). “Linear Network Coding over + Rings, Part II: Vector Codes and Non-Commutative Alphabets.” + arXiv:1608.01737. [arXiv](https://arxiv.org/abs/1608.01737v2) + + Kept coding over the actual coefficient structure separate from a + convenient field-rank count. Acquisition, transport, scatter and erasure + therefore stay in the causal interface and the invoice. + +19. **Mark Braverman and Zhongtian He** (2026). “The Multiple Unicast + Conjecture is False.” arXiv:2610.10108. + [arXiv](https://arxiv.org/abs/2610.10108v1) + + Motivated timed, operator-valued carriers with causal local processing. + Its characteristic-three code and throughput factors were not transferred. + +20. **Vadim Kostrykin and Robert Schrader** (2001). “The Generalized Star + Product and the Factorization of Scattering Matrices on Graphs.” *Journal + of Mathematical Physics* 42, 1563–1598. + [arXiv:math-ph/0008022](https://arxiv.org/abs/math-ph/0008022) + + Suggested composing through retained interface variables in earlier work. + It remained an analogy; feedback inverses and recursive costs could not be + omitted. + +## BIT synthesis, state bounds and the H branch + +21. **Ketan N. Patel, Igor L. Markov and John P. Hayes** (2008). “Efficient + Synthesis of Linear Reversible Circuits.” *Quantum Information and + Computation* 8(3–4), 282–294. + [arXiv:quant-ph/0302002](https://arxiv.org/abs/quant-ph/0302002) + + Motivated explicit binary ADD/SWAP synthesis with shared linear forms. It + was used for exact F₂ synthesis and never for characteristic-zero complex + arithmetic. The archived BIT note cites the preprint under another title. + +22. **Andrea Visconti, Chiara V. Schiavo and René Peralta** (2018). “Improved + upper bounds for the expected circuit complexity of dense systems of linear + equations over GF(2).” *Information Processing Letters* 137. + [NIST](https://www.nist.gov/publications/improved-upper-bounds-expected-circuit-complexity-dense-systems-linear-equations-over) + + Suggested keeping shared dense sums alive until all consumers finish. The + resulting deferred reconstruction did not remove the physical frame or the + dirty baseline. + +23. **Alex Parent, Martin Roetteler and Krysta M. Svore** (2015). “Reversible + circuit compilation with space constraints.” arXiv:1510.00377. + [arXiv](https://arxiv.org/abs/1510.00377) + + Motivated joint lifetime and cleanup accounting. The helper-pair census + then rejected all five geometrically eligible coalescings; lifetime overlap + alone was not enough for native reuse. + +24. **Dmitry Chistikov, Szabolcs Iván, Anna Lubiw and Jeffrey Shallit** (2017). + “Fractional Coverings, Greedy Coverings, and Rectifier Networks.” + STACS 2017, LIPIcs 66, 23. + [doi:10.4230/LIPIcs.STACS.2017.23](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23) + + Motivated shared consumer maps and source-neighborhood covers. Rectifier + size does not pay for restoration or native transport. + +25. **G. David Forney Jr.** (2009; rev. 2010). “Minimal realizations of linear + systems: The ‘shortest basis’ approach.” arXiv:0910.4336. + [arXiv](https://arxiv.org/abs/0910.4336v3) + + Framed the question as the state needed across an actual time cut. The + nine-center rank checks then ruled out an eight-signal carrier at that cut, + leaving other cuts open. + +26. **Navin Kashyap** (2007). “On Minimal Tree Realizations of Linear Codes.” + arXiv:0711.1383. [arXiv](https://arxiv.org/abs/0711.1383) + + State requirements depend on the decomposition and consumer interface. This + scoped the same fixed-cut obstruction rather than a universal lower bound. + +27. **Clément Pernet and Arne Storjohann** (2018). “Time and space efficient + generators for quasiseparable matrices.” *Journal of Symbolic Computation* + 85, 224–246. [arXiv:1701.00396](https://arxiv.org/abs/1701.00396) + + Motivated compact exact state across cuts in earlier observer probes, whose + measured gaps exposed insufficient state in the tested contract. + +28. **Itziar Baragaña and Alicia Roca** (2019). “Fixed rank perturbations of + regular matrix pencils.” arXiv:1907.10657. + [arXiv](https://arxiv.org/abs/1907.10657v2) + + Distinguished prescribed-rank completion from a loose rank bound. Its + field-size hypotheses were not imported into F₂; the BIT completion was + proved and tested directly. + +29. **A. Amparan, I. Baragaña, S. Marcaida and A. Roca** (2023; rev. 2024). + “Row or column completion of polynomial matrices of given degree.” + arXiv:2306.06104. [arXiv](https://arxiv.org/abs/2306.06104v2) + + Suggested completing only the designated cheap output rows. Checking + invertibility instead of requiring an involution produced the three-row BIT + completion. Endpoint debt still blocks a higher claim. + +30. **Christophe Reutenauer** (1994). “A Survey on Noncommutative Rational + Series.” FPSAC 1994, 393–402. + [proceedings](https://fpsac-archive.github.io/FPSAC94/ARTICLES/45Reutenauer.pdf) + + Motivated testing the coefficient space of all boundary words jointly. The + resulting span obstruction rejects the universal two-passage model and + leaves geometry-aware coding open. + +31. **Jean Berstel and Christophe Reutenauer** (2010). “Minimization.” Ch. 2 of + *Noncommutative Rational Series with Applications*, Cambridge University + Press, 29–46. + [chapter](https://www.cambridge.org/core/books/abs/noncommutative-rational-series-with-applications/minimization/ACD53CC08C3CB9451564D3488D24FE22) + + Supplied the minimal-realization viewpoint for the same test. The native + obstruction has its own direct proof. + +## Retained outer inputs + +32. **Volker Strassen** (1969). “Gaussian elimination is not optimal.” + *Numerische Mathematik* 13, 354–356. + [doi:10.1007/BF02165411](https://doi.org/10.1007/BF02165411) + + The retained boundary-matrix algorithm. The exact comparison + `7^16 < 2^45` permits the conservative exponent 45/16 in the outer ledger. + +33. **Timothy S. Trudgian** (2014). “Updating the error term in the prime + number theorem.” arXiv:1401.2689. + [arXiv](https://arxiv.org/abs/1401.2689v2) + + Theorem 2 is a cited input to the inherited fine-prime padding argument. + The local verifier does not re-prove it. + +Apache-2.0 for this commentary; cited works and retained third-party sources +keep their respective rights. diff --git a/research/kappa-0023-productive-jobs/PR-BODY.md b/research/kappa-0023-productive-jobs/PR-BODY.md new file mode 100644 index 0000000000..ba794d6de8 --- /dev/null +++ b/research/kappa-0023-productive-jobs/PR-BODY.md @@ -0,0 +1,612 @@ +**Conditional candidate with a complete local replay; awaiting external mathematical review.** + +The winning uniform templates are finite and specified explicitly by +algorithm, but at astronomically large ranks: `2^276540184361` for complex and +`2^6552928412` for BIT before covering. Their instruction lists and constants +have not been materialized. **This is an asymptotic existence result, not a +practical algorithm.** + +## Result + +$$ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=0.0023. +$$ + +| Reference point | κ | Gain from this PR | +|---|---:|---:| +| [#1094](https://github.com/CrocSwap/integer-mult-bounds/pull/1094), selected frontier (maintainer-reviewed) | 0.001672660197 | +37.51% | +| [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181), highest conditional claim located in public PRs | 0.001730980256 − 10⁻⁴⁶ − 10⁻⁷⁰ | **+32.87%** | +| **This PR** | **0.0023** | | + +The increase over [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181) is 5.690 × 10⁻⁴ in the exponent saving. The +percentages compare those endpoints, not measured running times. + +Recent conditional gains have been 10⁻⁹ to 10⁻⁷ per step, obtained by +refilling the same staged words. [#4091](https://github.com/CrocSwap/integer-mult-bounds/pull/4091) proved the pooled ceiling for those +words, and [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181) attained it, noting that any further gain needs a changed word +or inventory. This PR changes what the recursion counts as useful output. +Helpers that finish holding a completed transform of their own arbitrary input +keep that work instead of undoing it, so one local unit produces jobs of several +ranks. A finite integer compiler then turns that mixed rule back into an +ordinary uniform supplier, so the inherited outer theorem can use it. + +## The key idea: count the work helpers already do + +In the uniform-supplier interface every local unit returns its helpers to the +identity. Each helper's arbitrary input is borrowed, transformed in passing and +then restored. Often, though, a helper ends a controller holding a *known, +completed* transform of its own arbitrary input. Restoring that input pays twice +for work that could be delivered as output. + +We keep these **productive jobs** at whatever rank they happen to have and +price the controller by its complete mixed balance. With `W_k` independent +productive outputs of rank `k` and `p_r` paid recursive calls of rank `r`, the +mixed saving `C` solves + +$$ +\sum_r p_r\, r^{1-C}=\sum_k W_k\, k^{1-C}. +$$ + +For example, a helper that has completed a width-8 transform of its own input +earns nothing under uniform accounting. It must be restored, or forced into a +full rank-57 output, before it counts. Under mixed accounting it is simply a +rank-8 productive job and contributes `8^{1−C}` to the right-hand side. The p3 +invoice below books 1,570 such helper jobs of ranks 6–9 beside its 672 main +outputs. + +That change of accounting changed which controllers were worth building. A +**pK controller** is a path of `K` call pairs, `2K + 1` local calls in all. An +**idle window** is a stretch of stages in which a bank of 224 fields sits +untouched by the main calls, so helper work can be scheduled there. Even the +first complete p2 cover beats [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181)'s supplier endpoint (0.0017310): + +| Controller (complex) | What changed | Mixed saving `C` | +|---|---|---:| +| p2, complete identity cover | Helpers priced at their own ranks | 0.0017730 | +| p2, initial-prefix deletion | Early idle windows removed literally | 0.0020133 | +| **p3, seven calls** | Integral boundary factorization; adapters replace idle paths | **0.0023051** | + +The p3 controller keeps four idle windows: two of rank 16 on stages 0/1 and +5/6, and two of rank 8 on stages 2 and 4. An exact, integral factorization of +its boundary operator removes two more. Exact input/output scalar adapters +take the place of those whole idle paths, which did no useful work, and that is +what yields rank-25 and rank-57 main outputs. The emitted word +records every normalizer, absolute phase, source complement, owned center and +helper endpoint. + +**How the controllers were found.** Candidates were discovered with +floating-point LP column generation. Each reported construction was then +reconstructed as exact rational patterns, and its inventory, chronology, +target mass and paid ranks were checked again without the optimizer. Larger +menus did not beat p3: paired controllers through p6, 64 p3 single-field masks +and 123,904 commuting stage allocations. These are finite search results, not +optimality claims. + +A favorable balance alone proves nothing. Productive jobs have different ranks, +exist at different times and live in different physical frames. The remaining +work converts this local tariff into a κ. + +## From a mixed tariff to κ + +### 1. Compile mixed jobs into one uniform supplier + +The inherited outer theorem needs a *uniform* supplier, in which every output +has the same rank. The obvious fix is to rotate the mixed roles so that each +array takes every role once, and it fails. Averaging replaces `Σ W_k k^τ` by a +smaller concave expression, and the mixed saving disappears. The compiler +instead keeps the hierarchy of sources, complements and exits. + +It builds on the mixed-target lockstep flow of [@jacobalansussman](https://github.com/jacobalansussman)'s +Theorem 1G ([#4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268)). At a fixed top rank `N`, a Pascal-class bijection matches +each paid round's jobs to target roles of a child class at exactly equal integer +ranks. Missing demands become top-level sources of rank `s`, and the deepest +classes emit strictly smaller exit jobs of rank `e`. As in Theorem 1G, each +source array splits as `s + (N − s)`, and its complementary `N − s` axes are +completed recursively. Here the already-compiled smaller uniform program pays +for them, so every array ends with the full rank-`N` transform. With `L(n)` the +paid rank-one leaves per productive array, `w_s` normalized source weights and +`x_e` exit counts normalized by the number of source arrays, + +$$ +L(n)=\sum_s w_s\,L(n-s)+\sum_e x_e\,L(e),\qquad \sum_s w_s=1 . +$$ + +At `τ = 1 − C` the source and exit potentials are equal. Strict child/target +separation (`r_max < k_min`) makes exit ranks sublinear in `n`, so the exits +become negligible at any slightly larger exponent `z₀`. Concavity of the +complements then gives `L(n) ≤ A n^{z₀}`. Some explicit finite rank therefore +satisfies + +$$ +P_N < B_N\,N^{z}\qquad(z_0 25,600,000 N^5`, and every bad class +pays its full fallback bill. + +### 3. Re-derive precision and storage for the new template + +A compiled word this large cannot inherit the old supplier's small constants. +[`FINITE-TEMPLATE-CLOSURE.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md) derives new ones from the template's +own finite counts. The semantic precision guard stays linear in depth, `C0·d`, +because children return completed values on one fine grid. Their temporaries +are never simultaneously unfinished parent operands. The simultaneous reserve +becomes `q0(h,p) ≤ C(log₂h + h^0.003 (log₂p)^3 + 1)`, with reserved bits +deferred to later shrinking phases. + +### 4. Pay the complete outer bill + +Every retained outer cost has exponent saving strictly above 0.0023: + +| Outer cost | Saving | +|---|---:| +| Main transform | 0.00230197698 | +| Global coordinate routing | 0.0023025 | +| Fused Gaussian work | 0.00231193094 | +| Group BIT routing | 0.002312430925 | +| Boundary matrix work | 0.00230297698 | +| Sparse union movement | 0.40855297698 | +| Group scans | 0.99998 | +| Polynomial products | 0.99999 | +| Microbox face corrections | 77.00004 | + +The minimum slack is exactly `98849/50000000000 = 0.00000197698`. A separate +outer auditor recomputes all nine rows and the changed functional inequalities. +Boundary matrix work uses Strassen with the exact comparison `7^16 < 2^45`. + +## Complete invoices + +The counts below belong to each **local p3 mixed unit**, and their balance gives +the mixed root. BIT figures are before the cover. + +| Local mixed unit | Complex | BIT | +|---|---|---| +| Productive jobs `{rank: count}` | `{6:192, 7:56, 8:93, 9:1229, 25:448, 57:224}` | `{8:3073, 22:444, 50:222}` | +| Paid calls `{rank: count}` | `{1:17623, 2:5065, 3:1738, 4:714, 5:254}` | `{1:9261, 2:9394, 3:553, 4:308, 5:714, 6:77, 7:1465}` | +| Independent productive fields | 2,242 | 3,739 | +| Total paid calls | 25,394 | 21,772 | +| Scalar allowance | 1,147,304 real primitives | 174,908 ADDs (swap = 3 ADDs) | +| Mixed root `C` | 0.0023050901948… | 0.0027698302… | + +The compiler turns each unit into a uniform supplier with only rank-one paid +leaves. Its own counts are finite but enormous, and they are not the unit counts +above. + +| Compiled uniform supplier | Complex | BIT | +|---|---|---| +| Explicit uniform rank | `2^276540184361` | `2^6552928412` | +| Uniform saving at that rank | 0.002304 | 0.002760 (before cover) | +| Saving entering the outer bill | **0.002304** (binding) | 0.0027 (after cover fallback and conversion) | + +The exact root enclosures are `[0.002305090194876170622919094755, +0.002305090194876170622919094756]` (complex) and `[0.002769830250, +0.002769830251]` (BIT). + +## Verification + +Based on canonical main `12fa8c5c844be21bfefb38a19503fbd867a11244` (priority +revision 7). Python standard library only: + +```sh +python3 -B research/kappa-0023-productive-jobs/verify.py --check-only +python3 -B research/kappa-0023-productive-jobs/verify.py --out /absolute/path/to/new-replay +``` + +- A fresh sequential replay outside the checkout passes **23 scientific stages + and 59 artifact comparisons**. It covers native reconstruction, phases, owners + and bills, the finite compiler, the delayed BIT cover, the new constants, the + explicit ranks and the independent outer audit. Logs are in + [`research/kappa-0023-productive-jobs/evidence/`](https://github.com/zxv/integer-mult-bounds/tree/codex/conditional-kappa-0023/research/kappa-0023-productive-jobs/evidence). +- The review runner fixes a stale-pin ordering defect in the original runner and + compares receipts across hosts under declared metadata rules. No mathematical + field is waived, and four mutation controls, such as a changed target κ, are + rejected. +- The import binds 365 unchanged delivered files plus 16 canonical-main inputs. + The critical replay does not rerun every historical negative search. + +**Where to start reading** + +- [`OUTER-TRANSFER-AUDIT.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md): proof entry point and the complete outer bill. +- [`FINITE-PRECOMPILATION.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md): the finite integer compiler (step 1). +- [`DELAYED-COVER-BRIDGE.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md): the delayed BIT cover and atom conversion (step 2). +- [`EXPLICIT-TEMPLATE-BOUNDS.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.md): the explicit winning ranks. +- [`FINITE-TEMPLATE-CLOSURE.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md): new precision and storage constants (step 3). +- [`VALIDATION.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-0023-productive-jobs/VALIDATION.md): replay runner, comparison rules and import details. + +## Scope + +- **Conditional.** The claim keeps the inherited analytic decomposition, exact + recovery, ordered movement, fixed-tape execution and prime-padding premises. +- **Not yet reviewed.** The new universal arguments are written proofs backed by + finite checks: the compiler, native adapters, common-weight BIT interface and + support-local outer transfer. They are neither formally verified nor + externally accepted. +- **Not materialized.** The winning templates' full words and constants have not + been generated. The materialized rank-57 complex and rank-50 BIT examples check + completion, batching and ownership, but neither reaches the limiting saving. +- The selected frontier and shared verification code are unchanged. + +## Outstanding research directions + +The most direct route past 0.0023 is the **H branch**. The identity `H = I + BC` +with `CB = −2I` gives `H² = I`, which suggests cheap reversible completions over +the legal ring. On the [#589](https://github.com/CrocSwap/integer-mult-bounds/pull/589) BIT word, a minimum three-row invertible completion +cuts the local correction to three rank-one calls, and phase-correct complex +owner maps pass exact checks. What it +lacks is a cheap way to close the controller's endpoints. Three interval +corrections remain, costing 672 rank-one requests in complex arithmetic and 666 +in BIT, so even the cheapest BIT version falls 21 short. Generic cancellation, +register sharing and same-cut compression have been ruled out. A +geometry-aware decoder has not been. Details are in +[`LITERATURE-AND-PROBES.md`](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/LITERATURE-AND-PROBES.md) and the +[architecture overview](https://github.com/zxv/integer-mult-bounds/blob/codex/conditional-kappa-0023/research/kappa-rebuild/architecture/README.md). + +## Open next questions + +- **Review the chain as one argument.** Start with the integer source/exit + matching and all-column native adapters, then the common-weight BIT interface + and the support-local outer transfer. +- **Shrink the templates.** Tighten the paid-leaf superpotential or find smaller + explicit successful ranks without giving up integer batching or complete + constants. +- **Build a geometry-aware H decoder.** Start from the actual query subspaces, + the phase-correct complex owners and the three-row BIT completion. A survivor + must put preparation, routes, reads, restoration and all endpoints in one + invoice. +- **Improve the complex tariff.** The complex mixed root now binds, so more BIT + saving alone cannot advance this point. A changed native word or + productive-job contract must improve its complete moment. + +None of these questions is used to justify the 0.0023 claim. + +## Literature + +These sources shaped the construction above, grouped by the part they informed. +Each note says what the source suggested and what we actually used; the notes +describe our use, not claims by the cited authors. Sources behind unsuccessful +branches are included because they explain the turns the search took. + +### Productive work and mixed accounting + +1. **Jacob Alan Sussman** (2026). “Theorem 1G: mixed-target recursion.” + [PR #4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268) · + [pinned proof](https://github.com/jacobalansussman/wht-power-saving-lean/blob/563b4e6e913625637f36ba7869bf5fcd0dd4975a/research/record-track/proofs/theorem-1g.md) + + The direct algorithmic predecessor. Its integer Pascal-class lockstep flow + matches unequal target jobs with smaller paid demands under an exact moment + balance, splitting each source as `s + (n − s)` and paying the remainder + recursively. It is the core of our compiler. Our contribution is realizing + that mechanism as a finite integer compiler with native adapters, an explicit + winning rank and the outer transfer. [#4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268) claims a DFT improvement and no κ; its Lean results + do not verify this compiler or the multiplication claim. + +2. **Arnold Schönhage** (1981). “Partial and Total Matrix Multiplication.” + *SIAM Journal on Computing* 10(3), 434–455. + [doi:10.1137/0210032](https://doi.org/10.1137/0210032) + + The classic precedent for assembling useful partial computations into a + larger complete one, which is the shape of the productive-job idea. The + tensor theorem served only as motivation; the native source/exit schedule + and paid complements carry the actual bridge. + +3. **Péter Vrana** (2024). “Partial and weighted matrix multiplication.” + arXiv:2408.15728. [arXiv](https://arxiv.org/abs/2408.15728v1) + + Reinforced that unequal, weighted useful work deserves its own accounting. + It encouraged us to keep mixed ranks rather than round them to a uniform + job. Its support-rank theorem is not applied to transform programs. + +4. **Robert Robere and Jeroen Zuiddam** (2021). “Amortized Circuit Complexity, + Formal Complexity Measures, and Catalytic Algorithms.” ECCC TR21-035. + [ECCC](https://eccc.weizmann.ac.il/report/2021/035/) + + Motivated one production-and-consumption ledger with an explicit price for + every resource. Its repeated-output setting reuses a single input, so it + grants no independent-output credit here. + +### Building the controller over the legal coefficient ring + +5. **J. J. Koliha** (2001). “Block diagonalization.” *Mathematica Bohemica* + 126(1), 237–246. [EMIS](https://emis.de/ft/45438) + + Resolutions of the identity by idempotents pointed us at the operator + coefficients of the whole controller rather than its individual calls. That + view informed the exact coefficient factorizations tested for p3, including + the integral boundary factorization that removes two idle windows. + +6. **Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel** (2023). + “Dyadic linear programming and extensions.” arXiv:2309.04601. + [arXiv](https://arxiv.org/abs/2309.04601) + + Made the denominator restriction a first-class constraint. Power-of-two + balancing and shear factorizations then avoided an unavailable division + by 13. Its general LP algorithm was not implemented. + +7. **Ahmad Abdi, Gérard Cornuéjols, Bertrand Guenin and Levent Tunçel** + (2021; rev. 2022). “Total dual dyadicness and dyadic generating sets.” + arXiv:2111.05749. [arXiv](https://arxiv.org/abs/2111.05749v2) + + Separated a rational feasibility witness from a realization over the + allowed dyadic ring. This shaped the search's rejection gates. + +8. **Ingrid Daubechies and Wim Sweldens** (1998). “Factoring Wavelet Transforms + into Lifting Steps.” *Journal of Fourier Analysis and Applications* 4(3), + 247–269. [author manuscript](https://9p.io/who/wim/papers/factor/factor.pdf) + + Lifting suggested reversible triangular factors for the six-bank transfer in + place of generic scalar elimination. Frame contacts, coefficients, + preparation and restoration were still checked natively. + +9. **Christopher M. Brislawn** (2024). “The Causal Complementation Algorithm + for Lifting Factorization of Perfect Reconstruction Multirate Filter Banks.” + arXiv:2408.07970. [arXiv](https://arxiv.org/abs/2408.07970v1) + + Suggested factors constrained by *when* boundary data become available, not + unconstrained matrix elimination. We used exact legal shears and verified + the actual contact times. + +10. **Christopher M. Brislawn** (2019; final rev. 2024). “Factoring Perfect + Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine + Approach.” *Journal of Computational Algebra* 12, 100024. + [arXiv:1902.09040](https://arxiv.org/abs/1902.09040v5) + + Sharpened the same causal-divisibility question in an earlier handoff. It + never licensed an unavailable contact or an uncharged inverse. + +11. **Hyung-Ju Park** (1995). *A Computational Theory of Laurent Polynomial + Rings and Multidimensional FIR Systems.* PhD thesis, UC Berkeley, + UCB/ERL M95/39. + [report](https://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/ERL-95-39.pdf) + + The factor `I − P + zP` (§7.4) guided exact projector/delay + representations, checked with rational oblique projectors and their full + coefficient identities. No orthogonal rotations were imported. + +12. **Zinnur Doğanata, P. P. Vaidyanathan and Truong Q. Nguyen** (1988). + “General synthesis procedures for FIR lossless transfer matrices, for + perfect-reconstruction multirate filter bank applications.” *IEEE Trans. + Acoustics, Speech, and Signal Processing* 36(10). + [doi:10.1109/29.7544](https://doi.org/10.1109/29.7544) · + [Caltech copy](https://authors.library.caltech.edu/records/mv71c-vqj75) + + Inspired a transfer-matrix view of the controller with an explicit boundary + basis. Its lossless lattices do not license irrational rotations or free + frame changes. + +13. **John Urschel** (2023). “Representing the Special Linear Group with Block + Unitriangular Matrices.” arXiv:2305.00485. + [arXiv](https://arxiv.org/abs/2305.00485v2) + + Motivated short block-shear controllers in the earlier redesign. Its + field-level factor counts were leads only; dyadic coefficients, contacts + and the recursive bill were checked separately. + +### Arbitrary dirty state, restoration and causal flow + +14. **Jean-Guillaume Dumas and Bruno Grenet** (2024). “In-place accumulation + of fast multiplication formulae.” ISSAC 2024; arXiv:2307.12712. + [arXiv](https://arxiv.org/abs/2307.12712v3) + + Suggested borrowing completed output arrays as dirty scratch and restoring + them exactly. Every native contact and reverse operation stayed priced. + +15. **Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff and Florian + Speelman** (2014). “Computing with a full memory: Catalytic space.” + ECCC TR14-053. [ECCC](https://eccc.weizmann.ac.il/report/2014/053/) + + Supplied the discipline for borrowed scratch: it starts arbitrary and must + return exactly. Productive jobs meet a different contract, finishing with + the declared transform of their arbitrary input. Keeping the two + requirements apart is what lets the construction credit productive work. + It also motivated all-column dirty-bank tests rather than zero-initialized + examples. + +16. **Antoine Vinciguerra** (2026). “An Operator Approach to Register Programs + for Catalytic Computing.” arXiv:2609.18692. + [arXiv](https://arxiv.org/abs/2609.18692v1) + + Motivated stating restoration as a whole-operator identity and checking + whole arbitrary register columns. Its access counts were not imported. Its + polynomial-degree and passive-output assumptions differ from our linear + jobs, and its matrix lifting and coefficient divisions need not stay + within the permitted dyadic coefficients. + +17. **Marten Folkertsma, Ian Mertz, Sergii Strelchuk and Sathyawageeswar + Subramanian** (2026). “Fast Cliffords When Your Quantum Memory Is Full.” + arXiv:2609.40144. [arXiv](https://arxiv.org/abs/2609.40144v1) + + Prompted a dirty-workspace cycling probe, which was rejected. Its + restored-workspace premise fails for our accumulating dirty logger. + +18. **Joseph Connelly and Kenneth Zeger** (2016). “Linear Network Coding over + Rings, Part II: Vector Codes and Non-Commutative Alphabets.” + arXiv:1608.01737. [arXiv](https://arxiv.org/abs/1608.01737v2) + + Kept coding over the actual coefficient structure separate from a + convenient field-rank count. Acquisition, transport, scatter and erasure + therefore stay in the causal interface and the invoice. + +19. **Mark Braverman and Zhongtian He** (2026). “The Multiple Unicast + Conjecture is False.” arXiv:2610.10108. + [arXiv](https://arxiv.org/abs/2610.10108v1) + + Motivated timed, operator-valued carriers with causal local processing. + Its characteristic-three code and throughput factors were not transferred. + +20. **Vadim Kostrykin and Robert Schrader** (2001). “The Generalized Star + Product and the Factorization of Scattering Matrices on Graphs.” *Journal + of Mathematical Physics* 42, 1563–1598. + [arXiv:math-ph/0008022](https://arxiv.org/abs/math-ph/0008022) + + Suggested composing through retained interface variables in earlier work. + It remained an analogy; feedback inverses and recursive costs could not be + omitted. + +### BIT synthesis, state bounds and the H branch + +21. **Ketan N. Patel, Igor L. Markov and John P. Hayes** (2008). “Efficient + Synthesis of Linear Reversible Circuits.” *Quantum Information and + Computation* 8(3–4), 282–294. + [arXiv:quant-ph/0302002](https://arxiv.org/abs/quant-ph/0302002) + + Motivated explicit binary ADD/SWAP synthesis with shared linear forms. It + was used for exact F₂ synthesis and never for characteristic-zero complex + arithmetic. The archived BIT note cites the preprint under another title. + +22. **Andrea Visconti, Chiara V. Schiavo and René Peralta** (2018). “Improved + upper bounds for the expected circuit complexity of dense systems of linear + equations over GF(2).” *Information Processing Letters* 137. + [NIST](https://www.nist.gov/publications/improved-upper-bounds-expected-circuit-complexity-dense-systems-linear-equations-over) + + Suggested keeping shared dense sums alive until all consumers finish. The + resulting deferred reconstruction did not remove the physical frame or the + dirty baseline. + +23. **Alex Parent, Martin Roetteler and Krysta M. Svore** (2015). “Reversible + circuit compilation with space constraints.” arXiv:1510.00377. + [arXiv](https://arxiv.org/abs/1510.00377) + + Motivated joint lifetime and cleanup accounting. The helper-pair census + then rejected all five geometrically eligible coalescings; lifetime overlap + alone was not enough for native reuse. + +24. **Dmitry Chistikov, Szabolcs Iván, Anna Lubiw and Jeffrey Shallit** (2017). + “Fractional Coverings, Greedy Coverings, and Rectifier Networks.” + STACS 2017, LIPIcs 66, 23. + [doi:10.4230/LIPIcs.STACS.2017.23](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23) + + Motivated shared consumer maps and source-neighborhood covers. Rectifier + size does not pay for restoration or native transport. + +25. **G. David Forney Jr.** (2009; rev. 2010). “Minimal realizations of linear + systems: The ‘shortest basis’ approach.” arXiv:0910.4336. + [arXiv](https://arxiv.org/abs/0910.4336v3) + + Framed the question as the state needed across an actual time cut. The + nine-center rank checks then ruled out an eight-signal carrier at that cut, + leaving other cuts open. + +26. **Navin Kashyap** (2007). “On Minimal Tree Realizations of Linear Codes.” + arXiv:0711.1383. [arXiv](https://arxiv.org/abs/0711.1383) + + State requirements depend on the decomposition and consumer interface. This + scoped the same fixed-cut obstruction rather than a universal lower bound. + +27. **Clément Pernet and Arne Storjohann** (2018). “Time and space efficient + generators for quasiseparable matrices.” *Journal of Symbolic Computation* + 85, 224–246. [arXiv:1701.00396](https://arxiv.org/abs/1701.00396) + + Motivated compact exact state across cuts in earlier observer probes, whose + measured gaps exposed insufficient state in the tested contract. + +28. **Itziar Baragaña and Alicia Roca** (2019). “Fixed rank perturbations of + regular matrix pencils.” arXiv:1907.10657. + [arXiv](https://arxiv.org/abs/1907.10657v2) + + Distinguished prescribed-rank completion from a loose rank bound. Its + field-size hypotheses were not imported into F₂; the BIT completion was + proved and tested directly. + +29. **A. Amparan, I. Baragaña, S. Marcaida and A. Roca** (2023; rev. 2024). + “Row or column completion of polynomial matrices of given degree.” + arXiv:2306.06104. [arXiv](https://arxiv.org/abs/2306.06104v2) + + Suggested completing only the designated cheap output rows. Checking + invertibility instead of requiring an involution produced the three-row BIT + completion. Endpoint debt still blocks a higher claim. + +30. **Christophe Reutenauer** (1994). “A Survey on Noncommutative Rational + Series.” FPSAC 1994, 393–402. + [proceedings](https://fpsac-archive.github.io/FPSAC94/ARTICLES/45Reutenauer.pdf) + + Motivated testing the coefficient space of all boundary words jointly. The + resulting span obstruction rejects the universal two-passage model and + leaves geometry-aware coding open. + +31. **Jean Berstel and Christophe Reutenauer** (2010). “Minimization.” Ch. 2 of + *Noncommutative Rational Series with Applications*, Cambridge University + Press, 29–46. + [chapter](https://www.cambridge.org/core/books/abs/noncommutative-rational-series-with-applications/minimization/ACD53CC08C3CB9451564D3488D24FE22) + + Supplied the minimal-realization viewpoint for the same test. The native + obstruction has its own direct proof. + +### Retained outer inputs + +32. **Volker Strassen** (1969). “Gaussian elimination is not optimal.” + *Numerische Mathematik* 13, 354–356. + [doi:10.1007/BF02165411](https://doi.org/10.1007/BF02165411) + + The retained boundary-matrix algorithm. The exact comparison + `7^16 < 2^45` permits the conservative exponent 45/16 in the outer ledger. + +33. **Timothy S. Trudgian** (2014). “Updating the error term in the prime + number theorem.” arXiv:1401.2689. + [arXiv](https://arxiv.org/abs/1401.2689v2) + + Theorem 2 is a cited input to the inherited fine-prime padding argument. + The local verifier does not re-prove it. + +## Attribution + +| PR | Submitter | Used here | +|---|---|---| +| [#23](https://github.com/CrocSwap/integer-mult-bounds/pull/23) | [@jacklightChen](https://github.com/jacklightChen) | Semantic precision and bulk resampling | +| [#555](https://github.com/CrocSwap/integer-mult-bounds/pull/555) | [@eumemic](https://github.com/eumemic) | Punctured BIT supplier predecessor | +| [#589](https://github.com/CrocSwap/integer-mult-bounds/pull/589) | [@DreamingOfClouds](https://github.com/DreamingOfClouds) | Retained BIT word and transport improvements | +| [#1094](https://github.com/CrocSwap/integer-mult-bounds/pull/1094) | [@rohanarun](https://github.com/rohanarun) | Selected conditional result | +| [#1673](https://github.com/CrocSwap/integer-mult-bounds/pull/1673) | [@Vlado-V](https://github.com/Vlado-V) | Capped phase-cell fusion outer wrapper | +| [#1758](https://github.com/CrocSwap/integer-mult-bounds/pull/1758) | [@jeremykothe2](https://github.com/jeremykothe2) | Gauge-free interiors and radix-distributed union schedule | +| [#1786](https://github.com/CrocSwap/integer-mult-bounds/pull/1786) | [@chafreaky](https://github.com/chafreaky) | Fine source-prime padding | +| [#3945](https://github.com/CrocSwap/integer-mult-bounds/pull/3945) | [@Dugongue](https://github.com/Dugongue) | Staged544 owner-fill predecessor | +| [#4076](https://github.com/CrocSwap/integer-mult-bounds/pull/4076) | [@rohanarun](https://github.com/rohanarun) | Three staged helper entries and T360 fill | +| [#4085](https://github.com/CrocSwap/integer-mult-bounds/pull/4085) | [@zxv](https://github.com/zxv) | Receiver sharing on the staged T360 supplier | +| [#4088](https://github.com/CrocSwap/integer-mult-bounds/pull/4088) | [@1337isnot1337](https://github.com/1337isnot1337) | Mixed fourth staged entry | +| [#4091](https://github.com/CrocSwap/integer-mult-bounds/pull/4091) | [@RossRichardson](https://github.com/RossRichardson) | Fixed-inventory pooled ceiling | +| [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181) | [@rohanarun](https://github.com/rohanarun) | Pooled-ceiling fill; comparison point | +| [#4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268) | [@jacobalansussman](https://github.com/jacobalansussman) | Mixed-target recursion (Theorem 1G), basis of the compiler; claims no κ | + +Earlier geometry, native checker, owner, compiler and circuit contributions are +credited in the frozen packages and source notices. + +Prepared with substantial AI assistance, including ChatGPT Pro (Astra 6.0), +GPT-6.1 Sol and Opus 5.5. Apache-2.0 for new material; original notices and +third-party licenses are retained. diff --git a/research/kappa-0023-productive-jobs/README.md b/research/kappa-0023-productive-jobs/README.md new file mode 100644 index 0000000000..dd2b1653e4 --- /dev/null +++ b/research/kappa-0023-productive-jobs/README.md @@ -0,0 +1,81 @@ +# Proposed conditional κ = 0.0023: local review and submission draft + +The proposed conditional exponent saving is **23/10000**, approximately +**32.8727% above** @rohanarun's [#4181](https://github.com/CrocSwap/integer-mult-bounds/pull/4181), +the leading conditional κ claim located in the checked public PR records. +This compares exponent-saving endpoints. This package records a local finite +replay and a draft submission; independent mathematical acceptance remains open. + +Read the [PR description](PR-BODY.md) for the construction story, exact invoices, +useful equations, all nine outer costs, literature, attribution and next steps. +The [standalone bibliography](LITERATURE.md) covers the full delivered research +trail and distinguishes direct use from motivation and unsuccessful probes. + +## Scientific entry points + +- [Frozen claim and exact source bindings](../kappa-rebuild/CLAIM.json). +- [Outer transfer argument](../kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md). +- [Finite integer compiler](../kappa-rebuild/architecture/FINITE-PRECOMPILATION.md). +- [Delayed BIT cover and common weight](../kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md). +- [Explicit finite ranks](../kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.md). +- [New precision and storage constants](../kappa-rebuild/transfer-audit/FINITE-TEMPLATE-CLOSURE.md). +- [Local validation receipt](evidence/VALIDATION.json) and [review notes](VALIDATION.md). + +The finite uniform ranks `2^276540184361` (complex) and `2^6552928412` +(BIT before cover) specify enormous fixed templates by algorithm. Their full +words and constants have not been materialized. Small rank-57/rank-50 examples +check ownership and complements; they do not achieve the limiting savings. + +## Reproduce the supported critical replay + +From a checkout containing the committed inputs: + +```sh +python3 -B research/kappa-0023-productive-jobs/verify.py --check-only +python3 -B research/kappa-0023-productive-jobs/verify.py --out /absolute/path/to/new-replay +``` + +Python standard library only. The output directory must be new and outside the +checkout. The runner authenticates the import, copies its exact inputs, executes +the original 23 stages sequentially in dependency order and compares all 59 +regenerated artifacts. It leaves the original checkpoint untouched. Read +[VALIDATION.md](VALIDATION.md) for the stale-pin ordering repair and narrowly +scoped host-metadata/compression comparison rules. + +## Import scope and preservation + +[IMPORT.json](IMPORT.json) binds 365 unchanged delivered files, including all +240 current-round files, and 16 canonical-main inputs. The remaining imported +files supply the critical dependency closure and attribution. One historical +predecessor README conflicted with main; its bytes are retained under +`predecessor-notices/`, leaving the canonical file unchanged. + +The complete 3,454-path cumulative overlay remains in the user's Downloads. +This focused contribution is not an assertion that every predecessor package +or historical search is reproduced by its subset. No frozen source, shared +verification code, selected certificate or frontier record is modified. +Scoped `.gitattributes` additions preserve the original research bytes in Git. +Seven dependencies are stored privately under this review package and mapped +to their historical destinations only in the fresh replay, keeping the existing +`kappa-next-regime` package untouched. +The original [delivery receipt](DELIVERY-VERIFICATION.json) is attributed supplied +evidence; its full-base application was not rerun because that exact ZIP is absent. +The reported historical 332-entry recovery discrepancy remains unresolved. + +`PR-BODY.md` is the complete PR description. Its literature section mirrors +`LITERATURE.md`; edit both together. + +## Open next questions + +1. Independently review the compiler, native lift, delayed cover and outer + support argument together; begin with the source/exit recurrence and + common-weight condition. External acceptance and inherited assumptions remain open. +2. Tighten the superpotential or find smaller successful fixed ranks without + losing integer batching and complete constants; the current templates are impractical. +3. Test a geometry-aware H boundary decoder against the actual owner/query + spaces and a complete bill. Direct endpoint correction currently loses. +4. Change the binding complex job tariff; more BIT saving alone does not + advance this checkpoint. The [PR description](PR-BODY.md#open-next-questions) gives the first tests and blockers. + +Prepared with substantial OpenAI/Codex assistance. Apache-2.0 for new +review material; original notices and third-party licenses are retained. diff --git a/research/kappa-0023-productive-jobs/VALIDATION.md b/research/kappa-0023-productive-jobs/VALIDATION.md new file mode 100644 index 0000000000..bf69dca2b5 --- /dev/null +++ b/research/kappa-0023-productive-jobs/VALIDATION.md @@ -0,0 +1,163 @@ +# Local validation scope and findings + +This is a fresh execution of supplied scientific programs and a local review +of their claim boundary. The separately implemented auditors were part of the +delivered research. Running them here does not create independent human +mathematical approval or a formal theorem proof. + +## Source and result + +Canonical main was fetched and the isolated `codex/conditional-kappa-0023` +branch rebased onto `12fa8c5c844be21bfefb38a19503fbd867a11244`. +The selected record continues to identify #1094 and its conditional +κ = 0.001672660197076918; its construction was not changed or cold-replayed. + +The three delivered-file fingerprints match +[ARCHIVE-INTEGRITY.json](evidence/ARCHIVE-INTEGRITY.json). +Safe extraction authenticated 3,450 research paths in the cumulative overlay, +and all 240 current-round files matched the smaller new-research ZIP exactly. +The frozen `verify_scope.py` passes its 239 entries and 26 claim-source bindings. +Loose duplicate/downloaded files are not substituted for the authoritative ZIP. + +The fresh scientific run passed all **23 stages**. All **59 regenerated +artifacts** have equal mathematical content under the comparison rules below. +The receipt is [VALIDATION.json](evidence/VALIDATION.json); its stage logs are +under [evidence/logs](evidence/logs). The host was macOS arm64, Python 3.14.5. +The receipt records the interpreter, platform, zlib, exact commands, scientific +source hashes, log hashes and each old/fresh artifact hash. + +## Original replay-order failure + +The delivered runner orders these stages as: + +```text +p3_native_cover -> p3_independent_phases -> p3_independent_owners +``` + +`p3_native_cover.py` records the existing phase-audit digest in its `sources`. +Regenerating the phase audit afterward changes that digest, through regenerated +source-receipt metadata and compressed-container differences. The owner audit +correctly rejects the stale pin. The actual failure is retained in +[original-runner-stale-pin.log](evidence/original-runner-stale-pin.log). + +The phase auditor reads the formal factorization and completed native receipts; +it does not require the generated p3 cover. Therefore the valid order is: + +```text +p3_independent_phases -> p3_native_cover -> p3_independent_owners +``` + +The review runner uses this order and runs every original stage exactly once, +sequentially in the focused fresh replay. No scientific source or assertion is +patched. The earlier full-overlay attempt and its resumed stages were diagnostic; +the published 23-stage receipt comes from the focused input copy. + +## Cross-environment comparison + +The original comparison permits four named timing keys and exact old-to-fresh +hash references of the 59 regenerated artifacts. We preserve those rules. +Gzip containers differ across environments; their decompressed JSON is compared. +After all stages passed, the only remaining differences were host descriptions +in these three files: + +- `complex/fresh-certificates/RESULT-H.json` +- `complex/fresh-certificates/RESULT-helpers.json` +- `complex/fresh-certificates/RESULT-logger.json` + +Only `/environment/python` and `/environment/platform` in those named files +are excluded from mathematical equality, and both original and fresh values +are retained in the receipt. Fresh values must equal the executing interpreter +and platform. No other environment key is excluded. The old receipts describe +Linux/Python 3.12.14; the local run describes macOS/Python 3.14.5. + +All 59 containers differ in this run, because source-receipt metadata and gzip +differences propagate through their exact hash references. Every remaining +mathematical field agrees. The final comparison was repeated after narrowing +these metadata exceptions; the successful scientific stages were not rerun +merely to change comparison policy. The receipt discloses that repair. + +Four controls challenge that policy on the actual regenerated outputs: +changing the target κ, making a functional gap negative, adding an undeclared +environment field, and supplying a false fresh-interpreter description are +all rejected. Positive comparison succeeds before and after the challenges; +temporary output bytes are restored. See +[COMPARISON-CONTROLS.json](evidence/COMPARISON-CONTROLS.json). + +```sh +python3 -B research/kappa-0023-productive-jobs/check_comparison.py \ + --repo /absolute/path/to/new-replay/repo \ + --report /absolute/path/to/comparison-controls.json +``` + +## Focused publication inputs + +[IMPORT.json](IMPORT.json) binds the unchanged checkpoint, its executed inputs, +attribution and inherited written proof notes. It preserves all 240 current-round +files. Historical experiments outside this closure remain in the complete +cumulative Downloads archive. The focused replay needs no private archive. + +The first replay copied 301 delivered files and 16 main inputs. Publication +then relocated an unexecuted historical README to preserve main's existing +README and added 65 unchanged inherited proof notes for review. Seven +`kappa-next-regime` dependency/notice files were moved under this review package; +`replay_path` maps them to their historical names inside the temporary execution +directory. These are byte-preserving layout changes; executed scripts and their +inputs are unchanged. [PUBLICATION-INPUTS.json](evidence/PUBLICATION-INPUTS.json) +records the mapping. The fresh execution hash and current publication manifest +hash are kept separately in the validation receipt. + +Scoped root attributes disable text conversion only for the eight newly +introduced research prefixes. Without them Git would normalize CRLF in inherited +proof notes and invalidate their source hashes. Existing frozen packages and +unrelated source paths retain their prior attributes. + +## Public context and attribution checks + +The cached overlap search was run for `independent-moment`: its remote data was +from October 18 and was explicitly labelled stale. A live search of recent +conditional claims, relevant PR bodies, active Discussions and the board's +start-here post supplemented it. This is a scoped overlap check, not a novelty +certificate. Relevant live heads and authors are retained in +[PUBLIC-PREDECESSORS.json](evidence/PUBLIC-PREDECESSORS.json). + +The draft uses @rohanarun's #4181 as its comparison point, giving approximately +32.8727% more exponent saving. This is the leading conditional κ claim located +in the checked public records. The chronologically newer #4270 claims the lower +κ = 1/580; #4326 improves a BIT supplier's c rather than claiming a higher κ. +The validation receipt already contains the exact comparison against #4181. +#4268's DFT figure is not an integer-multiplication κ. +Discussion #4337 is checking that PR's literal half-cut binding; this proposal +uses its own complete p3 words and adds a different finite compiler/outer argument. +The [PR submitter audit](evidence/PR-ATTRIBUTION-AUDIT.json) checks all 14 +repository PRs linked in the newly authored narrative and bibliography against +GitHub's `author.login`. In particular, #4085 was submitted by @zxv, #1758 by +@jeremykothe2 and #1786 by @chafreaky. The draft uses verified handles for PR +submission credits and preserves source-authorship notices separately. +The initial draft incorrectly substituted Douglas Colkitt for @zxv and assigned +preparation of the submission to Douglas Colkitt without a verified basis. +Those unsupported attributions have been removed from the review narrative. + +## Unrun obligations and limits + +- Universal finite compilation, native tensor lift, common-weight cover, + semantic precision and support-local transfer are written mathematical + arguments supported by finite checks. They require external review. +- The inherited analytic decomposition, exact integer recovery, native ordered + movement, fixed-tape model and prime-padding inputs remain conditional. +- Winning astronomical uniform words, covers and constants were not materialized. + No practical cutoff or runtime advantage was measured. +- The exact private base ZIP is absent from Downloads. Its full application and + idempotence receipt is preserved as supplied evidence, not independently rerun. +- No Lean, Docker, hosted CI, selected-frontier cold replay or complete replay + of every auxiliary H/negative-search menu was run. Their retained receipts + retain their original author-checked status. +- The historical reported 332-entry recovery gap is not resolved by fresh + reconstruction. Missing originals are not claimed to have been recovered. + +## Open next questions + +The [PR description](PR-BODY.md#open-next-questions) gives feasible starting points and +blockers for independent review, smaller templates, geometry-aware boundary +decoding and an improved binding complex tariff. + +Prepared with substantial OpenAI Codex assistance. Apache-2.0. diff --git a/research/kappa-0023-productive-jobs/check_comparison.py b/research/kappa-0023-productive-jobs/check_comparison.py new file mode 100644 index 0000000000..3accf2b183 --- /dev/null +++ b/research/kappa-0023-productive-jobs/check_comparison.py @@ -0,0 +1,73 @@ +#!/usr/bin/env python3 +"""Challenge metadata normalization on actual fresh scientific artifacts. + +Uses an already completed temporary replay. Each mutation is restored in a +finally block; frozen source files are never modified. Apache-2.0; Codex-assisted. +""" +import argparse +import gzip +import json +from pathlib import Path +import verify as V + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--repo', type=Path, required=True) + parser.add_argument('--report', type=Path, required=True) + args = parser.parse_args() + root = args.repo.resolve() + V.need(root.is_dir() and not root.is_relative_to(V.ROOT), 'Use the separate temporary replay') + V.need(not args.report.resolve().is_relative_to(V.ROOT), 'Write control receipt outside source tree') + outputs = set() + for stage in V.load_stages(): + for pattern in stage['outputs']: + found = [p for p in (root / V.SCOPE).glob(pattern) if p.is_file()] + V.need(found, 'Complete replay output: ' + pattern) + outputs.update(p.relative_to(root).as_posix() for p in found) + V.need(len(outputs) == 59, 'All 59 outputs present') + V.compare_outputs(root, outputs) + outer = str(V.SCOPE / 'transfer-audit/OUTER-TRANSFER-AUDIT.json') + helper = str(V.SCOPE / 'complex/fresh-certificates/RESULT-helpers.json') + def wrong_kappa(d): + d['conditional_assembly']['kappa'] = '24/10000' + def wrong_gap(d): + d['conditional_assembly']['functional_interface_gaps']['global_gamma_spare_power'] = '-1' + def extra_environment(d): + d['environment']['unexpected_scientific_field'] = 'must not be ignored' + def wrong_host(d): + d['environment']['python'] = 'fabricated interpreter' + controls = [] + for name, filename, change in ( + ('altered_target_kappa', outer, wrong_kappa), + ('negative_functional_gap', outer, wrong_gap), + ('undeclared_environment_field', helper, extra_environment), + ('false_fresh_host_metadata', helper, wrong_host)): + path = V.checked(root, filename) + original = path.read_bytes() + data = V.read_payload(path) + change(data) + reason = None + try: + encoded = (json.dumps(data, sort_keys=True) + '\n').encode() + path.write_bytes(gzip.compress(encoded, mtime=0) if path.suffix == '.gz' else encoded) + try: + V.compare_outputs(root, outputs) + except ValueError as error: + reason = str(error) + V.need(reason is not None, 'Control must be rejected: ' + name) + finally: + path.write_bytes(original) + controls.append({'control': name, 'path': filename, 'rejected': True, 'reason': reason}) + print('REJECTED ' + name, flush=True) + V.compare_outputs(root, outputs) + result = {'status': 'PASS_SCIENTIFIC_COMPARISON_CONTROLS', 'positive_artifacts': 59, + 'controls': controls, 'fresh_artifacts_restored': True, + 'comparator_sha256': V.digest(Path(V.__file__)), + 'control_source_sha256': V.digest(Path(__file__))} + args.report.write_text(json.dumps(result, indent=2, sort_keys=True) + '\n') + print(result['status']) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/LICENSE b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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Two adjacent targets read one common side query from the same original + carrier at a shared codimension-two frame. They then move to their own + hyperplanes and read their exclusive contributions. This reduces the + observed side functions from ten pair-partials per target to three + functions per target pair, without private side copies or extra raw + rank backtracking. +2. For one isolated target pair, the common original-input gather is + uncomputed immediately after those two reads. Its arbitrary dirty + carrier is then reused for an exclusive query. This is a literal + three-to-two helper reduction with all original inputs and helpers + restored, no CLEAR operation, and no downward frame transitions. + +**No new multiplication kappa is admitted.** The all-port consumer has a +fully paid but large reference producer. The isolated two-helper fragment +does not yet compose into an all-port source router. Exact profile screens +show the quantitative opportunity if a compatible compressed producer is +found; they are labeled as acquisition targets throughout. + +## What was proved and checked + +For targets T=K+{a} and U=K+{b}, with |K|=4, let C be their common side +neighbor sum and D_T,D_U the exclusive sums. Their outputs are -(C+D_T) +and -(C+D_U). C has rational source span +F=(span{t_T,t_U}) perpendicular under H=I-(2/25)J. Its rank is h-2 and +its normal Gram matrix is [[3,2],[2,3]]. The exclusive sums also span +explicit nondegenerate rank-(h-2) frames. + +Both targets first read C at F. Only afterward do they move to their +distinct final hyperplanes. The proofs establish these inclusions for +every h>=9 and check all prototype source lines and exact rational +projectors at h9 and h19. The ambient form is indefinite at h19; +nondegeneracy, rather than an incorrect positivity assumption, is used. + +| Certificate | Exact verification | +|---|---| +| All-port reference at h9 | 5,418 independent scalar roles; full operator for both signs, 29,354,724 coefficients per sign | +| Literal native phase schedule | All 33,120 ADDs checked at owner/query-specific common frames; 16,038 recursive frame changes reproduce the rank histogram | +| Dirty-helper controls | Four omitted-observation words are rejected | +| Two-helper isolated pair at h9 and h19 | Complete integer operator for both signs; every original input and both arbitrary dirty helpers restored | +| h19 two-helper fragment | 10,512 unit ADDs, 135,303,424 matrix coefficients determined by exact sparse rows; no private copies or extra rank backtracking | +| Global routing control | Explicit source requiring two distinct final hyperplanes rejects naive composition of the isolated fragments, even with delayed final retirement | +| Primitive moment screens | Exact rational log/exp enclosures; floating arithmetic only locates candidate endpoints | + +The all-port scalar identity uses F3. The isolated pair identity holds +over the integers and therefore every scalar characteristic. Scalar ADD +counts exclude additional linear copying, retirement, routing and tape +work; recursive transforms on copies are charged in the rank profile. +The results concern native local phases under the retained stream contract +and shared padded cover. They do not certify a new outer arbitrary-precision +multiplication engine. + +## The h19 numbers and what they mean + +Here V=11,628, c=171, and the inherited cover width is m=57. A is the +number of persistent helper streams. The primitive moment parameter a is +not automatically the multiplication exponent improvement kappa. + +| Model | Persistent helpers A | Status | Passing primitive a | +|---|---:|---|---:| +| One occurrence per source per query | 30,651,408 | Complete paid reference | 0.000005084 | +| Three query carriers per target pair, plus centers | 17,613 | Terminal-only acquisition target | 0.004471641 | +| Two carriers per target pair through early reuse, plus centers | 11,799 | Global acquisition target; isolated fragment constructed | 0.005369309 | + +The complete paid reference is intentionally a simple construction that +makes every acquisition and restoration explicit. Its helper count is +larger than the previously retained joint-DAG reference; it is not a +numerical improvement over that reference or over the selected record. + +For the first compressed target, the actual terminal profile is + +\[ +(5V/2+2c)H_{h-2}+(V/2+c)H_2+3V H_1+V H_{h-1}. +\] + +Under the retained global profile 3 H_local + 2V H2, its h19 normalized +moment at a=.004 is **0.9993395917423624**, rigorously below one. A +compressed producer has room for at most **6,128 additional ideal H19 +helpers**, or 23,741 helpers total, if it adds no other recursive work. +A separate V-source bank exceeds this first allowance. + +The second target uses the proven isolated reuse pattern and has profile + +\[ +(3V/2+2c)H_{h-2}+(3V/2)H_{h-1}+(5V/2)H_1+cH_2. +\] + +It has moment **0.998138608664116** at .004 and room for at most +**14,815 additional ideal H19 helpers**, or 26,614 total. This is a +larger design allowance. It remains conditional on a global producer: +the common inputs must remain available for early uncomputation, all +source routes must be compatible, and actual input acquisition fragments +the helper paths. The receipt separately prices several such fragmented +paths; the single-H19 allowance is an optimistic screen. + +| Extra-helper trajectory at h19 | Three-carrier target: maximum extras | Reused-carrier target: maximum extras | +|---|---:|---:| +| One H19, optimistic | 6,128 | 14,815 | +| H1+H16+H2, acquired input reaching a codimension-two frame | 4,109 | 9,934 | +| 2H1+H17, acquired input reaching a target hyperplane | 4,456 | 10,773 | + +Consequently, even the reused-carrier target cannot simply add a whole +V=11,628 bank with either displayed below-I input trajectory. The original +input reuse and the global frame-aware acquisition must be designed together. + +Both models retain the split target path H(h-2)+H1, all center copies, +all helper retirements, both data endpoints, and the cover connectors. +Neither treats a source sum as a free input. + +## Why the isolated reuse is an actual change + +Before acquisition, the helper dirt z,w is compensated at frame zero: +Y_T receives 2z and Y_U receives z+w. The common helper then holds z+C, +which is subtracted from both targets at F. Reversing the common gather +while its original inputs remain at F returns the helper exactly to z. +That same carrier next holds z+D_T at H_T and is read again. The other +helper handles w+D_U at H_U. Reversing both exclusive gathers restores +the helper inputs. + +This changes when the first carrier becomes available. Its signal is +removed by an explicit inverse computation while its parents remain +live. It does not assume an arbitrary dirty value can be cleared. + +The exact global counterexample identifies the next difficulty: source +mask 31 would need to visit the distinct rank-18 hyperplanes for target +masks 241 and 227 in two different matched pairs. Those hyperplanes are +incomparable. A global producer must encode, combine, or route source +information before making such incompatible commitments. + +## What the literature contributes + +The primary sources in [LITERATURE.md](LITERATURE.md) support three +specific ideas: eager uncomputation while original inputs remain live; +in-place linear matrix realization without an output register bank; and +factorizations into single-register assignments. The first is implemented +by the two-helper pair. The latter two motivate a joint mutable producer, +but their unrestricted register model does not pay rational address +changes or enforce native common frames. + +The next bounded experiment is a frame-aware linear basis quotient. For +each live query, choose a frame between its source span and the +intersection of its remaining target hyperplanes. After an early read, +that upper constraint can grow. Bring compatible live states to one +allowed frame, synthesize an invertible basis change using independent +dirty directions, and retain only the observations still needed. The +two-helper word is a positive test case; the incompatible-hyperplane +witness is a required rejection case. Such a producer must fit the exact +moment allowance before it can be turned into a kappa claim. + +## Reproduce + +Use ordinary Python 3 with its standard library. No external packages, +network access, inherited paths, or installed repository checkout are +needed to execute these checks. The source compiler and cover documents +are proof dependencies identified in SOURCES.json, rather than runtime +imports. The interval/matrix helper file under reference/ is an exact +copy of the prior frozen program and is hashed in each receipt. + +```bash +python -B paired_consumer.py +python -B borrowed_pair.py +python -B verify.py +``` + +The last command authenticates the declared file inventory, makes a fresh +relocated copy, deletes both generated receipts, regenerates them, and +compares their exact bytes. It writes VALIDATION.json. By default the +temporary replay tree is removed after successful validation. + +Files: + +- `PROOF.md`: general algebra, rational geometry, phase paths, both cost + ledgers, exact scope of the negative routing witness, and the next + synthesis problem. +- `PAIRED-CONSUMER.json`: rational frames, full all-port scalar replay, + labeled native phases, fully charged reference invoices, conditional + terminal profiles, and exact moment/helper-budget gates. +- `BORROWED-PAIR.json`: two-helper integer operators, all actual native + transitions, negative controls, and explicit routing witnesses. +- `SOURCES.json` and `LITERATURE.md`: source pins, primary papers and + model distinctions. No exhaustive novelty claim is made. +- `REPLAY-PLAN.json`, `INTEGRITY.json`, `VALIDATION.json`: reproducibility + plan, canonical byte inventory and relocated verification receipt. + +Prior frozen files in bit-output and kappa-resume are unchanged. diff --git a/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/bit-compression/carrier-reuse/reference/joint_pair_dag.py b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/bit-compression/carrier-reuse/reference/joint_pair_dag.py new file mode 100644 index 0000000000..8b40135a84 --- /dev/null +++ b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/bit-compression/carrier-reuse/reference/joint_pair_dag.py @@ -0,0 +1,731 @@ +#!/usr/bin/env python3 +"""Joint side/pair-total monotone DAG with a fully paid copied output interface. + +The tensor recurrence and equal-support interning follow the pinned upstream +ternary_side.py / disjoint_tensor.py construction. The additional total query, +shared observation word, explicit enclosure frames and invoice are this probe. +No installed numerical packages and no external files are needed at replay. +""" +from array import array +from collections import Counter +from functools import cache +from fractions import Fraction as Q +from hashlib import sha256 +from itertools import combinations +from math import comb, exp, log +from pathlib import Path +import argparse +import json +import sys + + +def require(p, message): + if not p: + raise ValueError(message) + + +def mask(t): + return sum(1 << i for i in t) + + +@cache +def subsets(points, size): + return tuple(mask(t) for t in combinations(points, size)) + + +@cache +def tensor_plan(n, a, b): + require(0 <= a <= n and 0 <= b <= n, "Invalid tensor size") + if a+b > n: + return 0, "empty" + if not a: + return 0, "copy" + if not b: + return comb(n, a)-1, "sum" + best = comb(n, b)*(comb(n-b, a)-1), "rows" + for left in range(1, n//2+1): + right = n-left + cost = 0 + for j in range(max(0, b-right), min(left, b)+1): + parts = 0 + for i in range(max(0, a-right), min(left, a)+1): + if i+j > left or a-i+b-j > right: + continue + A = tensor_plan(left, i, j)[0] + B = tensor_plan(right, a-i, b-j)[0] + cost += min(comb(right, a-i)*A+comb(left, j)*B, + comb(left, i)*B+comb(right, b-j)*A) + parts += 1 + if parts: + cost += comb(left, j)*comb(right, b-j)*(parts-1) + if cost < best[0]: + best = cost, left + return best + + +class TripleTemplate: + """Cancellation-free triple queries, augmented by the complete total.""" + def __init__(self, h): + self.inputs = list(subsets(tuple(range(h)), 3)) + self.support = [0]+[1 << i for i in range(len(self.inputs))] + self.args = [None]*len(self.support) + self.lookup = {s: i for i, s in enumerate(self.support)} + values = {s: i+1 for i, s in enumerate(self.inputs)} + self.outputs = self.transform(tuple(range(h)), 3, 3, values) + self.total_root = self.total(list(values.values())) + self.active = set() + todo = [*self.outputs.values(), self.total_root] + while todo: + node = todo.pop() + if not node or node in self.active: + continue + self.active.add(node) + if self.args[node]: + todo.extend(self.args[node]) + for node in sorted(self.active): + if self.args[node]: + a, b = self.args[node] + require(a < node and b < node, "Noncausal template edge") + require(not self.support[a] & self.support[b], "Overlapping summands") + require(self.support[node] == self.support[a] | self.support[b], + "Wrong template support") + all_inputs = (1 << len(self.inputs))-1 + require(self.support[self.total_root] == all_inputs, "Wrong total query") + for t, node in self.outputs.items(): + expected = sum(1 << i for i, s in enumerate(self.inputs) if not s & t) + require(self.support[node] == expected, "Wrong disjoint-triple query") + + def add(self, a, b): + if not a or not b: + return a or b + require(not self.support[a] & self.support[b], "Addition is not cancellation-free") + s = self.support[a] | self.support[b] + node = self.lookup.get(s) + if node is None: + node = len(self.support) + self.lookup[s] = node + self.support.append(s) + self.args.append((a, b)) + return node + + def total(self, nodes): + if not nodes: + return 0 + if len(nodes) == 1: + return nodes[0] + m = len(nodes)//2 + return self.add(self.total(nodes[:m]), self.total(nodes[m:])) + + def transform(self, points, a, b, values): + n = len(points) + _, method = tensor_plan(n, a, b) + outputs = subsets(points, b) + if method == "empty": + return {t: 0 for t in outputs} + if method == "copy": + return {t: values[0] for t in outputs} + if method == "sum": + return {0: self.total(list(values.values()))} + if method == "rows": + return {t: self.total([values[s] for s in subsets(points, a) if not s & t]) + for t in outputs} + left, right = points[:method], points[method:] + l, r = len(left), len(right) + terms = {t: [] for t in outputs} + for j in range(max(0, b-r), min(l, b)+1): + for i in range(max(0, a-r), min(l, a)+1): + if i+j > l or a-i+b-j > r: + continue + A = tensor_plan(l, i, j)[0] + B = tensor_plan(r, a-i, b-j)[0] + if comb(r, a-i)*A+comb(l, j)*B <= comb(l, i)*B+comb(r, b-j)*A: + middle = {s: self.transform(left, i, j, + {u: values[u | s] for u in subsets(left, i)}) + for s in subsets(right, a-i)} + for t in subsets(left, j): + row = self.transform(right, a-i, b-j, + {s: v[t] for s, v in middle.items()}) + for u, node in row.items(): + terms[t | u].append(node) + else: + middle = {s: self.transform(right, a-i, b-j, + {u: values[u | s] for u in subsets(right, a-i)}) + for s in subsets(left, i)} + for t in subsets(right, b-j): + row = self.transform(left, i, j, + {s: v[t] for s, v in middle.items()}) + for u, node in row.items(): + terms[t | u].append(node) + return {t: self.total(nodes) for t, nodes in terms.items()} + + +def array_digest(*arrays): + out = sha256() + for a in arrays: + if a.itemsize > 1 and sys.byteorder != "little": + a = array(a.typecode, a) + a.byteswap() + out.update(a.tobytes()) + return out.hexdigest() + + +def build(h, exact_support=False): + require(type(h) is int and 8 <= h <= 24, "Probe budget is 8 <= h <= 24") + require(not exact_support or h <= 9, "Dense coefficient certificate budget exceeded") + template = TripleTemplate(h) + triples = template.inputs + pairs = list(subsets(tuple(range(h)), 2)) + labels = list(subsets(tuple(range(h)), 5)) + V, c = len(labels), len(pairs) + source_id = {t: i+1 for i, t in enumerate(labels)} + pair_id = {t: i for i, t in enumerate(pairs)} + left = array("I", [0])*(V+1) + right = array("I", [0])*(V+1) + common = array("I", [0]+labels) + union = array("I", [0]+labels) + roots, targets = array("I"), array("I") + centers = array("I") + exact = [0]+[1 << i for i in range(V)] if exact_support else None + table = {} + nodes = sorted(template.active) + + def allocate(a, b): + require(0 < a < len(left) and 0 < b < len(left) and a != b, "Invalid DAG input") + node = len(left) + require(node < 2**32, "DAG index overflow") + left.append(a) + right.append(b) + common.append(common[a] & common[b]) + union.append(union[a] | union[b]) + require(common[node].bit_count() >= 2, "Source enclosure lost the common pair") + if exact is not None: + require(not exact[a] & exact[b], "Overlapping exact source supports") + exact.append(exact[a] | exact[b]) + return node + + for pair in pairs: + good = sum(1 << i for i, t in enumerate(triples) if not t & pair) + image = array("I", [0])*len(template.args) + inter = array("I", [0])*len(template.args) + uni = array("I", [0])*len(template.args) + within = {} + for i, t in enumerate(triples): + if not t & pair: + image[i+1] = source_id[pair | t] + inter[i+1] = uni[i+1] = t + for x in nodes: + if not template.args[x]: + continue + u, v = template.args[x] + a, b = image[u], image[v] + if not a or not b: + image[x] = a or b + inter[x] = inter[u] if a else inter[v] + uni[x] = uni[u] if a else uni[v] + continue + I, U = inter[u] & inter[v], uni[u] | uni[v] + inter[x], uni[x] = I, U + support = template.support[x] & good + require(I.bit_count() <= 2, "Two distinct triples have three common points") + if not I: + y = within.get(support) + if y is None: + y = allocate(a, b) + within[support] = y + else: + if I.bit_count() == 1: + residual = 0 + bits = support + while bits: + bit = bits & -bits + bits ^= bit + residual |= 1 << pair_id[triples[bit.bit_length()-1] ^ I] + else: + residual = U ^ I + key = ((pair | I) << c) | residual + y = table.get(key) + if y is None: + y = allocate(a, b) + table[key] = y + require(common[y] == common[a] & common[b] and + union[y] == union[a] | union[b], "Interned enclosure mismatch") + if exact is not None: + require(exact[y] == exact[a] | exact[b] and not exact[a] & exact[b], + "Interned unequal formal source sums") + image[x] = y + for t, x in template.outputs.items(): + if t & pair: + continue + y = image[x] + target = t | pair + require(y != 0, "Empty side output") + roots.append(y) + targets.append(source_id[target]-1) + if exact is not None: + expected = sum(1 << i for i, s in enumerate(labels) if s & target == pair) + require(exact[y] == expected, "Wrong complete side coefficient row") + center = image[template.total_root] + centers.append(center) + require(common[center] == pair and union[center] == (1 << h)-1, + "Pair total is not at its h-2 enclosure") + if exact is not None: + expected = sum(1 << i for i, s in enumerate(labels) if s & pair == pair) + require(exact[center] == expected, "Wrong pair-total coefficient row") + require(len(roots) == 10*V and len(centers) == c, "Missing query outputs") + require(Counter(targets) == {i: 10 for i in range(V)}, "Wrong side target multiplicity") + active = bytearray(len(left)) + for root in [*roots, *centers]: + active[root] = 1 + additions = 0 + for node in range(len(left)-1, V, -1): + if active[node]: + additions += 1 + active[left[node]] = active[right[node]] = 1 + require(sum(active[1:V+1]) == V, "A source is missing") + return dict(h=h, V=V, c=c, labels=labels, pairs=pairs, left=left, right=right, + common=common, union=union, roots=roots, targets=targets, centers=centers, + active=active, additions=additions, exact=exact, + template_active_additions=sum(template.args[i] is not None for i in template.active), + template_total_augmented=True) + + +def frame_rank(I, U): + k = I.bit_count() + require(I & ~U == 0 and 2 <= k <= 5, "Invalid enclosure") + if k == 5: + require(I == U, "A common five-subset can only be a singleton") + return 1 + return U.bit_count()-k + + +def compile_native(g, retain=False): + h, V, c = g["h"], g["V"], g["c"] + L, R, active = g["left"], g["right"], g["active"] + size = len(L) + uses = array("I", [0])*size + lo = array("I", [0])*size + ro = array("I", [0])*size + for node in range(V+1, size): + if active[node]: + lo[node] = uses[L[node]] + uses[L[node]] += 1 + ro[node] = uses[R[node]] + uses[R[node]] += 1 + query_roots = [*g["roots"], *g["centers"]] + ordinals = array("I") + for node in query_roots: + ordinals.append(uses[node]) + uses[node] += 1 + pivot = array("I", [0])*size + first = array("I", [0])*size + frames = [] + hist = Counter() + moves = 0 + scalar_L = [] if retain else None + scalar_L_count = 0 + source_slots = [] + transition_hash = sha256() + unique_enclosures = set() + + def slot(node, ordinal): + return pivot[node] if ordinal == 0 else first[node]+ordinal-1 + + def move(s, node): + nonlocal moves + old = frames[s] + if old == node: + return + I, U = g["common"][node], g["union"][node] + new_rank = frame_rank(I, U) + if old: + old_I, old_U = g["common"][old], g["union"][old] + require(I & ~old_I == 0 and old_U & ~U == 0, + "Physical forward edge does not have nested enclosures") + old_rank = frame_rank(old_I, old_U) + else: + old_rank = 0 + d = new_rank-old_rank + require(d >= 0, "Negative monotone rank increment") + if d: + hist[d] += 1 + moves += 1 + frames[s] = node + if retain: + unique_enclosures.add((I, U)) + transition_hash.update(f"{s}:{old}>{node}:{d};".encode()) + + for node in range(1, size): + if not active[node]: + continue + require(uses[node] > 0, "Active node has no user") + if node <= V: + p = len(frames) + frames.append(0) + ins = [p] + source_slots.append((node-1, p)) + else: + ins = [slot(L[node], lo[node]), slot(R[node], ro[node])] + p = ins[0] + pivot[node] = p + first[node] = len(frames) + new_slots = range(len(frames), len(frames)+uses[node]-1) + frames.extend([0]*(uses[node]-1)) + outs = [p, *new_slots] + require(len(set(ins+outs)) == len(ins)+len(outs)-1, "Pivot aliasing error") + for s in [*ins, *outs[1:]]: + move(s, node) + for s in ins[1:]: + scalar_L_count += 1 + if retain: + scalar_L.append((p, s, 1)) + for s in outs[1:]: + scalar_L_count += 1 + if retain: + scalar_L.append((s, p, 1)) + A = len(frames) + require(A == g["additions"]+10*V+c, "Wrong reversible role count") + output_slots = [slot(node, ordinal) for node, ordinal in zip(query_roots, ordinals)] + require(len(set(output_slots)) == len(output_slots), "Query outputs share a physical slot") + center_slots = output_slots[10*V:] + side_slots = output_slots[:10*V] + # Query reads happen after the producer and before any inverse update. + # Center original streams stay put; each private copy is discarded after its read. + for s, center in zip(center_slots, g["centers"]): + require(frames[s] == center, "Center original was consumed before its read") + r = frame_rank(g["common"][center], g["union"][center]) + require(r == h-2, "Center rank is not h-2") + hist[r] += 1 + moves += 1 + # Side output uses meet the actual target at its fixed hyperplane. + side_checks = 0 + for s, target in zip(side_slots, g["targets"]): + node = frames[s] + I, U = g["common"][node], g["union"][node] + T = g["labels"][target] + if I.bit_count() == 5: + require((I & T).bit_count() == 2, "Singleton side is not orthogonal") + else: + # All coordinates on I equal a; total is 5a. Only a pair of I + # occurs in the target; its other coordinates are outside U. + require((I & T).bit_count() == 2 and (T & U) == (T & I), + "Output enclosure is not below the target hyperplane") + d = h-1-frame_rank(I, U) + require(d >= 0, "Side output rank exceeds its hyperplane") + if d: + hist[d] += 1 + moves += 1 + frames[s] = -1 # distinct actual target hyperplane, rank h-1 + side_checks += 1 + for node in frames: + r = h-1 if node == -1 else frame_rank(g["common"][node], g["union"][node]) + d = h-r + require(d >= 0, "Helper endpoint rank exceeds I") + if d: + hist[d] += 1 + moves += 1 + # Data source Pi -> I; data target 0 -> I-Pi. + hist[h-1] += 2*V + moves += 2*V + mass = sum(r*n for r, n in hist.items()) + minimum = (2*V+A)*h-2*V + require(mass == minimum+c*(h-2), "Joint native rank telescoping failed") + global_hist = Counter({r: 3*n for r, n in hist.items()}) + global_hist[2] += 2*V + W, m = 2*V+A, 3*h + global_mass = sum(r*n for r, n in global_hist.items()) + delta = W*m-global_mass + require(delta == 2*V-3*c*(h-2), "Padded cover saving failed") + out = dict(persistent_auxiliaries=A, data_roles=2*V, W=W, m=m, + local_profile=dict(sorted(hist.items())), local_rank_mass=mass, + monotone_endpoint_minimum=minimum, local_excess=c*(h-2), + physical_rank_moves=moves, forward_mixer_primitives=scalar_L_count, + complete_scalar_primitives=4*scalar_L_count+42*V, + side_output_inclusions_checked=side_checks, + central_private_copies=c, central_copy_rank=h-2, + global_profile=dict(sorted(global_hist.items())), + global_rank_mass=global_mass, global_output_mass=W*m, + global_rank_saving=delta, + physical_transition_sha256=transition_hash.hexdigest()) + if retain: + out["_program"] = dict(L=scalar_L, sources=source_slots, side=side_slots, + centers=center_slots, enclosures=sorted(unique_enclosures)) + return out + + +def f3add(a, b, coefficient=1): + u, v = a + x, y = b + coefficient %= 3 + if coefficient == 0: + return a + if coefficient == 2: + x, y = y, x + return ((u & ~(x | y)) | (x & ~(u | v)) | (v & y), + (v & ~(x | y)) | (y & ~(u | v)) | (u & x)) + + +def scalar_certificate(g, ledger, sign=1, omit=None): + p = ledger["_program"] + V, A, W = g["V"], ledger["persistent_auxiliaries"], ledger["W"] + values = [(1 << i, 0) for i in range(W)] + operations = 0 + + def add(dst, src, coefficient): + nonlocal operations + values[dst] = f3add(values[dst], values[src], coefficient) + operations += 1 + + def mixer(inverse=False): + ops = reversed(p["L"]) if inverse else p["L"] + for dst, src, coefficient in ops: + add(2*V+dst, 2*V+src, -coefficient if inverse else coefficient) + + def observation(k, which): + if omit != which+"_center": + for pair, slot in zip(g["pairs"], p["centers"]): + for i, label in enumerate(g["labels"]): + if label & pair == pair: + add(V+i, 2*V+slot, k) + if omit != which+"_side": + for i, slot in zip(g["targets"], p["side"]): + add(V+i, 2*V+slot, -k) + + mixer() + observation(-sign, "old") + mixer(True) + for i, s in p["sources"]: + add(2*V+s, i, 1) + mixer() + observation(sign, "new") + mixer(True) + for i, s in reversed(p["sources"]): + add(2*V+s, i, -1) + expected = [(1 << i, 0) for i in range(W)] + for i in range(V): + expected[V+i] = f3add(expected[V+i], expected[i], sign) + changed_rows = sum(a != b for a, b in zip(values, expected)) + changed_coefficients = 0 + for a, b in zip(values, expected): + d = f3add(a, b, -1) + changed_coefficients += (d[0] | d[1]).bit_count() + helper_rows = sum(values[i] != expected[i] for i in range(2*V, W)) + if omit is None: + require(changed_rows == 0, "Complete scalar operator failed") + require(operations == ledger["complete_scalar_primitives"], "Scalar count mismatch") + else: + require(changed_coefficients > 0, "Negative control was not detected") + return dict(sign=sign, independent_roles=W, all_operator_coefficients=W*W, + scalar_primitives=operations, changed_rows=changed_rows, + changed_coefficients=changed_coefficients, changed_helper_rows=helper_rows, + arbitrary_dirty_helper_columns=True, omitted_phase=omit) + + +def eye(n): + return [[Q(i == j) for j in range(n)] for i in range(n)] + + +def mm(A, B): + return [[sum(a*b for a, b in zip(row, col)) for col in zip(*B)] for row in A] + + +def transpose(A): + return [list(x) for x in zip(*A)] + + +def enclosure_projector(h, I, U): + H = [[Q(i == j)-Q(2, 25) for j in range(h)] for i in range(h)] + k = I.bit_count() + if k == 5: + basis = [[Q(bool(I >> i & 1))] for i in range(h)] + inverse_gram = [[Q(1, 3)]] + else: + outside = [i for i in range(h) if U >> i & 1 and not I >> i & 1] + basis = [[Q(i == j)+Q(bool(I >> i & 1), 5-k) for j in outside] + for i in range(h)] + alpha = Q(k-2, (5-k)**2) + n = len(outside) + inverse_gram = [[Q(i == j)-alpha/(1+n*alpha) for j in range(n)] for i in range(n)] + gram = mm(mm(transpose(basis), H), basis) + require(mm(gram, inverse_gram) == eye(len(gram)), "Closed-form Gram inverse failed") + E = mm(mm(mm(basis, inverse_gram), transpose(basis)), H) + require(mm(E, E) == E and mm(transpose(E), H) == mm(H, E), "Invalid rational projector") + require(sum(E[i][i] for i in range(h)) == frame_rank(I, U), "Projector rank mismatch") + return E + + +def exact_enclosure_certificate(g, ledger): + h = g["h"] + enclosures = ledger["_program"]["enclosures"] + digest = sha256() + for I, U in enclosures: + E = enclosure_projector(h, I, U) + digest.update(json.dumps([[str(x) for x in row] for row in E], + separators=(",", ":")).encode()) + return dict(h=h, distinct_enclosures=len(enclosures), + exact_rational_projector_entries=len(enclosures)*h*h, + idempotence_checked=True, H_self_adjointness_checked=True, + closed_form_Gram_inverse_checked=True, + projector_sha256=digest.hexdigest(), + all_large_frame_inclusions="Checked as exact mask constraints; proved by the enclosure equations") + + +@cache +def log_interval(n, d=1, terms=50): + """Rational bounds from atanh, reducing the argument to [1,2).""" + x = Q(n, d) + require(x >= 1, "Log interval currently expects x >= 1") + k = 0 + while x >= 2: + x /= 2 + k += 1 + + def unit(y): + z = (y-1)/(y+1) + z2 = z*z + term = z + lower = Q(0) + for j in range(terms): + lower += 2*term/(2*j+1) + term *= z2 + upper = lower+2*term/((2*terms+1)*(1-z2)) + return lower, upper + + a, b = unit(x) + c, d2 = unit(Q(2)) + return dyadic_enclose(a+k*c, b+k*d2) + + +def dyadic_enclose(lo, hi, bits=128): + scale = 1 << bits + lower = (lo.numerator*scale)//lo.denominator + upper = -((-hi.numerator*scale)//hi.denominator) + return Q(lower, scale), Q(upper, scale) + + +def exp_interval(lo, hi, terms=20): + require(0 <= lo <= hi < 1, "Exp argument outside the fixed probe interval") + def partial(t): + s = Q(1) + term = Q(1) + for k in range(1, terms+1): + term *= t/k + s += term + return s, term*t/(terms+1) + lower = partial(lo)[0] + s, next_term = partial(hi) + upper = s+next_term/(1-hi/(terms+2)) + return dyadic_enclose(lower, upper) + + +def moment_interval(ledger, a): + require(0 <= a <= Q(1, 100), "Only small primitive exponents are screened") + lo = hi = Q(0) + W, m = ledger["W"], ledger["m"] + for r, count in ledger["global_profile"].items(): + lower, upper = log_interval(m, r) + lower, upper = exp_interval(a*lower, a*upper) + weight = Q(count*r, W*m) + lo += weight*lower + hi += weight*upper + return lo, hi + + +def quotient(x): + return f"{x.numerator}/{x.denominator}" + + +def moment_gate(ledger): + # Floating arithmetic locates a candidate only. Every pass/fail below is + # proved independently by rational interval arithmetic. + W, m = ledger["W"], ledger["m"] + prof = ledger["global_profile"] + f = lambda a: sum(count*r*exp(a*log(m/r)) for r, count in prof.items())/(W*m) + if f(0) >= 1: + candidate = Q(0) + else: + lo, hi = 0.0, 0.01 + for _ in range(64): + mid = (lo+hi)/2 + if f(mid) < 1: + lo = mid + else: + hi = mid + candidate = Q(int(lo*10**9), 10**9) + targets = sorted({candidate, candidate+Q(1, 10**9), Q(1, 1000), Q(1, 250)}) + gates = [] + for a in targets: + lo, hi = moment_interval(ledger, a) + gates.append(dict(a=quotient(a), moment_lower=float(lo), moment_upper=float(hi), + lower_exact=quotient(lo), upper_exact=quotient(hi), + strict_pass=hi < 1, strict_fail=lo > 1, + enclosure_method="Exact rational atanh/log and Taylor/exp tails")) + if candidate > 0: + require(gates[targets.index(candidate)]["strict_pass"], "Rounded candidate not certified") + require(gates[targets.index(candidate+Q(1, 10**9))]["strict_fail"], "Endpoint bracket not certified") + return dict(certified_primitive_a=quotient(candidate) if candidate > 0 else None, + candidate_location_only="Float bisection; never used as certificate", + targets=gates, multiplication_kappa_claim=False) + + +def summarize(g, ledger): + return dict(h=g["h"], V=g["V"], c=g["c"], + active_additions=g["additions"], allocated_additions=len(g["left"])-g["V"]-1, + side_query_uses=len(g["roots"]), center_query_uses=len(g["centers"]), + tensor_total_augmented=g["template_total_augmented"], + template_active_additions=g["template_active_additions"], + DAG_sha256=array_digest(g["left"], g["right"], g["common"], g["union"]), + query_sha256=array_digest(g["roots"], g["targets"], g["centers"]), + active_sha256=sha256(g["active"]).hexdigest(), + exact_formal_supports_expanded=g["exact"] is not None, + native={k: v for k, v in ledger.items() if not k.startswith("_")}, + primitive_moment=moment_gate(ledger)) + + +def main(): + ap = argparse.ArgumentParser() + ap.add_argument("--h", type=int, nargs="+", default=[8, 14, 19]) + ap.add_argument("--receipt", type=Path, default=Path(__file__).with_name("JOINT-PAIR-DAG.json")) + ap.add_argument("--skip-projector-matrices", action="store_true") + args = ap.parse_args() + result = dict(schema="joint-pair-total-monotone-native-DAG/1", + status="PASS_PAID_NATIVE_REFERENCE", new_admitted_kappa=False, + scalar_field="F3; rational geometry is separate", + geometry="H=I-(2/25)J, five-subset lines of norm 3", + word=["L@0", "-J@0", "L^-1@0", "V@P_i", "L@nested E(I,U)", + "J_center via immutable copies E_pair->0", "J_side@(I-P_i)", + "L^-1@I", "-V@I"], + upstream_algorithm_pin="12fa8c5c844be21bfefb38a19503fbd867a11244", + scalar_count_scope=("Counts pointwise scalar ADDs in the algebraic word. Complete stream " + "copying, retirement, routing and tape work are additional linear work; " + "recursive copy transforms are included in the rank profile."), + runtime_scope=("Native local/cover phase construction and exact moment gate under the " + "retained complete-stream contract; no new outer arbitrary-precision " + "engine or multiplication-runtime certificate is supplied."), + cases=[], scalar_checks=[], negative_controls=[], rational_projectors=[]) + for h in args.h: + exact = h == 8 + g = build(h, exact_support=exact) + ledger = compile_native(g, retain=exact) + if exact: + for sign in (1, -1): + result["scalar_checks"].append(scalar_certificate(g, ledger, sign)) + for omit in ("old_center", "new_center", "old_side", "new_side"): + result["negative_controls"].append(scalar_certificate(g, ledger, omit=omit)) + if not args.skip_projector_matrices: + result["rational_projectors"].append(exact_enclosure_certificate(g, ledger)) + case = summarize(g, ledger) + result["cases"].append(case) + print(json.dumps({"h": h, "V": g["V"], "A": ledger["persistent_auxiliaries"], + "delta": ledger["global_rank_saving"], + "certified_a": case["primitive_moment"]["certified_primitive_a"]}), flush=True) + result["program_sha256"] = sha256(Path(__file__).read_bytes()).hexdigest() + result["scope"] = ("The paid local F3 word has standard data/helper endpoints. The global profile uses the " + "previously proved shared padded 3h cover. This is a large-state reference, not a new " + "selected primitive or multiplication record; no complex supplier claim is made.") + args.receipt.parent.mkdir(parents=True, exist_ok=True) + args.receipt.write_text(json.dumps(result, indent=2)+"\n") + print(json.dumps({"receipt": str(args.receipt), "status": result["status"]})) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/README.md b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/README.md new file mode 100644 index 0000000000..91b0158e0d --- /dev/null +++ b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/README.md @@ -0,0 +1,167 @@ +# Native observations, a complete local consumer, and a terminal decoder + +Substantial OpenAI assistance; Apache-2.0. The source QQ16 construction, +retained mixed supplier and current-frame affine-map conventions are credited +through the pinned source files and the cumulative package's notices. + +## Results + +This directory closes two concrete operator contracts. Neither is a new +multiplication bound. + +| Construction | Exact result | Native child bill | Owned full-cover fields | +|---|---|---:|---:| +| One current ordinary-Y observation | An owned completed-frame copy of the original hole-framed Y; source preserved | 1 H1 per copied port | 1 per live output | +| Fused current-pair acquisition and consumer | The complete repair `H_(s,t) B_t F_s(x,y)=(-y,x+y)` for the changed 104-port local pair | 208 H1 per pair | 17, all returned zero | +| Complete terminal content decoder | `D_final Q^-1 D_final^-1` for the literal nine-call, four-bank-pair content schedule | 832 H1 | 8, all returned zero | + +The second construction is the most useful new closure. The sibling +`weighted-sideinfo` program acquires the needed current queries for 104 H1. +The complete consumer costs another 104 H1: eight selected X-to-Y transfers +and 96 ordinary Y corrections. It uses the same 17-field stock by consuming +selected observations early. There is no extra eight-field saved-Y bank. +See [PROOF.md](PROOF.md) for the complete formula and chronology. + +The full literal local program has, **per pair**, 208 positive H1 children, +416 line-wrapper monomial maps, 8,800 further physical affine maps, and +42,016 real scalar primitives. These counts include the query acquisition, +all selected and ordinary output corrections, and retirement of all owned +fields. They exclude the producer's existing local core, the producer's +separate new wrappers, and the still-unconstructed global hosting and idle +profile. Every routed read is lowered to a map on its source, a distinct-target +scalar ADD, and the exact inverse source map. Sources can change during that +three-operation view and are restored when the view ends. + +The terminal construction takes a different approach: normalize all current X +fields into the common final frame, invert the entire content word, then put +all X fields back. There are four pairs, hence **416 X fields**, not 104. +Two adapters on each of them cost 832 H1. This word has 1,664 line-wrapper +maps, 6,656 further physical maps, and 185,568 real scalar primitives. Its +complete schedule is bound to the retained owner template, including bridges +and the signed terminal exchange. + +## What the paid bill says + +The earlier changed-producer wrappers cost 64 H1 across the nine-call screen. +Both explicit routes therefore reach an extra native bill of **896 H1** when +the four local pair consumers are counted, or when the terminal decoder is +used. This numerical equality is not a proof that four local repairs can be +spliced into the retained nine-call controller. Later local calls lie on +different bank pairs and are separated by bridges. The terminal decoder +includes that full content schedule; the local-pair repair still needs a +global composition. + +The inherited *relaxed* controller allowances from the frozen package are: + +| Primitive target C | Inherited allowance | New 896-H1 bill exceeds it by | +|---|---:|---:| +| 0.001745 | 571.332681182781… | 324.667318817218… | +| 0.004 | 155.819896738978… | 740.180103261021… | + +These are screens based on the old weight and idle accounting, not certified +moments for the changed 104-port geometry. They already reject the explicit +new native bill before new ownership or scalar/address-work obligations. +The retained outer conversion is `κ=C−10^-46−10^-70`; the ordinary +`C/(1+C)` conversion is not used by this retained package. + +The 104-H1 query acquisition remains a real improvement over the earlier +16-H8 acquisition at this interface. However, keeping its four-pair bill of +416 H1 and the 64 producer wrappers leaves only 91.332681… at C=.001745 for +all consumers and other new costs. An equally distributed consumer allowance +would be less than 22.833171 H1 per pair, compared with the explicit 104. +More than about 78.045% of that consumer's native bill must disappear even +before hosting. At C=.004, the acquisition and wrapper bill alone fails. +That target needs an additional source, geometry, or normalized-operator +change, not just faster retirement of these query fields. + +A more specific next target is to retain the eight selected transfers per +pair. They consume 32 of the remaining 91.332681… units, leaving less than +14.833171 H1 per pair for the ordinary response. If that response could be +realized through eight native rank-one channels, the total screen would be +544 H1 with 27.332681… left for other new native costs. This is an unproved +budget target. A candidate must realize the full operator-valued response +`K_s,U−K_t,U M^-1` in all distinct Y hole frames; scalar rank eight alone +does not establish such an implementation. + +## Physical frame facts + +Let `F_t` denote the completed rank `9+8t` frame. At a local repair boundary, +X is in `F_t` and Y at port p is in `F_t C_qp^-1`. A copied Y observation +therefore needs one positive `C_qp` to become a completed-frame field. +A completed correction needs one negative `C_qp` before adding to that Y. +The signed line recipes use positive H1 children with explicit linear and +translation wrappers; their complete 512-column local kernels are checked. + +Owner routes are actual 73-dimensional binary matrices. They are never +collapsed to a single scalar owner. An owner map g is given its actual +translation δ so that its complete Z4 phase polynomial preserves `F_t`. +The inverse map includes its correct affine translation. This is why +completed-frame query synthesis is legal and ordinary uncorrected route +substitution is not. + +The final named X and Y banks are not in identical physical frames. The +terminal exchange leaves X in `F_final C_qp` and Y in `F_final`, where +`F_final` is the completed rank-73 frame. Thus a content +correction must be conjugated by `D_final`; merely emitting `Q^-1` in the +current physical registers is not valid. + +There is also an exact, restricted whole-word obstruction to moving away an +X-only subset of those final adapters. On the owner-constant all-ones payload +character, all actual corrected owner routes act identically, while every +port line acts by i. The dyadic Q matrix has one connected support component +containing all 416 X and all 416 Y fields; 173,400 off-diagonal nonzeros are +witnessed modulo 65,521. A commuting X-only adapter mask would have to be a +union of components, so no nonempty such mask works. This rejects a specific +commute-and-cancel strategy. It is not a lower bound of 832 children for +arbitrary new decoder synthesis or changed source encodings. + +## Evidence and reproduction + +Python 3.12 or newer, standard library only. The scripts support isolated +mode. From this directory: + +```sh +python3 -I -O -B observation_probe.py --output /tmp/OBSERVATION-NEW.json --word-output /tmp/OBSERVATION-WORD-NEW.json +python3 -I -O -B consumer_probe.py --output /tmp/CONSUMER-NEW.json --word-output /tmp/CONSUMER-WORD-NEW.json +python3 -I -O -B controller_probe.py --output /tmp/CONTROLLER-NEW.json --word-output /tmp/CONTROLLER-WORD-NEW.json +``` + +The cumulative replay compares parsed outputs with the committed receipts. +The consumer needs the sibling `weighted-sideinfo` directory; its code, +literal word and receipt are byte-pinned in `sources/QUERY-SOURCE-PINS.json`. +The two underlying QQ16 data files are byte-pinned in `native_common.py`. +The complete retained owner template is independently byte-pinned in the +terminal decoder, and the extracted recipe must match it exactly. + +| Program | Receipt and literal word | Main verification | +|---|---|---| +| `observation_probe.py` | `receipts/OBSERVATION.json`, `OBSERVATION-WORD.json` | Signed kernels, all new completed-frame maps, dirty-source and wrong-line controls | +| `consumer_probe.py` | `receipts/CONSUMER.json`, `CONSUMER-WORD.json` | Four stages, 208 independent current input fields, actual owner products, independent free-query consumer, 32 semantic controls | +| `controller_probe.py` | `receipts/CONTROLLER.json`, `CONTROLLER-WORD.json` | All 27 low-rank macros, both literal scalar inverse orders, both whole scalar-sector inverse orders, signed endpoint kernels, whole-Q commutation screen | + +The consumer's operator proof is compositional. The acquisition prefix is +expanded on all independent data and initial owned dirt. The suffix is +expanded on 16 independent free query fields; substitution of the proven +prefix gives the data-dependent result. The actual same-port unit identities +then prove composition with the changed producer. The terminal proof likewise +composes individually checked transvections and elementary inverse words. No +receipt claims to have expanded an exponentially large 73-dimensional +whole-supplier matrix. + +## Research direction after this closure + +The ranked literature and its quantitative relevance are in +[PRIMARY-LITERATURE.md](PRIMARY-LITERATURE.md). The highest-value next target is +a functional correction interface: produce or consume the final low-rank +response without separately normalizing every ordinary Y target. A concrete +candidate is to inject the correction through an already paid compatible +scatter inside the rebased clean-J core. Such a proposal must match the +query's current physical gauge, every owner route, timing, arbitrary helper +restoration and actual final scatter. That compatibility is not proved here. + +Another viable search changes the producer or the normalized code so the +required response already lives in the consumer's hole frames. Coherent +changes of coordinates alone do not alter a fixed normalized operator, and +scalar collapse of owner routes cannot establish this interface. The new +local closure provides exact targets and a complete invoice against which to +judge a replacement. diff --git a/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/native_common.py b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/native_common.py new file mode 100644 index 0000000000..df276cc71f --- /dev/null +++ b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/native_common.py @@ -0,0 +1,176 @@ +#!/usr/bin/env python3 +"""Exact owner/phase utilities for terminal QQ16 decoder probes. + +Apache-2.0; OpenAI assistance. Phase and owner conventions follow the pinned +QQ16 source and ywz65's qq16-current-frame-consumer/native_support.py. +No predecessor executable is imported; all source data are byte-pinned. +""" +from collections import Counter +from fractions import Fraction as Q +from functools import lru_cache +from hashlib import sha256 +from pathlib import Path +import gzip +import json + +HERE = Path(__file__).resolve().parent +AXES = tuple(1 << i for i in range(73)) +PINS = { + "certificate.json.gz": "a5ed85fac5ebf3c091482a113ab0d351eb5a7892fe2841a35e16017ec1ff49f7", + "DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa", +} + + +def need(ok, why): + if not ok: + raise ValueError(why) + + +def load_sources(): + result = [] + for name, digest in PINS.items(): + raw = (HERE / "sources" / name).read_bytes() + need(sha256(raw).hexdigest() == digest, "source pin: " + name) + result.append(json.loads(gzip.decompress(raw) if name.endswith(".gz") else raw)) + packet, phases = result + need(len(phases["recipes"]) == 112 and phases["m"] == 73, "native dimensions") + need(all(r["port"] == p for p, r in enumerate(phases["recipes"])), "port ordering") + return packet, phases + + +def bits(x): + while x: + low = x & -x + yield low.bit_length() - 1 + x ^= low + + +def act(g, x): + out = 0 + for j in bits(x): + out ^= g[j] + return out + + +@lru_cache(None) +def product(g, h): + return tuple(act(g, v) for v in h) + + +def frame(entries): + result = Counter() + for axis, coefficient in entries: + if axis: + result[axis] = (result[axis] + coefficient) % 4 + return tuple((a, c) for a, c in sorted(result.items()) if c) + + +@lru_cache(None) +def polynomial(entries): + linear, quadratic = [0] * 73, [0] * 73 + for axis, c in entries: + support = tuple(bits(axis)) + for i in support: + linear[i] = (linear[i] + c) % 4 + if c % 2: + for j in support: + if j > i: + quadratic[i] ^= 1 << j + return tuple(linear), tuple(quadratic) + + +def completed(stage): + need(0 <= stage <= 8, "completed stage") + return frame((a, 1) for a in AXES[:9 + 8 * stage]) + + +def move_frame(f, g, shift=0): + return frame([(act(g, a), c) for a, c in f] + [(shift, 2)]) + + +def corrected_shift(g, f): + """Return the actual affine correction, or fail when no translation works.""" + want, got = polynomial(f), polynomial(move_frame(f, g)) + need(want[1] == got[1], "a translation cannot repair a quadratic mismatch") + delta = tuple((a - b) % 4 for a, b in zip(want[0], got[0])) + need(all(v in (0, 2) for v in delta), "a translation cannot repair an odd phase") + shift = sum((v // 2) << i for i, v in enumerate(delta)) + need(polynomial(move_frame(f, g, shift)) == want, "complete affine phase identity") + return shift + + +def source_route(phases, stage, port): + if stage == 0: + return AXES + recipe = phases["recipes"][port] + basis = recipe["idle_children"][0]["orthogonal_axes"][:8] + swaps = [b ^ AXES[9 + 8 * (stage - 1) + i] for i, b in enumerate(basis)] + g = tuple(e ^ act(swaps, sum(((e & v).bit_count() % 2) << i + for i, v in enumerate(swaps))) for e in AXES) + need(product(g, g) == AXES, "source route involution") + need(act(g, recipe["q"]) == recipe["q"], "source route fixes its own port normal") + return g + + +def matrix_rows(columns): + n = len(columns) + return [sum(((columns[j] >> i) & 1) << j for j in range(n)) for i in range(n)] + + +def address_action(rows, address): + return sum(((r & address).bit_count() % 2) << j for j, r in enumerate(rows)) + + +def line_recipe(q, sign): + """Positive H1 and two literal monomial wrappers; inverse uses T_q C_q.""" + need(0 < q < 512 and sign in (-1, 1), "local signed line") + pivot = (q & -q).bit_length() - 1 + columns = (q,) + tuple(1 << j for j in range(9) if j != pivot) + inv = {act(columns, v): v for v in range(512)} + need(len(inv) == 512, "invertible nine-axis line wrapper") + return {"rank": 1, "left_rows": matrix_rows(columns), + "left_offset": 0 if sign == 1 else q, + "right_rows": matrix_rows(tuple(inv[1 << i] for i in range(9)))} + + +def line_column(recipe, address): + middle = address_action(recipe["right_rows"], address) + a = address_action(recipe["left_rows"], middle) ^ recipe["left_offset"] + b = address_action(recipe["left_rows"], middle ^ 1) ^ recipe["left_offset"] + return {a: (1, 1), b: (1, -1)} # common denominator 2 + + +def sparse_add(target, source, coefficient=Q(1)): + for key, value in source.items(): + v = target.get(key, Q(0)) + coefficient * value + if v: + target[key] = v + else: + target.pop(key, None) + + +def routed(g, row): + result = {} + for (variable, h), c in row.items(): + key = variable, product(g, h) + result[key] = result.get(key, Q(0)) + c + return {key: c for key, c in result.items() if c} + + +def scalar_cost(kind, coefficient=Q(1)): + if kind == "clear": + return 2 + a, b = abs(coefficient.numerator), coefficient.denominator + need(a > 0 and not b & (b - 1), "nonzero dyadic coefficient") + if kind == "add": + return 2 * (2 + a.bit_length() + a.bit_count() + b.bit_length() - 1) + need(kind == "scale" and not a & (a - 1), "power-of-two scale") + return 2 * (abs(a.bit_length() - b.bit_length()) + (coefficient < 0)) + + +def write_json(value, output=None): + text = json.dumps(value, indent=2, sort_keys=True) + "\n" + if output is None: + print(text, end="") + else: + Path(output).write_text(text) diff --git a/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json new file mode 100644 index 0000000000..d9a4f84211 --- /dev/null +++ b/research/kappa-0023-productive-jobs/dependencies/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json @@ -0,0 +1 @@ +{"cover_correction":{"complete_small_cover_checks":[{"n":1,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":1},{"n":2,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":2},{"n":3,"nonzero_full_kernel_corrections":0,"orthogonal_matrices":6},{"n":4,"nonzero_full_kernel_corrections":24,"orthogonal_matrices":48},{"n":5,"nonzero_full_kernel_corrections":600,"orthogonal_matrices":720}],"formula":"delta(g)_i = (weight(row_i(g))-1)/2 mod 2; apply X_delta(g) to EVERY final live stream in cover class g.","mandatory":true,"reason":"P_g C_all P_g^-1 = C_all X_delta(g) exactly. Orthogonality cancels all quadratic cross terms; both sides fix the uniform vector."},"global_phase_argument":"Every C_q and every linear address permutation fixes the uniform vector. Equality of the exact Z4 Fourier multiplier therefore fixes the scalar phase as 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b/research/kappa-0023-productive-jobs/evidence/logs/bit_allsize_parameters.log @@ -0,0 +1 @@ +{"status": "PASS_EXACT_PARAMETERS_CONDITIONAL_ON_FINITE_UNIFORM_PRECOMPILATION", "C0_before_cover": "69/25000", "C1_after_cover": "11/4000", "ordinary_saving_before_final_relaxation": "27418653797/10000000000000", "ordinary_saving_admitted": "27/10000"} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/bit_delayed_cover.log b/research/kappa-0023-productive-jobs/evidence/logs/bit_delayed_cover.log new file mode 100644 index 0000000000..016bcd864e --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/bit_delayed_cover.log @@ -0,0 +1 @@ +{"status": "PASS_EXACT_LOCAL_RING_AUDIT", "cover_rows": 3888, "exact_matches": 7776, "affine_high_row_checks": 7776, "runtime_seconds": 0.4358917500358075} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_bridge.log b/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_bridge.log new file mode 100644 index 0000000000..17bad5f278 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_bridge.log @@ -0,0 +1 @@ +{"status": "PASS_NATIVE_ROLES_COMPLETE_DAG_ALL_COLUMN_PAIRING_COMMON_WEIGHT_AND_PAID_COVER", "native_productive_fields": 3739, "paid_COPY_calls": 63, "uniform50_paid1": 186725, "nonintegral_cover_checks": 7776, "ordinary_saving": "27418653797/10000000000000"} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_controller.log b/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_controller.log new file mode 100644 index 0000000000..c68a4eb1c9 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/bit_independent_controller.log @@ -0,0 +1 @@ +{"status": "PASS", "arbitrary_function_input_columns": 1536, "scalar_matrix_coefficients_compared": 2359296, "seconds": 0.014202125021256506} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/bit_native_and_mixed_controller.log 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444, "50": 222}, "paid": {"1": 9261, "2": 9394, "3": 553, "4": 308, "5": 714, "6": 77, "7": 1465}, "scalar_operation_counts": {"unit_ADDs_including_arithmetic_final_swaps": 174908, "source_owned_COPY_episodes": 63, "COPY_or_ERASE_records": 126, "nonidentity_port_right_routes_including_inverses": 2664, "maximum_owned_center_fibers_per_base_unit": 1}, "runtime_seconds": 0.7925338330096565} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complete_outer_parameters.log b/research/kappa-0023-productive-jobs/evidence/logs/complete_outer_parameters.log new file mode 100644 index 0000000000..a5a27a1856 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complete_outer_parameters.log @@ -0,0 +1 @@ +{"inherited_crosscheck": {"all_inherited_checks": 63, "matched_cost_rows": 9, "matched_named_checks": 50, "native_admission_not_inferred_from_parameter_function": true}, "smallest_cost_margin": {"decimal": "0.00000197698", "rational": "98849/50000000000"}, "source_sha256": "2c9144f576645ea1f71460758ec80580960fc4bd4c09ff73ef0912827f39ae45", "status": "PASS_EXACT_ARITHMETIC_AND_SOURCE_BOUND_MIXED_MOMENTS", "strict_checks": 54, "target": "23/10000"} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_composition_contracts.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_composition_contracts.log new file mode 100644 index 0000000000..447bd38630 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_composition_contracts.log @@ -0,0 +1 @@ +PASS_EXACT_C_H_NEGATIVE_C_AND_SOURCE_BOUND_NATIVE_COMPOSITION diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_dyadic_balance.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_dyadic_balance.log new file mode 100644 index 0000000000..d08862761d --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_dyadic_balance.log @@ -0,0 +1,4 @@ +PASS balanced THREE-base H8H1+3504; logger8H9+3440 +PASS balanced THREE-extra H8H1+3504; logger8H9+3440 +PASS balanced FOUR-base H8H1+3504; logger8H9+3440 +PASS balanced FOUR-extra H8H1+3504; logger8H9+3440 diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_full_identity.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_full_identity.log new file mode 100644 index 0000000000..6c7a16362e --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_full_identity.log @@ -0,0 +1,4 @@ +PASS full identity THREE-base {1: 2487, 2: 713, 3: 232, 4: 102, 5: 28} +PASS full identity THREE-extra {1: 2487, 2: 713, 3: 232, 4: 102, 5: 28} +PASS full identity FOUR-base {1: 2489, 2: 713, 3: 235, 4: 99, 5: 28} +PASS full identity FOUR-extra {1: 2489, 2: 713, 3: 235, 4: 99, 5: 28} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_h_reconstruction.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_h_reconstruction.log new file mode 100644 index 0000000000..b7f8c2e2df --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_h_reconstruction.log @@ -0,0 +1,4 @@ +PASS H112 THREE-base 8 H1 + 4336; both directions +PASS H112 THREE-extra 8 H1 + 4336; both directions +PASS H112 FOUR-base 8 H1 + 4336; both directions +PASS H112 FOUR-extra 8 H1 + 4336; both directions diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_helper_reconstruction.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_helper_reconstruction.log new file mode 100644 index 0000000000..28ce89c7fa --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_helper_reconstruction.log @@ -0,0 +1,4 @@ +PASS helpers THREE-base {5: 28, 6: 18, 7: 56, 8: 93, 9: 213} +PASS helpers THREE-extra {5: 28, 6: 18, 7: 56, 8: 93, 9: 213} +PASS helpers FOUR-base {5: 28, 6: 18, 7: 57, 8: 92, 9: 213} +PASS helpers FOUR-extra {5: 28, 6: 18, 7: 57, 8: 92, 9: 213} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_logger_reconstruction.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_logger_reconstruction.log new file mode 100644 index 0000000000..b899810b40 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_logger_reconstruction.log @@ -0,0 +1,4 @@ +PASS logger THREE-base 8 H9 + 4224 +PASS logger THREE-extra 8 H9 + 4224 +PASS logger FOUR-base 8 H9 + 4224 +PASS logger FOUR-extra 8 H9 + 4224 diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_native_extensions.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_native_extensions.log new file mode 100644 index 0000000000..ebd16720dd --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_native_extensions.log @@ -0,0 +1,4 @@ +PASS extensions THREE-base embedded0 extra children; wrapped H20192 extra scalars +PASS extensions THREE-extra embedded0 extra children; wrapped H20192 extra scalars +PASS extensions FOUR-base embedded0 extra children; wrapped H20192 extra scalars +PASS extensions FOUR-extra embedded0 extra children; wrapped H20192 extra scalars diff --git a/research/kappa-0023-productive-jobs/evidence/logs/complex_phase_contacts.log b/research/kappa-0023-productive-jobs/evidence/logs/complex_phase_contacts.log new file mode 100644 index 0000000000..b557bd2a8e --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/complex_phase_contacts.log @@ -0,0 +1 @@ +{"H_stage_identification": "At both contacts the normalized logical map is D_s H D_s, with D_s diagonal in port index and each owner route squaring to identity.", "all_affine_maps_have_explicit_checked_inverses": true, "all_owner_routes_involutive": true, "backward_orientation": "The physical views are real permutations/translations; conjugating the checked identity proves the dual orientation.", "contact_count": 2016, "determining_addresses_per_unique_phase_pattern": 2702, "generator_sha256": "9c9de9d5d954a7d0680655a10fa7c1ecc6b865e05146c1cfc12688b3654600d1", "input_generator_sha256": "b4f8c79450edca57f3a359e09d1206a253e399308f01e89e469044149116c2e8", "input_receipt_sha256": "f63986aa1448ddcee29a0b60e8402550861fa129fc13bf6e61f035333ed1bbf5", "negative_control": {"contact": "preY", "mutated_shift_xor": 1, "nonzero_phase_Z4": 2, "port": 0, "rejecting_address": 1, "stage": 1}, "no_new_kappa_claimed": true, "phase_packet_sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa", "proof_scope": "Exact finite frame/phase contact admission only. It does not remove terminal monodromy or establish a complete caller.", "schema": "independent-direct-symbol-H-contact-audit/1", "seconds": 5.657380374963395, "status": "PASS_ALL_2016_CONTACTS_AND_INVERSES_WITH_GLOBAL_PHASE", "unique_phase_patterns": 1855} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/explicit_finite_ranks.log b/research/kappa-0023-productive-jobs/evidence/logs/explicit_finite_ranks.log new file mode 100644 index 0000000000..d8a0c4d3c4 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/explicit_finite_ranks.log @@ -0,0 +1 @@ +{"status": "PASS_EXACT_SUPERPOTENTIAL_AND_COMPRESSED_RANK_BOUNDS", "families": [{"family": "complex", "uniform_saving": "36/15625", "n0": {"encoding": "2^H0", "H0": 60000040}, "N": {"encoding": "2^H1", "H1": 276540184361}, "final_strict_power_gap": "1/2000000"}, {"family": "BIT", "uniform_saving": "69/25000", "n0": {"encoding": "2^H0", "H0": 10666731}, "N": {"encoding": "2^H1", "H1": 6552928412}, "final_strict_power_gap": "9/2000000"}]} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/finite_integer_compiler.log b/research/kappa-0023-productive-jobs/evidence/logs/finite_integer_compiler.log new file mode 100644 index 0000000000..700afe0ba1 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/finite_integer_compiler.log @@ -0,0 +1 @@ +{"status": "PASS_FINITE_DAG_FIELD_OWNERS_BATCH_CLEARING_AND_FULL_RANK_COMPLETION", "toy_ranks": [4, 7, 12, 16, 28, 40], "actual_full_rank": 41, "actual_width": 1397, "actual_paid_rank1": 57149} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/independent_outer_transfer.log b/research/kappa-0023-productive-jobs/evidence/logs/independent_outer_transfer.log new file mode 100644 index 0000000000..fc278ee166 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/independent_outer_transfer.log @@ -0,0 +1 @@ +{"status": "PASS_EXPLICIT_RANKS_NEW_NATIVE_INTERFACES_AND_CONDITIONAL_OUTER_POINT", "kappa": "23/10000", "limiting_margin": "98849/50000000000", "explicit_N_exponents": {"complex": 276540184361, "BIT": 6552928412}} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/new_precision_and_stock.log b/research/kappa-0023-productive-jobs/evidence/logs/new_precision_and_stock.log new file mode 100644 index 0000000000..e30f12b4ec --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/new_precision_and_stock.log @@ -0,0 +1 @@ +{"status": "PASS_SOURCE_BOUND_CHARGE_EFFECTIVE_LINEAR_GUARD_AND_COMPLETE_ROWS", "p3_complete_elementary_charge_bound": 1494376, "general_substitution": {"Z": "W+O", "O": "at most U, conservatively U", "G": "1494376*U", "E": "64*(Z+N+P+G+1)^3", "B": "P+E", "C0": "32*N*B^2", "C1": 1, "semantic_recurrence": "A(e) <= A(floor(e/N))+P*floor(e/N)+E", "induction": "A(e) <= 2*B*e", "full_layer": "A_layer <= (2*B+18)*d < C0*d"}} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/p2_native_cover.log b/research/kappa-0023-productive-jobs/evidence/logs/p2_native_cover.log new file mode 100644 index 0000000000..3c565da592 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/p2_native_cover.log @@ -0,0 +1 @@ +{"contacts": 2240, "fields": 1397, "paid": {"1": 12565, "2": 3621, "3": 1217, "4": 510, "5": 169}, "receipt_sha256": "5e2868b6d8538eb8effb377d9c15b7da7d69e7374c4d01158842ae887e71c77d", "scalar": 796520, "status": "PASS_FINITE_IDEAL_NATIVE_OWNER_PHASE_AND_COMPLETE_INVOICE", "targets": {"6": 146, "9": 803, "41": 448}} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_owners.log b/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_owners.log new file mode 100644 index 0000000000..406ca731d6 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_owners.log @@ -0,0 +1 @@ +{"status": "PASS_ALL2856_HELPERS_CAUSAL_WINDOWS_MAIN_NORMALIZER_ORDER_AND_COMPLETE_BILL", "targets": {"6": 192, "7": 56, "8": 93, "9": 1229, "25": 448, "57": 224}, "paid": {"1": 17623, "2": 5065, "3": 1738, "4": 714, "5": 254}, "scalar": 1147304, "uniform_full57_paid1": 127570} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_phases.log b/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_phases.log new file mode 100644 index 0000000000..5038e72d52 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/p3_independent_phases.log @@ -0,0 +1 @@ +{"status": "PASS_LITERAL_LAURENT_NATIVE_CONTACTS_AND_ALL672_MAIN_OUTPUT_PHASES", "contacts": 3136, "targets": {"25": 448, "57": 224}, "endpoint_phase_controls": 336, "sha256": "d455ddb20ea83658854a1d83aed99d5302f9e27a86da4ed8064a7ee3252d19a6"} diff --git a/research/kappa-0023-productive-jobs/evidence/logs/p3_native_cover.log b/research/kappa-0023-productive-jobs/evidence/logs/p3_native_cover.log new file mode 100644 index 0000000000..e47d097f7b --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/logs/p3_native_cover.log @@ -0,0 +1 @@ +{"paid": {"1": 17623, "2": 5065, "3": 1738, "4": 714, "5": 254}, "root_bracket": {"lower": "0.002305090194876170622919094755", "margin_at_lower": {"lower": "0.00000000000000000000000001304634154", "upper": "0.00000000000000000000000001304634155"}, "margin_at_upper": {"lower": "-0.00000000000000000000000008370970220", "upper": "-0.00000000000000000000000008370970219"}, "uniqueness": "The paid/target ratio is strictly increasing in C because every paid rank is smaller than every target rank.", "upper": "0.002305090194876170622919094756"}, "scalar": {"boundary_negations_per_port": 8, "boundary_unit_ADD_per_port": 22, "bridge_unit_ADD_per_port": 12, "conservative_real_scalar_primitive_allowance": 1147304, "coordinate_and_phase_moves": "Explicit separate fixed movement operations; not scalar ADDs and not paid recursive calls.", "native_real_scalar_allowance": 1106984, "terminal_signed_swap_unit_ADD_per_port": 9}, "sha256": "1c86359ceea1b08366c6dbc9f98e9a95cc19fd1ff405cb9b27243b969f29b33b", "status": "PASS_COMPLETE_FINITE_NATIVE_CONTROLLER_OWNER_PHASE_INVOICE", "targets": {"6": 192, "7": 56, "8": 93, "9": 1229, "25": 448, "57": 224}} diff --git a/research/kappa-0023-productive-jobs/evidence/original-runner-stale-pin.log b/research/kappa-0023-productive-jobs/evidence/original-runner-stale-pin.log new file mode 100644 index 0000000000..b19d414616 --- /dev/null +++ b/research/kappa-0023-productive-jobs/evidence/original-runner-stale-pin.log @@ -0,0 +1,8 @@ +Traceback (most recent call last): + File "/private/tmp/kappa-0023-replay/research/kappa-rebuild/transfer-audit/p3_cover_audit.py", line 163, in + main() + File "/private/tmp/kappa-0023-replay/research/kappa-rebuild/transfer-audit/p3_cover_audit.py", line 38, in main + need(digest(ROOT/path) == expected, 'Pinned source '+path) + File "/private/tmp/kappa-0023-replay/research/kappa-rebuild/transfer-audit/p3_cover_audit.py", line 22, in need + raise AssertionError(message) +AssertionError: Pinned source research/kappa-rebuild/transfer-audit/P3-NATIVE-AUDIT.json.gz diff --git a/research/kappa-0023-productive-jobs/predecessor-notices/kappa-next-regime-README.md b/research/kappa-0023-productive-jobs/predecessor-notices/kappa-next-regime-README.md new file mode 100644 index 0000000000..88f47fbd03 --- /dev/null +++ b/research/kappa-0023-productive-jobs/predecessor-notices/kappa-next-regime-README.md @@ -0,0 +1,88 @@ +# Next κ regime: exact interfaces and architectural probes + +This continuation adds cheaper current-output query acquisition, complete +local native consumers, rational source codes, and a concrete future-code +routine. Its BIT follow-up proves large algebraic compression, implements +an early paired consumer and an isolated two-helper fragment, and gives +exact .004 acquisition budgets. **No new multiplication κ is admitted.** + +Start with [REPORT.md](REPORT.md) for a programmer-oriented explanation of +the remaining gap and [LITERATURE.md](LITERATURE.md) for the ranked primary +sources and their actual use. Each component has a local proof, generator +and reproducible receipt. New work includes substantial OpenAI assistance; +independent agent checks are not independent human review or formal proof +verification. + +## Components + +| Directory | What it contains | +|---|---| +| `weighted-sideinfo/` | 104-H1 current query acquisition, 17-field basis reuse, all-subset small-read exclusion, exact target budgets | +| `direct-decoder/` | Literal signed line observation, complete local pair consumer, and a whole-controller terminal decoder contract | +| `architecture-history/` | Pinned repository history and live PR review, rational lifting, local rational source code, 444-H1 common-frame future code, independent receiver/consumer audits | +| `bit-output/` | Source-bank causal echo, joint side/centre native DAG, exact primitive moments and helper-compression targets | +| `bit-literature/` | Exact finite restrictions of five published disjointness factorizations, scalar edge counts and native frame screens | +| `bit-coded-search/` | Möbius pooling, exact modular rank 3,876, exact own-X minrank 171, and scoped decoder/donor obstructions | +| `bit-compression/carrier-reuse/` | Early paired outputs, fully charged reference producer, isolated two-helper integer word, routing counterexample and exact .004 targets | + +The 104-port query/consumer branch and the full 112-port rational source +branch are alternatives. Their savings have not been composed into one +supplier and must not be added together. + +## Reproduce + +From the repository root, using Python 3.12, NumPy and Node.js: + +```sh +python3 research/kappa-next-regime/validate.py --output /absolute/path/to/kappa-next-validation.json +``` + +The default run authenticates this package, regenerates its declared +results in a fresh relocated copy, compares exact proof-relevant JSON +including decompressed native words, and runs the retained predecessor's +scoped 21-stage replay. It does not fetch network data, edit frozen source, +run the historical selected multiplication release engine, or update a +selected result. Do not pass `-O`: one retained lifting probe uses assertions. + +Use `--new-only` only when deliberately omitting the predecessor replay. +`REPLAY-PLAN.json` enumerates every regenerated result and its claim scope. +`INTEGRITY.json` covers source and reference files; `VALIDATION.json` records +the delivered fresh run. Gzip container hashes are not substituted for +comparisons of their complete decompressed mathematical payloads. +Every stage deletes its declared outputs before execution, compares the +fresh outputs, then restores their frozen bytes for downstream source +bindings. This avoids mistaking gzip container metadata or display-only +floating formatting for a change in a mathematical input. + +## Cumulative preservation + +The enclosing content-addressed overlay includes the supplied cumulative +archive, all 188 unchanged files from the preceding native continuation, +and this new package. The companion review patch includes both unmerged +continuation scopes. The original compact repository ZIP is the comparison +baseline. The enclosing manifest and safe apply tool specify exact paths, +preimages, hashes and modes. + +The known later literature ZIP was reported in recovered context but its +bytes were unavailable; neither continuation claims to reconstruct it. +The precise recovery limitation remains in [PREDECESSOR.json](PREDECESSOR.json) +and the previous package's `RECOVERY.md`. No work has been merged or posted. + +## Open next questions + +1. **Global BIT acquisition for early consumption.** Supply the paired + common/exclusive queries within the exact .004 budget. Synthesize + source encodings, observation order and native frames jointly; retain + the proved isolated reuse and the negative global routing witness. +2. **Direct complex final-function synthesis.** The explicit consumer costs + 104 extra H1 per pair. An eight-channel ordinary-output response would + fit an optimistic 544-H1 controller subtotal, but no such native word + is yet supplied. The actual owner-valued output operator is the target. +3. **Changing source codes across the controller.** Place or improve the + new 444-H1 future-code routine at real controller cuts. Price its + differently framed inputs, map tariffs, owned accumulator lifetime and + residual terminal code before composing the global word. +4. **Admit a complete supplier only after a competitive word exists.** Build + the changed geometry's actual owner fill, helper lifetimes, idle profile, + bit/precision guards and outer composition. A smaller port count does + not authorize scaling the old weight or reusing its idle schedule. diff --git a/research/kappa-0023-productive-jobs/verify.py b/research/kappa-0023-productive-jobs/verify.py new file mode 100644 index 0000000000..7dad0b79ab --- /dev/null +++ b/research/kappa-0023-productive-jobs/verify.py @@ -0,0 +1,234 @@ +#!/usr/bin/env python3 +"""Replay the frozen 0.0023 checkpoint sequentially in a new private directory. + +Only the dependency order changes: the independent phase audit must precede +the cover that records its hash. No scientific predicate or frozen source is +patched. Apache-2.0; substantial OpenAI Codex assistance. +""" +import argparse +import gzip +import hashlib +import importlib.util +import json +import os +from pathlib import Path, PurePosixPath +import platform +import shutil +import subprocess +import sys +import tempfile +import time +import zlib +from fractions import Fraction + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SCOPE = Path('research/kappa-rebuild') +TIMING = {'runtime', 'runtime_seconds', 'elapsed_seconds', 'seconds'} +ENVIRONMENT_RECEIPTS = {str(SCOPE / 'complex/fresh-certificates' / ('RESULT-' + tag + '.json')) + for tag in ('H', 'helpers', 'logger')} + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def digest(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def checked(root, name): + p = PurePosixPath(name) + need(not p.is_absolute() and '..' not in p.parts and str(p) == name, + 'Canonical relative input path: ' + name) + target = root + for part in p.parts: + target = target / part + need(not target.is_symlink(), 'No source symlink: ' + name) + need(target.is_file(), 'Source exists: ' + name) + return target + + +def read_payload(path): + data = path.read_bytes() + return json.loads(gzip.decompress(data) if path.suffix == '.gz' else data) + + +def normalize(value, hashes): + if isinstance(value, dict): + return {k: normalize(v, hashes) for k, v in value.items() if k not in TIMING} + if isinstance(value, list): + return [normalize(v, hashes) for v in value] + return hashes.get(value, value) if isinstance(value, str) else value + + +def compare_outputs(repo, outputs): + hashes = {} + for name in sorted(outputs): + old, new = digest(checked(ROOT, name)), digest(checked(repo, name)) + need(old not in hashes or hashes[old] == new, 'Equal source artifacts replay equally') + hashes[old] = new + comparisons = [] + for name in sorted(outputs): + old, new = checked(ROOT, name), checked(repo, name) + exact = old.read_bytes() == new.read_bytes() + left, right = normalize(read_payload(old), hashes), normalize(read_payload(new), {}) + environment_changes = {} + if name in ENVIRONMENT_RECEIPTS: + # These exact metadata leaves identify the executing host. Preserve + # both values in the receipt; no other environment field is waived. + for key, actual in (('python', sys.version), ('platform', platform.platform())): + need(right['environment'][key] == actual, 'Fresh environment metadata is accurate') + a, b = left['environment'].pop(key), right['environment'].pop(key) + if a != b: + environment_changes[key] = {'reference': a, 'fresh': b} + need(left == right, 'Scientific artifact differs beyond declared metadata/hash substitution: ' + name) + comparisons.append({'path': name, 'reference_sha256': digest(old), + 'fresh_sha256': digest(new), 'exact_bytes': exact, + 'environment_changes': environment_changes, + 'equal_after_declared_normalization': True}) + return comparisons + + +def authenticate(): + manifest = json.loads((HERE / 'IMPORT.json').read_text()) + names, destinations = set(), set() + for row in manifest['imported_files'] + manifest['canonical_main_inputs']: + name = row['path'] + need(name not in names, 'Unique imported input: ' + name) + names.add(name) + destination = row.get('replay_path', name) + relative = PurePosixPath(destination) + need(not relative.is_absolute() and '..' not in relative.parts and str(relative) == destination, + 'Canonical replay destination: ' + destination) + need(destination not in destinations, 'Unique replay destination: ' + destination) + destinations.add(destination) + path = checked(ROOT, name) + need(path.stat().st_size == row['bytes'] and digest(path) == row['sha256'], + 'Exact input bytes: ' + name) + if 'mode' in row: + need(path.stat().st_mode & 0o777 == row['mode'], 'Exact source mode: ' + name) + claim = json.loads((ROOT / SCOPE / 'CLAIM.json').read_text()) + need(claim['kappa'] == '23/10000', 'Exact claimed kappa') + need(claim['official_selected_frontier_modified'] is False, 'Conditional claim scope') + target = Fraction(claim['kappa']) + savings = {k: Fraction(v) for k, v in claim['nine_outer_savings'].items()} + need(len(savings) == 9 and min(savings.values()) - target == Fraction(98849, 50000000000), + 'All nine outer costs and exact limiting slack') + previous = Fraction('0.001730980255989036959962') - Fraction(1, 10**46) - Fraction(1, 10**70) + return manifest, { + 'kappa': str(target), 'previous_pr': 4181, 'previous_conditional_kappa': str(previous), + 'relative_exponent_saving_increase': str(target / previous - 1), + 'limiting_outer_slack': str(min(savings.values()) - target), + 'source_scope_files': len(manifest['imported_files']), + 'canonical_main_inputs': len(manifest['canonical_main_inputs']), + } + + +def load_stages(): + path = ROOT / SCOPE / 'replay_critical.py' + spec = importlib.util.spec_from_file_location('frozen_kappa_replay', path) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + stages = [*module.COMPLEX, *module.BIT, *module.ASSEMBLY] + phase = next(s for s in stages if s['name'] == 'p3_independent_phases') + stages.remove(phase) + index = next(i for i, s in enumerate(stages) if s['name'] == 'p3_native_cover') + stages.insert(index, phase) + need(len(stages) == 23 and len({s['name'] for s in stages}) == 23, + 'Exactly the original 23 distinct scientific stages') + return stages + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--check-only', action='store_true') + parser.add_argument('--out', type=Path, help='New, nonexistent directory for replay, logs and receipt') + args = parser.parse_args() + need(args.check_only != (args.out is not None), 'Choose --check-only or --out NEW_DIRECTORY') + manifest, arithmetic = authenticate() + if args.check_only: + subprocess.run([sys.executable, '-B', str(ROOT / SCOPE / 'verify_scope.py')], + cwd=ROOT, check=True) + print(json.dumps({'status': 'PASS_FROZEN_IMPORT_AND_CLAIM_ARITHMETIC', **arithmetic}, sort_keys=True)) + return + out = args.out.resolve() + need(not out.exists(), 'Replay output must not already exist') + need(not out.is_relative_to(ROOT), 'Keep generated replay outside the contribution checkout') + out.mkdir(parents=True) + repo = out / 'repo' + logs = out / 'logs' + logs.mkdir() + rows = manifest['imported_files'] + manifest['canonical_main_inputs'] + for row in rows: + source = checked(ROOT, row['path']) + destination = repo / row.get('replay_path', row['path']) + destination.parent.mkdir(parents=True, exist_ok=True) + shutil.copy2(source, destination) + stages = load_stages() + started = time.monotonic() + result = { + 'schema': 'conditional-kappa-0023-local-replay/1', 'status': 'RUNNING', + 'base_commit': manifest['base_commit'], 'import_manifest_sha256': digest(HERE / 'IMPORT.json'), + 'review_runner_sha256': digest(Path(__file__)), + 'environment': {'python': sys.version, 'executable': sys.executable, + 'platform': platform.platform(), 'zlib': zlib.ZLIB_RUNTIME_VERSION}, + 'execution': 'Sequential, in a fresh directory; original scientific scripts are unchanged.', + 'order_repair': 'p3_independent_phases precedes p3_native_cover, which pins its regenerated hash.', + 'arithmetic': arithmetic, 'stages': [], 'comparisons': [], + 'limits': 'Finite executable checks and quantitative interfaces; written universal proofs and inherited analytic/native assumptions require external review. Winning uniform templates are not materialized.', + } + receipt = out / 'VALIDATION.json' + environment = os.environ.copy() + environment.pop('PYTHONOPTIMIZE', None) + environment.pop('PYTHONPATH', None) + try: + with tempfile.TemporaryDirectory(prefix='kappa-0023-bytecode-') as cache: + environment['PYTHONPYCACHEPREFIX'] = cache + for stage in stages: + script = checked(repo, stage['script']) + command = [sys.executable, '-B', '-X', 'pycache_prefix=' + cache, + stage['script'], *stage['arguments']] + logfile = logs / (stage['name'] + '.log') + began = time.monotonic() + with logfile.open('wb') as output: + completed = subprocess.run(command, cwd=repo, env=environment, + stdout=output, stderr=subprocess.STDOUT, + timeout=600, check=False) + entry = {'stage': stage['name'], 'command': command, + 'source_sha256': digest(script), 'returncode': completed.returncode, + 'seconds': time.monotonic() - began, + 'log': str(logfile.relative_to(out)), 'log_sha256': digest(logfile)} + result['stages'].append(entry) + print(json.dumps({k: entry[k] for k in ('stage', 'returncode', 'seconds')}), flush=True) + need(completed.returncode == 0, 'Stage failed; inspect ' + str(logfile)) + outputs = set() + for stage in stages: + for pattern in stage['outputs']: + found = [p for p in (repo / SCOPE).glob(pattern) if p.is_file()] + need(found, 'Nonempty declared output: ' + pattern) + outputs.update(p.relative_to(repo).as_posix() for p in found) + result['comparisons'] = compare_outputs(repo, outputs) + need(len(outputs) == 59, 'Exactly the checkpoint\'s 59 declared artifacts') + # Recheck every frozen input after execution; regenerated files live elsewhere. + authenticate() + result.update(status='PASS_FRESH_23_STAGE_CONDITIONAL_REPLAY', + ignored_timing_keys=sorted(TIMING), + environment_metadata_exceptions={name: ['environment/python', 'environment/platform'] + for name in sorted(ENVIRONMENT_RECEIPTS)}, + source_hash_substitution='Exact old-to-fresh hashes of the 59 regenerated artifacts only; no other fields are waived.', + exact_artifacts=sum(r['exact_bytes'] for r in result['comparisons'])) + except Exception as error: + result.update(status='FAIL', error=repr(error)) + raise + finally: + result['runtime_seconds'] = time.monotonic() - started + receipt.write_text(json.dumps(result, indent=2, sort_keys=True) + '\n') + print(json.dumps({'status': result['status'], 'stages': len(result['stages']), + 'artifacts': len(result['comparisons']), 'exact_artifacts': result['exact_artifacts'], + 'receipt': str(receipt)}, sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-accounted-composition/alias/CONSTANT-CHART-AUDIT.json b/research/kappa-accounted-composition/alias/CONSTANT-CHART-AUDIT.json new file mode 100644 index 0000000000..3b1587017d --- /dev/null +++ b/research/kappa-accounted-composition/alias/CONSTANT-CHART-AUDIT.json @@ -0,0 +1,2563 @@ +{ + "cases": [ + { + "all_initial_columns_retained": true, + "dimension": 6, + "input_order": [ + "X0_U", + "X0_A", + "Y0_U", + "Y0_A", + "d0", + "d1" + ], + "max_abs_endpoint_coefficient": 1872, + "observation_rank": 5, + "observations": "All final selected A, all F caches, and all completed ordinary Y.", + "ordinary_error_gap_eight_channels": 0, + "ordinary_error_gap_per_channel": 0, + "pairs": 1, + "schedule": [ + [ + "F", + 0, + 0 + ], + [ + "B", + 0 + ], + [ + "F", + 0, + 1 + ] + ], + "whole_matrix": [ + [ + 157, + 13, + -13, + -1, + -12, + 0 + ], + [ + -1872, + -155, + 156, + 12, + 143, + 0 + ], + [ + -169, + -14, + 14, + 1, + 12, + -1 + ], + [ + 0, + 0, + 0, + 0, + 0, + -1 + ], + [ + 13, + 1, + 0, + 0, + 0, + 0 + ], + [ + -169, + -14, + 13, + 1, + 12, + 0 + ] + ], + "whole_rank": 6, + "zero_observation_nonzero_error_witnesses": [] + }, + { + "all_initial_columns_retained": true, + "dimension": 11, + "input_order": [ + "X0_U", + "X0_A", + "Y0_U", + "Y0_A", + "X1_U", + "X1_A", + "Y1_U", + "Y1_A", + "d0", + "d1", + "d2" + ], + "max_abs_endpoint_coefficient": 3900, + "observation_rank": 9, + "observations": "All final selected A, all F caches, and all completed ordinary Y.", + "ordinary_error_gap_eight_channels": 8, + "ordinary_error_gap_per_channel": 1, + "pairs": 2, + "schedule": [ + [ + "F", + 0, + 0 + ], + [ + "B", + 0 + ], + [ + "bridge", + 1, + 0, + 1 + ], + [ + "F", + 0, + 1 + ], + [ + "bridge", + 1, + 0, + -1 + ], + [ + "B", + 1 + ], + [ + "F", + 1, + 2 + ], + [ + "bridge", + 0, + 1, + 1 + ] + ], + "whole_matrix": [ + [ + 326, + 27, + -26, + -2, + -13, + -2, + 13, + 1, + -24, + 0, + 0 + ], + [ + -3900, + -323, + 312, + 24, + 156, + 24, + -156, + -12, + 287, + 0, + 0 + ], + [ + -169, + -14, + 14, + 1, + 13, + 1, + 0, + 0, + 12, + -1, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + -1, + 0 + ], + [ + -169, + -14, + 13, + 1, + 1, + 1, + -13, + -1, + 12, + 0, + 0 + ], + [ + 2028, + 168, + -156, + -12, + -13, + -12, + 156, + 12, + -144, + 0, + 0 + ], + [ + 169, + 14, + -13, + -1, + 0, + -1, + 14, + 1, + -12, + -1, + -1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + -1, + -1 + ], + [ + 13, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + -169, + -14, + 13, + 1, + 13, + 1, + 0, + 0, + 12, + 0, + 0 + ], + [ + 169, + 14, + -13, + -1, + 0, + -2, + 13, + 1, + -12, + 0, + 0 + ] + ], + "whole_rank": 11, + "zero_observation_nonzero_error_witnesses": [ + { + "initial_state": [ + -12, + 156, + 13, + -182, + 12, + 13, + 0, + -143, + 0, + 13, + 0 + ], + "ordinary_errors": [ + 13, + -13, + -13, + 0 + ] + } + ] + }, + { + "all_initial_columns_retained": true, + "dimension": 16, + "input_order": [ + "X0_U", + "X0_A", + "Y0_U", + "Y0_A", + "X1_U", + "X1_A", + "Y1_U", + "Y1_A", + "X2_U", + "X2_A", + "Y2_U", + "Y2_A", + "d0", + "d1", + "d2", + "d3" + ], + "max_abs_endpoint_coefficient": 5928, + "observation_rank": 13, + "observations": "All final selected A, all F caches, and all completed ordinary Y.", + "ordinary_error_gap_eight_channels": 16, + 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+ -13, + 0, + 0, + 0, + 0, + 0 + ] + }, + { + "initial_state": [ + -12, + 156, + 0, + 0, + 12, + 0, + 0, + 0, + -12, + 0, + 13, + -182, + 12, + 13, + 0, + -143, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 13, + 0, + 0 + ], + "ordinary_errors": [ + 13, + 0, + -13, + 0, + 13, + -13, + -13, + 0, + 0, + 0 + ] + }, + { + "initial_state": [ + 12, + -156, + 0, + 0, + -12, + 0, + 0, + 0, + 12, + 0, + 0, + 0, + -12, + 0, + 13, + -182, + 12, + 13, + 0, + -143, + 0, + 0, + 0, + 0, + 13, + 0 + ], + "ordinary_errors": [ + -13, + 0, + 13, + 0, + -13, + 0, + 13, + -13, + -13, + 0 + ] + } + ] + } + ], + "new_kappa_claimed": false, + "schema": "independent-alias-constant-chart/1", + "scope": "Exact constant-owner scalar active-channel model K=1,L=13. Native phase maps, changing owner charts, cache restoration and global output-credit assignment are not established by this receipt.", + "source_sha256": "c2f110fff2e4588c4e8558aa5efa39e23eedf70c8d746f6f4615f2022ac38d0b", + "status": "PASS_EXACT_INVERTIBILITY_AND_SCOPED_OBSERVER_GAP" +} diff --git a/research/kappa-accounted-composition/alias/FREEZE.json b/research/kappa-accounted-composition/alias/FREEZE.json new file mode 100644 index 0000000000..061d6130ce --- /dev/null +++ b/research/kappa-accounted-composition/alias/FREEZE.json @@ -0,0 +1,19 @@ +{ + "files": { + "CONSTANT-CHART-AUDIT.json": "18b78c0877364e75bd487bec9a2f9c31c08db3401deabc026fa5bc64e6be8425", + "PROOF.md": "868682c44a5de5e1e7de4aebb03bad0b95edd96142ba516fae1d161ae57061ce", + "README.md": "901ba6640d978fb0206b46c2829205563d9b7fa34bb0bb2a8d3755ec1f2d3d2c", + "alias_primitives.py": "e89b771460309b9b085e2a528fb310748d13e446d4e265ea41635627bd14fe69", + "constant_chart_audit.py": "c2f110fff2e4588c4e8558aa5efa39e23eedf70c8d746f6f4615f2022ac38d0b" + }, + "native_emitter_status": "WIP: frozen dependencies unavailable after executor reset; no native receipt produced", + "new_kappa_claimed": false, + "schema": "alias-recovery-snapshot/1", + "verified_outputs": [ + "CONSTANT-CHART-AUDIT.json" + ], + "verified_stage": [ + "python", + "checkpoint-tree/research/kappa-accounted-composition/alias/constant_chart_audit.py" + ] +} diff --git a/research/kappa-accounted-composition/alias/PROOF.md b/research/kappa-accounted-composition/alias/PROOF.md new file mode 100644 index 0000000000..d7b4ba2132 --- /dev/null +++ b/research/kappa-accounted-composition/alias/PROOF.md @@ -0,0 +1,112 @@ +# Exact scope of the alias probe + +The frozen geometry has K=L^T and LK=13I. The construction below uses only +dyadic scalar additions and signs. It assumes the previously verified H +kernel; the pending emitter is responsible for authenticating that input and +all physical boundary adapters. + +## Forward exchange + +At a completed source frame perform: + +1. A <- A+LU, forming v. +2. U <- U+KP-KA, giving U+K(d-v). +3. Exchange A and P using three shears and a sign. + +The result is `(U+K(d-v),d,v)`. From the output, recover + +\[ +d=A',\quad v=P',\quad U=U'-K(A'-P'),\quad A=P'-LU. +\] + +Thus arbitrary cache data are moved reversibly, not erased. With 416 signed +half coefficients in L, the wrapper has 1248 half additions, 24 unit +additions, and eight signs: `1248*10 + 24*8 + 8*2 = 12688` real scalar +allowance. Eight identity-to-full cache paths add eight rank-nine children. + +## Backward exchange + +Normalize selected original X_A to the Y entrance frame. Form `w=e+Lc` in +Y_A, then apply the three-shear signed exchange + +\[ +(X_A,Y_A)=(b,w)\longmapsto(w,-b). +\] + +Subtract Lc from Y_A and return X_A to its original line. The prefix therefore +maps `(a,b,c,e)` to `(a,e+Lc,c,-b-Lc)`. The backward H call subtracts + +\[ +H\begin{bmatrix}c\\-b-Lc\end{bmatrix} +=\begin{bmatrix}c+Kb\\-Lc-12b\end{bmatrix} +\] + +from X. This gives the stated whole-word formula. The prefix inverse is +`c=Y_U'`, `b=-(Y_A'+Lc)`, `e=X_A'-Lc`; its integrality and invertibility do not +depend on any data value. + +The scalar invoice is `832*10+24*8=8512`. The selected fields each make a paid +rank-one descent and ascent, for16 rank-one children. In the inherited +backward convention their actual input transform is conjugate(C_q); the +physical descent must be its adjoint, and the return its original operator. +The emitter checks both lowering monomials, the recursive Fourier, and +absolute phase. These checks remain pending while the frozen inputs are +unavailable. + +## A local cancellation is not a complete caller + +Immediately after a forward H call, ordinary target data are +`Y_U = y_U + x_U - K_s v`. The forward exchange gives +`X_U = x_U + K_s(d-v)` and `X_A=d`. If these enter the backward prefix without +an intervening ordinary mutation, its ordinary result is + +\[ +X_U-Y_U-K_t d=-y_U+(K_s-K_t)d. +\] + +Only equal charts make the last term vanish. The full caller also contains +bridges that act on both ordinary and selected fields. + +The independent audit restricts to one active constant-chart channel with +K=1,L=13. It constructs H shears literally, including all original selected +inputs and fresh forward-cache inputs. It uses the complete alternating +schedule and both bridge directions, then the signed terminal swap. Exact +rational elimination proves invertibility for p=1,...,5 and the scoped +observer gaps p-1. Other charts or additional observations are outside this +negative result. + +For p=2 an explicit witness, in initial order +`[X0_U,X0_A,Y0_U,Y0_A,X1_U,X1_A,Y1_U,Y1_A,d0,d1,d2]`, is + +\[ +(-12,156,13,-182,12,13,0,-143,0,13,0). +\] + +Every granted final selected A value, forward cache, and completed ordinary +Y value is zero. Ordinary endpoint errors in order +`[X0_U,Y0_U,X1_U,Y1_U]` equal `(13,-13,-13,0)`. Thus these observations alone +cannot determine the required correction, even though the full augmented +operator is invertible. + +## Physical helper normalization: a distinct useful lemma + +If a fully verified native word restores every helper logical input and has +zero helper-to-data and data-to-helper blocks, its whole physical operator is + +\[ +\operatorname{diag}(D,T_1,\ldots,T_h),\quad +T_i=F_{i,\mathrm{end}}F_{i,\mathrm{start}}^{-1}. +\] + +Boundary monomials on helper i can normalize T_i to its ordinary recursive +Fourier normal form without changing D or any internal ADD. This does not +retag each intermediate helper frame independently. It is a whole-operator +fact and can remove a spurious requirement to prepare every canonical helper +entrance in a new productive host. + +To use it, each T_i still needs an exact relative-frame lowering certificate: +the claimed rank, both adapters must be free monomials, and the absolute +phase must match. The available frozen `clifford_exact.lower_child` routine +can supply this certificate. Outer host packing and disjoint productive axes +must then be proved for the normalized complete blocks. This lemma does not +by itself supply a caller correction or a new kappa. diff --git a/research/kappa-accounted-composition/alias/README.md b/research/kappa-accounted-composition/alias/README.md new file mode 100644 index 0000000000..7848bb0cb5 --- /dev/null +++ b/research/kappa-accounted-composition/alias/README.md @@ -0,0 +1,141 @@ +# Selected-register aliases + +This directory contains an exact, independently replayed constant-chart audit +and a native-word emitter prepared for the frozen H112 sources. + +## Status + +`constant_chart_audit.py` and `CONSTANT-CHART-AUDIT.json` pass. They retain every +original data and cache input column, construct the full controller from +literal shears and bridges, and use exact rational elimination. For one to +five pairs the augmented matrices are invertible. Granting all final selected +data, all forward caches, and all completed ordinary Y data leaves respectively +0, 1, 2, 3, and 4 missing ordinary-error functions per active channel. Integer +null witnesses are included. This is a constant-owner calculation. + +`alias_primitives.py` is **not yet a verified native result**. A shared executor +disconnect interrupted its first emission. When the executor reconnected, the +workspace contained only this newly delivered patch; the frozen dependencies +were absent. Its run stopped at `ModuleNotFoundError: affine_core`. The emitter +must be replayed against the recovered, hash-pinned predecessor files before +its phase checks or full-column native checks can be claimed. No native receipt +has been produced in this directory. + +The queued emitter's known pre-H checkpoint issue has been fixed: owned C0 +must retain its arbitrary input symbol at the backward-prefix cut, because +the prefix never touches it. The inherited H body clears C0 later. This is a +checker correction, not evidence that the native replay has run. + +No new kappa is claimed by either artifact. + +## The proposed local contracts + +Let A be the eight selected original data ports, U the other 104, and +`v = A + L U`, with `K=L^T` and `L K=13 I`. The frozen kernel is + +\[ +H=\begin{bmatrix}I-KL&-K\\11L&12I\end{bmatrix},\qquad H^2=I. +\] + +The forward wrapper follows a complete H forward call, when its source X is +at the common full frame. Eight arbitrary cache fields P advance from identity +to full frame through eight rank-nine children. Three same-frame linear +steps and a swap give + +\[ +U'=U+K(d-v),\qquad A'=d,\qquad P'=v. +\] + +Here d is the arbitrary original P input. The original H target output is +unchanged. The wrapper does not clear or assume zero in any persistent field. + +The backward wrapper precedes an H backward call. It moves each of the eight +selected original X fields from its actual input line to the identity frame, +exchanges `(C Y, X_A)` with `(-X_A, C Y)`, and returns the selected field to its +input line. Here `C=[L,I]`. Each round trip costs two rank-one children. With +original X=(a,b), Y=(c,e), the complete prefix plus H backward word has output + +\[ +X_U'=a-c-Kb,\quad X_A'=e+2Lc+12b,\quad +Y_U'=c,\quad Y_A'=-b-Lc. +\] + +This is a pre-call operation. It does not assume that a backward source is +coframed after the call. + +## Proposed incremental invoice + +These counts follow from the literal wrappers and the frozen L geometry of +416 nonzero signed-half coefficients. They are predictions checked by the +emitter when its dependencies are available; they are not a replacement for +that replay. + +| Wrapper | Extra child calls over H | Extra real scalar allowance | New arbitrary persistent fields | +|---|---:|---:|---:| +| Forward post-call exchange | 8 rank-nine | 12,688 | 8 | +| Backward pre-call exchange | 16 rank-one | 8,512 | 0 | + +The forward cache output is a previous source function. It is not the Fourier +transform of that cache's own original input. Productive cache credit, outer +hosting, idle paths, and final return remain separate obligations. + +## Reproduction + +The dependency-free completed check is: + +```sh +python checkpoint-tree/research/kappa-accounted-composition/alias/constant_chart_audit.py +``` + +It writes `CONSTANT-CHART-AUDIT.json` deterministically. + +After the frozen complex subtree and all predecessor dependencies have been +recovered, the pending native stage is: + +```sh +python checkpoint-tree/research/kappa-accounted-composition/alias/alias_primitives.py +``` + +Its intended outputs are `native/RESULT.json`, `native/PHASES.json.gz`, and the +four providers' canonical/native word pairs. A successful native run must +verify all original 632 input columns, eight additional arbitrary inputs in +the forward direction, all 408 original helper returns, complete paid +histograms, all new child phases, meaningful omitted-gate controls, and exact +prefix coefficient bounds. + +The inherited H receipt is pinned to +`d14edeacd37e526f8723ebdfb22f6b8d470ebf1daa7ef7c87aacfe7bae00484c`. + +### Required recovered dependencies + +Restore these files from the preserved predecessor package, with their +transitive inputs. Do not recreate approximations under the frozen names. + +- `../complex/affine_core.py` +- `../complex/port_state_lift.py` +- `../complex/involution112.py` +- `../complex/FREEZE.json` +- `../complex/involution112-native/RESULT.json` and every canonical/native + artifact authenticated by that receipt +- `research/kappa-function-composition/complex/prune_native.py` +- `research/kappa-causal-continuation/buffer-borrowing/verify_buffers.py` +- `research/t74-native-fast/vendor/native/new/code/borrow/vendor/clifford/clifford_exact.py` + +The exact arbitrary-input checker SHA256 recorded before the reset is +`106b4d5e052427230df452f0c901d5390a0583782960b80a8af94d9a984022c7`. +The Clifford checker SHA256 is +`2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962`. +The frozen `involution112.py` SHA256 is +`e6dce64c7f284ad0aef5975f738b31d96ec0138d7690a269ddee61869ec2c33e`. +The remaining shared source hashes are read from the recovered +`complex/FREEZE.json`; they are not invented or weakened by this emitter. + +## The next boundary + +Same-chart adjacent F/B cancellation is only a local fact. Different stage +owners leave `(K_s-K_t)d`; bridges also change the selected state. Even in a +constant chart, the full multi-pair controller has the observer deficits +above. A completion needs an additional early observation, a different relay, +or a terminal action whose physical frames and dirty-state return are priced. + +See `PROOF.md` for the exact wrapper algebra and the independent witness. diff --git a/research/kappa-accounted-composition/alias/alias_primitives.py b/research/kappa-accounted-composition/alias/alias_primitives.py new file mode 100644 index 0000000000..8283e908c3 --- /dev/null +++ b/research/kappa-accounted-composition/alias/alias_primitives.py @@ -0,0 +1,323 @@ +#!/usr/bin/env python3 +"""Native selected-data alias exchanges around the frozen H112 kernel. + +Forward adds eight arbitrary productive-cache candidates with paid I->F9 +paths, then exchanges source function and cache at the completed X frame. +Backward moves each original selected X field to the Y entrance frame and +back, paying two rank-one children. No borrowed state is cleared. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction as Q +from pathlib import Path +import argparse +import json +import sys +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +COMPLEX = HERE.parent/'complex' +sys.path.insert(0, str(COMPLEX)) +import affine_core as A +import port_state_lift as PS +import involution112 as HN +P = A.P +HROOT = COMPLEX/'involution112-native' +H_RECEIPT_SHA = 'd14edeacd37e526f8723ebdfb22f6b8d470ebf1daa7ef7c87aacfe7bae00484c' + + +def add(target, source, q, frame): + q = Q(q) + return ['scalar', ['add', list(target), list(source), q.numerator, q.denominator], frame] + + +def forward_suffix(selected, L): + out = [['phase', 'alias_forward_H_complete']] + out += [['cache_child', j, 0, 1, 9] for j in range(8)] + out.append(['phase', 'alias_forward_cache_full']) + # Form v in the old A fields before changing U. + for a, row in zip(selected, L): + out += [add(('L', a), ('L', p), q, 1) for p, q in sorted(row.items())] + for j, (a, row) in enumerate(zip(selected, L)): + for p, q in sorted(row.items()): + out += [add(('L', p), ('P', j), q, 1), + add(('L', p), ('L', a), -q, 1)] + # Ordinary swap (v,d)->(d,v), using three shears and a sign. + for j, a in enumerate(selected): + out += [add(('L', a), ('P', j), 1, 1), + add(('P', j), ('L', a), -1, 1), + add(('L', a), ('P', j), 1, 1), + ['scalar', ['scale', ['P', j], -1, 1], 1]] + return out+[['phase', 'alias_forward_final']] + + +def backward_prefix(c, selected, L): + out = [['phase', 'alias_backward_entry']] + for j, (a, row) in enumerate(zip(selected, L)): + line = c['start'][a] + out.append(['state_child', a, line, 0, 1, f'{a}:down']) + # Current Y remains untouched in U while its selected function is used. + out += [add(('L', 112+a), ('L', 112+p), q, 0) for p, q in sorted(row.items())] + out += [add(('L', a), ('L', 112+a), 1, 0), + add(('L', 112+a), ('L', a), -1, 0), + add(('L', a), ('L', 112+a), 1, 0)] + out += [add(('L', 112+a), ('L', 112+p), -q, 0) for p, q in sorted(row.items())] + out.append(['state_child', a, 0, line, 1, f'{a}:up']) + return out+[['phase', 'alias_backward_prefix_complete']] + + +def paid(word): + out = Counter() + for e in word: + if e[0] == 'climb': out[e[5]] += 1 + elif e[0] in ('center', 'cache_child', 'state_child'): out[e[4]] += 1 + return out + + +def initial(v, forward): + rows = {('L', i): v.identity(('L', i)) for i in range(632)} + if forward: rows.update({('P', j): v.identity(('P', j)) for j in range(8)}) + rows['C', 0] = {} + return rows + + +def expected(v, selected, ordinary, L, H, direction, cut='final'): + forward = direction == 'forward' + rows = initial(v, forward) + ident = v.identity + if forward: + for p in range(112): + for q, c in H[p].items(): v.add(rows['L', 112+p], ident(('L', q)), c) + if cut in ('alias_forward_H_complete', 'alias_forward_cache_full'): return rows + for j, a in enumerate(selected): + rows['P', j] = ident(('L', a)) + for p, q in L[j].items(): v.add(rows['P', j], ident(('L', p)), q) + for p in ordinary: + for j, a in enumerate(selected): + q = L[j].get(p, Q(0)) + if q: + v.add(rows['L', p], ident(('P', j)), q) + v.add(rows['L', p], rows['P', j], -q) + for j, a in enumerate(selected): rows['L', a] = ident(('P', j)) + elif cut == 'alias_backward_prefix_complete': + # The prefix does not use owned C0. The inherited H body clears it + # later, so its arbitrary starting symbol still exists at this cut. + rows['C', 0] = ident(('C', 0)) + for j, a in enumerate(selected): + rows['L', a] = ident(('L', 112+a)) + rows['L', 112+a] = {} + v.add(rows['L', 112+a], ident(('L', a)), Q(-1)) + for p, q in L[j].items(): + v.add(rows['L', a], ident(('L', 112+p)), q) + v.add(rows['L', 112+a], ident(('L', 112+p)), -q) + else: + # Direct closed formula, independent of applying H to a simulated prefix. + for p in ordinary: + v.add(rows['L', p], ident(('L', 112+p)), Q(-1)) + for j, a in enumerate(selected): + if p in L[j]: v.add(rows['L', p], ident(('L', a)), -L[j][p]) + for j, a in enumerate(selected): + rows['L', a] = ident(('L', 112+a)) + v.add(rows['L', a], ident(('L', a)), Q(12)) + rows['L', 112+a] = {} + v.add(rows['L', 112+a], ident(('L', a)), Q(-1)) + for p, q in L[j].items(): + v.add(rows['L', a], ident(('L', 112+p)), 2*q) + v.add(rows['L', 112+a], ident(('L', 112+p)), -q) + return rows + + +def phase_receipt(c, selected): + ce = P.load(P.CLIFFORD, 'accounted_alias_clifford') + identity = ce.Op.identity(9) + def lower(op, rank): + child, _ = ce.lower_child(op) + P.need(child['rank'] == rank, 'Exact physical recursive rank') + result = P.monomial(ce, child['right'], 9).then(ce.fourier(9, range(rank))).then(P.monomial(ce, child['left'], 9)) + P.need(result.tab == op.tab and result.zero_state() == op.zero_state(), + 'Child and both monomial wrappers match the complete absolute phase') + return child + routes = {} + for a in selected: + P.need(c['frames'][c['start'][a]] == [c['ports'][a]], 'Selected physical entrance is its actual normal line') + actual = P.conjugate(ce, ce.cgate(9, c['ports'][a])) + down, up = actual.adjoint(), actual + for name, op, old, new in [('down', down, c['start'][a], 0), ('up', up, 0, c['start'][a])]: + routes[f'{a}:{name}'] = {'port': a, 'old_frame': old, 'new_frame': new, + 'child': lower(op, 1), 'actual_operator': 'conj(Cq)^-1' if name == 'down' else 'conj(Cq)'} + for product in (actual.then(down), down.then(up), up.then(down)): + P.need(product.tab == identity.tab and product.zero_state() == identity.zero_state(), + 'Selected state normalization and return are exact inverse physical operators') + wrong = actual.then(up) + P.need(wrong.tab != identity.tab or wrong.zero_state() != identity.zero_state(), + 'Wrong normalization orientation is rejected') + return {'backward_selected_routes': routes, 'forward_cache_I_to_F9': lower(ce.fourier(9), 9), + 'all_absolute_phases_checked': True, 'wrong_normalization_orientation_rejected': True, + 'backward_convention': 'The selected X entrance is the complex conjugate of actual C_q; the down child is its adjoint, and the return child is the original operator.'} + + +def replay(v, c, word, selected, ordinary, L, H, direction, phase): + forward = direction == 'forward' + rows = initial(v, forward) + # The owned input may initially contain arbitrary data, and must be cleared. + rows['C', 0] = v.identity(('C', 0)) + current = {('L', i): f for i, f in enumerate(c['start'])} + if forward: current.update({('P', j): 0 for j in range(8)}) + current['C', 0] = None + spaces = [v.span(f) for f in c['frames']] + dims = [len(s).bit_length()-1 for s in spaces] + scalar, profile, counts, cuts = 0, Counter(), Counter(), [] + max_l1, max_abs, max_den = Q(1), Q(1), 1 + witness = None + for position, e in enumerate(word): + kind = e[0]; counts[kind] += 1 + changed = None + if kind in ('climb', 'center', 'cache_child', 'state_child'): + if kind == 'climb': + _, _, index, old, new, rank = e; role = 'L', index + P.need(spaces[old] < spaces[new], 'Unchanged native climb remains nested') + else: + _, index, old, new, rank, *extra = e + role = ('C' if kind == 'center' else 'P' if kind == 'cache_child' else 'L'), index + P.need(spaces[old] < spaces[new] or spaces[new] < spaces[old], 'Paid normalization is a nested move') + if kind == 'cache_child': + P.need(forward and (old, new, rank) == (0, 1, 9), 'Paid arbitrary-cache whole local path') + if kind == 'state_child': + item = phase['backward_selected_routes'][extra[0]] + P.need(not forward and item['port'] == index and (old, new) == (item['old_frame'], item['new_frame']), + 'Exact selected original-state physical route') + P.need(current[role] == old and rank == abs(dims[new]-dims[old]), 'Paid edge actual source and rank') + current[role] = new; profile[rank] += 1 + elif kind == 'warp': + _, role, operator, old, new = e; role = tuple(role) + P.need(current[role] == old, 'Warp source frame') + rows[role] = v.warp(rows[role], operator, not forward); current[role] = new + elif kind == 'scalar': + g, f = e[1:]; target = tuple(g[1]); changed = target + if g[0] == 'clear': + P.need(target == ('C', 0), 'Only owned C0 may be cleared') + rows[target] = {}; current[target] = f; scalar += 1 + else: + P.need(current[target] == f, 'Scalar target frame') + if g[0] in ('add', 'copy'): + source = tuple(g[2]); P.need(current[source] == f, 'Scalar source frame') + if g[0] == 'copy': + P.need(target == ('C', 0) and not rows[target], 'Copy only into cleared owned field') + rows[target] = dict(rows[source]); scalar += 1 + else: + q = Q(g[3], g[4]); num, den = abs(q.numerator), q.denominator + P.need(target != source and num and not den & (den-1), 'Legal reversible dyadic ADD') + v.add(rows[target], rows[source], q) + scalar += 3+num.bit_length()-1+num.bit_count()+den.bit_length()-1 + else: + P.need(g[0] == 'scale', 'Known scalar operation') + q = Q(g[2], g[3]); num, den = abs(q.numerator), q.denominator + P.need(num and not num & (num-1) and not den & (den-1), 'Unit dyadic scale') + rows[target] = {k: q*x for k, x in rows[target].items()} + scalar += abs(num.bit_length()-den.bit_length())+int(q < 0) + elif kind == 'phase': + if e[1] in ('alias_forward_H_complete', 'alias_forward_cache_full', 'alias_forward_final', + 'alias_backward_prefix_complete'): + P.need(rows == expected(v, selected, ordinary, L, H, direction, e[1]), 'Alias cut operator: '+e[1]) + want_l = c['start'] if e[1] == 'alias_backward_prefix_complete' else c['final'] + P.need([current['L', i] for i in range(632)] == want_l, 'Alias cut L frames') + if forward: + pframe = 0 if e[1] == 'alias_forward_H_complete' else 1 + P.need(all(current['P', j] == pframe for j in range(8)), 'Alias cut cache frames') + cuts.append({'name': e[1], 'event': position, 'every_arbitrary_input_checked': True}) + else: + P.need(kind in ('adapter', 'residual_adapter'), 'Known inherited native phase event') + if changed is not None and rows[changed]: + l1 = sum(map(abs, rows[changed].values()), Q(0)) + if l1 > max_l1: max_l1, witness = l1, {'event': position, 'role': list(changed)} + max_abs = max(max_abs, *(abs(q) for q in rows[changed].values())) + max_den = max(max_den, *(q.denominator for q in rows[changed].values())) + P.need(rows == expected(v, selected, ordinary, L, H, direction), 'Complete alias endpoint operator mismatch') + P.need([current['L', i] for i in range(632)] == c['final'], 'All original endpoint frames retained') + P.need(not rows['C', 0], 'Owned C0 cleared at return') + if forward: P.need(all(current['P', j] == 1 for j in range(8)), 'All new cache outputs have full frame') + P.need(all(rows['L', i] == v.identity(('L', i)) for i in range(224,632)), 'All408 arbitrary original helpers restored') + P.need(profile == paid(word), 'Literal complete paid profile') + return {'direction': direction, 'all_original_632_inputs_arbitrary': True, + 'arbitrary_new_cache_inputs': 8 if forward else 0, 'total_arbitrary_persistent_inputs': 640 if forward else 632, + 'restored_original_helpers': 408, 'owned_temporary_peak_inherited_from_H': 1, + 'no_borrowed_clear_or_copy': True, 'all_owned_temporaries_cleared': True, + 'paid_profile': dict(sorted(profile.items())), 'paid_rank_mass': sum(r*n for r,n in profile.items()), + 'conservative_real_scalar_primitive_allowance': 2*scalar, + 'event_counts': dict(counts), 'cuts': cuts, + 'prefix_growth': {'max_exact_formal_row_l1': str(max_l1), 'l1_witness': witness, + 'max_abs_coefficient': str(max_abs), 'max_denominator': max_den, + 'scope': 'Exact logical prefix bound; not a rounded precision induction.'}} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path, default=HERE/'native') + args = parser.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + P.need(P.digest(HROOT/'RESULT.json') == H_RECEIPT_SHA, 'Frozen H112 receipt') + freeze = P.read(COMPLEX/'FREEZE.json') + for name in ('affine_core.py', 'port_state_lift.py', 'involution112.py'): + P.need(P.digest(COMPLEX/name) == freeze['files'][name], 'Frozen shared native source '+name) + P.need(P.digest(P.CHECKER) == P.CHECKER_SHA and P.digest(P.CLIFFORD) == P.CLIFFORD_SHA, + 'Pinned exact native and physical checkers') + v = P.load(P.CHECKER, 'accounted_alias_arbitrary_checker'); v.authenticate() + R, selected, ordinary, L = PS.geometry(v.read(v.PACKET)); H = HN.kernel(R, selected, ordinary, L) + source = P.read(HROOT/'RESULT.json'); providers, artifacts, phases = {}, {}, {} + for tag in P.TAGS: + cp, np = (HROOT/(tag+'-'+s+'.json.gz') for s in ('CANONICAL','NATIVE-WORD')) + for path in (cp,np): P.need(P.digest(path) == source['artifacts'][path.name], 'Frozen H artifact') + c, base = P.read(cp), P.read(np); phase = phase_receipt(c, selected) + phases[tag] = phase + word = {'forward': deepcopy(base['forward'])+forward_suffix(selected,L), + 'backward': backward_prefix(c,selected,L)+deepcopy(base['backward'])} + checks = {name: replay(v,c,w,selected,ordinary,L,H,name,phase) for name,w in word.items()} + for name, delta in [('forward',Counter({9:8})),('backward',Counter({1:16}))]: + P.need(paid(word[name]) == paid(base[name])+delta, 'Complete added recursive profile') + scalar = checks[name]['conservative_real_scalar_primitive_allowance']-source['providers'][tag]['checks'][name]['conservative_real_scalar_primitive_allowance'] + P.need(scalar == (12688 if name == 'forward' else 8512), 'Complete alias scalar bill') + for direction in ('forward','backward'): + bad = deepcopy(word[direction]) + if direction == 'forward': + index = max(i for i,e in enumerate(bad) if e[0]=='scalar' and e[1][0]=='scale') + else: + index = next(i for i,e in enumerate(bad) if e[0]=='state_child' and e[3]!=0) + del bad[index] + rejected = False + try: replay(v,c,bad,selected,ordinary,L,H,direction,phase) + except (ValueError,KeyError): rejected = True + P.need(rejected, 'Missing alias sign/selected return child negative control') + canonical = deepcopy(c) + canonical['local_profile_is_H_base_only'] = True + canonical['local_profiles_by_direction'] = {k: checks[k]['paid_profile'] for k in checks} + canonical['alias_extension'] = {'selected_ports': selected, 'extra_forward_P_fields':8, + 'P_start':0,'P_final':1,'complete_paid_profiles':{k:checks[k]['paid_profile'] for k in checks}, + 'state_child_grammar':'state_child,index,old,new,rank,exact_phase_route_id; applies the emitted actual recursive child to original L[index], retaining its logical value between declared frames.'} + paths=[] + for suffix,value in [('CANONICAL',canonical),('NATIVE-WORD',word)]: + path=args.out/(tag+'-'+suffix+'.json.gz');P.save(path,value) + artifacts[path.name]=P.digest(path);paths.append(path.name) + providers[tag]={'checks':checks,'baseline_H_checks':source['providers'][tag]['checks'], + 'extra_children_over_H':{'forward':{'9':8},'backward':{'1':16}}, + 'extra_scalar_over_H':{'forward':12688,'backward':8512}, + 'meaningful_omitted_gate_controls_rejected':True,'artifacts':paths} + print('PASS native selected alias',tag,'both orientations',flush=True) + path=args.out/'PHASES.json.gz';P.save(path,phases);artifacts[path.name]=P.digest(path) + result={'schema':'native-selected-state-alias/1','status':'PASS_LOCAL_F_POST_AND_B_PRE_ALIAS_WITH_COMPLETE_COST', + 'generator_sha256':P.digest(Path(__file__)),'frozen_H_receipt_sha256':H_RECEIPT_SHA, + 'shared_sources_sha256':{n:P.digest(COMPLEX/n) for n in ('affine_core.py','port_state_lift.py','involution112.py')}, + 'native_checker_sha256':P.CHECKER_SHA,'clifford_checker_sha256':P.CLIFFORD_SHA, + 'selected_ports':selected,'providers':providers,'artifacts':artifacts, + 'forward_contract':{'X_U':'a+K(d-v), v=b+L a','X_A':'d','P':'v','Y':'y+H x'}, + 'backward_prefix_contract':{'X_U':'a','X_A':'Y_A+L Y_U','Y_U':'c','Y_A':'-b-L c'}, + 'backward_whole_contract':{'X_U':'a-c-K b','X_A':'Y_A_in+2 L c+12 b','Y_U':'c','Y_A':'-b-L c'}, + 'local_same_pair_ordinary_composition':'If a forward source feeds the next backward target without another ordinary mutation, X_U_after_B=-Y_U_before_F+(K_s-K_t)d in the corresponding owner-aware notation. Only a matched scalar chart cancels the difference automatically.', + 'forward_wrapper_inverse':'a=a_prime-K(b_prime-P_prime); b=P_prime-L a; d=b_prime', + 'backward_wrapper_inverse':'c=Y_U_prime; b=-(Y_A_prime+L c); Y_A_in=X_A_prime-L c', + 'width_and_productive_scope':'Forward introduces eight independent arbitrary cache fields, paid I->F9. Their outputs are old source functions, not transforms of their own original data. Original cache inputs move into selected X data. No productive cache credit, global idle/hosting plan or final return is claimed. Backward introduces no persistent field.', + 'physical_scope':'Forward wrapper follows the full H word when X is coframed. Backward wrapper precedes H; every selected X datum makes a paid exact dual-normalization round trip to the Y identity frame. There is no free backward post-capture.', + 'new_kappa_claimed':False} + (args.out/'RESULT.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(result['status'],flush=True) + + +if __name__=='__main__': main() diff --git a/research/kappa-accounted-composition/alias/constant_chart_audit.py b/research/kappa-accounted-composition/alias/constant_chart_audit.py new file mode 100644 index 0000000000..725e62de75 --- /dev/null +++ b/research/kappa-accounted-composition/alias/constant_chart_audit.py @@ -0,0 +1,185 @@ +#!/usr/bin/env python3 +"""Independent exact audit of the selected-register alias controller. + +This file has no native-word dependencies. It checks a constant-owner active +channel with K=1,L=13, retaining all original data and cache input columns. +It is not a physical construction or a productive-cache credit certificate. +Apache-2.0; substantial OpenAI assistance. +""" +from fractions import Fraction as Q +from math import gcd, lcm +from pathlib import Path +import argparse +import hashlib +import json + + +def rref(rows): + a = [[Q(x) for x in row] for row in rows] + if not a: + return a, [] + pivots = [] + for col in range(len(a[0])): + k = len(pivots) + j = next((j for j in range(k, len(a)) if a[j][col]), None) + if j is None: + continue + a[k], a[j] = a[j], a[k] + q = a[k][col] + a[k] = [x/q for x in a[k]] + for j in range(len(a)): + if j != k and a[j][col]: + q = a[j][col] + a[j] = [x-q*y for x, y in zip(a[j], a[k])] + pivots.append(col) + if len(pivots) == len(a): + break + return a, pivots + + +def rank(rows): + return len(rref(rows)[1]) + + +def nullspace(rows): + a, pivots = rref(rows) + n = len(rows[0]) + out = [] + for free in range(n): + if free in pivots: + continue + v = [Q(0)]*n + v[free] = Q(1) + for j, p in enumerate(pivots): + v[p] = -a[j][free] + denominator = lcm(*(x.denominator for x in v)) + v = [int(x*denominator) for x in v] + divisor = gcd(*v) + out.append([x//divisor for x in v]) + return out + + +def dot(a, b): + return sum(x*y for x, y in zip(a, b)) + + +def audit(p): + n = 5*p+1 + identity = [[int(i == j) for j in range(n)] for i in range(n)] + s = [row[:] for row in identity] + cache = 4*p + schedule = [] + + def add(target, source, q): + s[target] = [x+q*y for x, y in zip(s[target], s[source])] + + def signed_swap(a, b): + add(a, b, 1) + add(b, a, -1) + add(a, b, 1) + + def forward(pair): + nonlocal cache + a, b, c, e = range(4*pair, 4*pair+4) + # Literal H shear: H=[[-12,-1],[143,12]]. + add(c, a, -12) + add(c, b, -1) + add(e, a, 143) + add(e, b, 12) + # Form v, translate U by d-v, then exchange raw A and P. + add(b, a, 13) + add(a, cache, 1) + add(a, b, -1) + signed_swap(b, cache) + s[cache] = [-x for x in s[cache]] + schedule.append(['F', pair, cache-4*p]) + cache += 1 + + def backward(pair): + a, b, c, e = range(4*pair, 4*pair+4) + # The native prefix normalizes b to the Y frame and returns it. + add(e, c, 13) + signed_swap(b, e) + add(e, c, -13) + # Literal negative H shear after the prefix. + add(a, c, 12) + add(a, e, 1) + add(b, c, -143) + add(b, e, -12) + schedule.append(['B', pair]) + + def bridge(source, target, q): + for k in (0, 1): + add(4*target+k, 4*source+k, q) + for k in (2, 3): + add(4*source+k, 4*target+k, -q) + schedule.append(['bridge', source, target, q]) + + forward(0) + backward(0) + for pair in range(1, p): + bridge(pair, pair-1, 1) + forward(pair-1) + bridge(pair, pair-1, -1) + backward(pair) + forward(p-1) + for pair in reversed(range(p-1)): + bridge(pair, pair+1, 1) + for pair in range(p): + for k in (0, 1): + a, b = 4*pair+k, 4*pair+k+2 + s[a], s[b] = s[b], [-x for x in s[a]] + assert cache == n + observations, errors = [], [] + for pair in range(p): + observations += [s[4*pair+1], s[4*pair+3], s[4*pair+2]] + for k in (0, 2): + errors.append([x-y for x, y in zip(s[4*pair+k], identity[4*pair+k])]) + observations += s[4*p:] + witnesses = [] + for v in nullspace(observations): + e = [dot(row, v) for row in errors] + if any(e): + assert not any(dot(row, v) for row in observations) + witnesses.append({'initial_state': v, 'ordinary_errors': e}) + full_rank = rank(s) + observed = rank(observations) + gap = rank(observations+errors)-observed + assert full_rank == n and gap == p-1 + assert bool(witnesses) == bool(gap) + return { + 'pairs': p, 'dimension': n, 'whole_matrix': s, + 'whole_rank': full_rank, 'all_initial_columns_retained': True, + 'input_order': [f'{bank}{pair}_{kind}' for pair in range(p) + for bank, kind in [('X','U'),('X','A'),('Y','U'),('Y','A')]] + +[f'd{j}' for j in range(p+1)], + 'schedule': schedule, + 'observations': 'All final selected A, all F caches, and all completed ordinary Y.', + 'observation_rank': observed, 'ordinary_error_gap_per_channel': gap, + 'ordinary_error_gap_eight_channels': 8*gap, + 'zero_observation_nonzero_error_witnesses': witnesses, + 'max_abs_endpoint_coefficient': max(abs(x) for row in s for x in row), + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path, default=Path(__file__).with_name('CONSTANT-CHART-AUDIT.json')) + args = parser.parse_args() + result = { + 'schema': 'independent-alias-constant-chart/1', + 'status': 'PASS_EXACT_INVERTIBILITY_AND_SCOPED_OBSERVER_GAP', + 'source_sha256': hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), + 'cases': [audit(p) for p in range(1,6)], + 'scope': 'Exact constant-owner scalar active-channel model K=1,L=13. Native phase maps, changing owner charts, cache restoration and global output-credit assignment are not established by this receipt.', + 'new_kappa_claimed': False, + } + args.out.parent.mkdir(parents=True, exist_ok=True) + args.out.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(result['status']) + for case in result['cases']: + print('p',case['pairs'],'rank',case['whole_rank'],'gap',case['ordinary_error_gap_per_channel']) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-FOREST.json b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-FOREST.json new file mode 100644 index 0000000000..9906ca0260 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-FOREST.json @@ -0,0 +1,564 @@ +{ + "cases": [ + { + "ambient_rank": 25, + "calls": 3, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-42.769174224631954373281806613734767960663259406", + "upper": "-42.769174224631954373281806613734767960663259405" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-138.695251913491363982573334607407511198700534698", + "upper": "-138.695251913491363982573334607407511198700534697" + } + } + ], + "host_assignment_sha256": "fbd420ed4b4cb7f5bcd1086ed1d0f0f9c848fb08c866376ad915c4dd1719bc3f", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "124.923223218592785011907471225278684871760470910", + "upper": "124.923223218592785011907471225278684871760470911" + }, + "raw_allowance": { + "lower": "148.923223218592785011907471225278684871760470910", + "upper": "148.923223218592785011907471225278684871760470911" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "28.601345501992670590502015199544234212245757101", + "upper": "28.601345501992670590502015199544234212245757102" + }, + "raw_allowance": { + "lower": "52.601345501992670590502015199544234212245757101", + "upper": "52.601345501992670590502015199544234212245757102" + } + } + ], + "lift": 3, + "name": "p1", + "productive_fields_per_controller": 408, + "rank_bill": "15576", + "rank_credit": 15800, + "rank_margin": "224", + "residual_profile_per_controller": { + "1": "129", + "2": "84" + }, + "role_match": true, + "strict_child_rank": true, + "templates": [ + [ + 168, + [ + 9, + 9, + 7 + ] + ], + [ + 54, + [ + 9, + 9, + 6 + ] + ], + [ + 84, + [ + 9, + 9, + 5 + ] + ], + [ + 27, + [ + 9, + 8, + 8 + ] + ], + [ + 75, + [ + 8, + 8, + 8 + ] + ] + ], + "total_fields_per_controller": 632 + }, + { + "ambient_rank": 25, + "calls": 3, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-42.519566363936157333822787441633480818295609919", + "upper": "-42.519566363936157333822787441633480818295609918" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-138.123982480176672070969743894441766441076930485", + "upper": "-138.123982480176672070969743894441766441076930484" + } + } + ], + "host_assignment_sha256": "c01e1c1e645a547c1e5cbe07531de7184517ee4446946b35212ed8080647ce3d", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "125.172831079288582051366490397379972014128120397", + "upper": "125.172831079288582051366490397379972014128120398" + }, + "raw_allowance": { + "lower": "149.172831079288582051366490397379972014128120397", + "upper": "149.172831079288582051366490397379972014128120398" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "29.172614935307362502105605912509978969869361314", + "upper": "29.172614935307362502105605912509978969869361315" + }, + "raw_allowance": { + "lower": "53.172614935307362502105605912509978969869361314", + "upper": "53.172614935307362502105605912509978969869361315" + } + } + ], + "lift": 3, + "name": "p1_every_host_has_width9", + "productive_fields_per_controller": 408, + "rank_bill": "15576", + "rank_credit": 15800, + "rank_margin": "224", + "residual_profile_per_controller": { + "1": "54", + "2": "9", + "3": "75" + }, + "role_match": true, + "strict_child_rank": true, + "templates": [ + [ + 168, + [ + 9, + 9, + 7 + ] + ], + [ + 54, + [ + 9, + 9, + 6 + ] + ], + [ + 9, + [ + 9, + 9, + 5 + ] + ], + [ + 75, + [ + 9, + 8, + 5 + ] + ], + [ + 102, + [ + 9, + 8, + 8 + ] + ] + ], + "total_fields_per_controller": 632 + }, + { + "ambient_rank": 41, + "calls": 5, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-22.794737821987735339199164939294080730881989163", + "upper": "-22.794737821987735339199164939294080730881989162" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-216.048573737889404882463586940351203614391975801", + "upper": "-216.048573737889404882463586940351203614391975800" + } + } + ], + "host_assignment_sha256": "632a4613fce22703d6465239677250ae9d18f41b5d47e7fade9cf449bb298d2f", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "256.692591250053496969449631459061673989824228029", + "upper": "256.692591250053496969449631459061673989824228030" + }, + "raw_allowance": { + "lower": "296.692591250053496969449631459061673989824228029", + "upper": "296.692591250053496969449631459061673989824228030" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "62.779088621250652739328662737901705403851843864", + "upper": "62.779088621250652739328662737901705403851843865" + }, + "raw_allowance": { + "lower": "102.779088621250652739328662737901705403851843864", + "upper": "102.779088621250652739328662737901705403851843865" + } + } + ], + "lift": 12, + "name": "p2", + "productive_fields_per_controller": 224, + "rank_bill": "27104", + "rank_credit": 27552, + "rank_margin": "448", + "residual_profile_per_controller": { + "1": "128", + "2": "130", + "3": "28", + "32": "20/3", + "5": "433/3", + "8": "29" + }, + "role_match": true, + "strict_child_rank": true, + "templates": { + "global_templates": [ + [ + "400/3", + "three 9s outside stage2; stage2=9" + ], + [ + "73", + [ + 8, + 8, + 9, + 8, + 8 + ] + ], + [ + "20/3", + [ + 0, + 0, + 9, + 0, + 0 + ] + ], + [ + "11", + [ + 9, + 9, + 0, + 9, + 9 + ] + ] + ], + "global_zero_positions": [ + 0, + 1, + 3, + 4 + ], + "original_pair_bins": [ + [ + 56, + [ + 9, + 7 + ] + ], + [ + 56, + [ + 7, + 9 + ] + ], + [ + 18, + [ + 9, + 6 + ] + ], + [ + 18, + [ + 6, + 9 + ] + ], + [ + 28, + [ + 9, + 5 + ] + ], + [ + 28, + [ + 5, + 9 + ] + ], + [ + 20, + [ + 8, + 8 + ] + ] + ] + }, + "total_fields_per_controller": 672 + }, + { + "ambient_rank": 41, + "calls": 5, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-22.733508650004101133377764684163866513952121367", + "upper": "-22.733508650004101133377764684163866513952121366" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-215.909497113006069224913826078560779160902612690", + "upper": "-215.909497113006069224913826078560779160902612689" + } + } + ], + "host_assignment_sha256": "f10ceaa7c57933f7ee91ebc48a2f8706007c5c491c6a8b43adf103ad1f2d6bc8", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "256.753820422037131175271031714191888206754095826", + "upper": "256.753820422037131175271031714191888206754095827" + }, + "raw_allowance": { + "lower": "296.753820422037131175271031714191888206754095826", + "upper": "296.753820422037131175271031714191888206754095827" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "62.918165246133988396878423599692129857341206975", + "upper": "62.918165246133988396878423599692129857341206976" + }, + "raw_allowance": { + "lower": "102.918165246133988396878423599692129857341206975", + "upper": "102.918165246133988396878423599692129857341206976" + } + } + ], + "lift": 1, + "name": "p2_219_hosts", + "productive_fields_per_controller": 219, + "rank_bill": "26899", + "rank_credit": 27347, + "rank_margin": "448", + "residual_profile_per_controller": { + "1": "128", + "2": "130", + "3": "28", + "5": "146", + "8": "29" + }, + "role_match": true, + "strict_child_rank": true, + "templates": { + "global_templates": [ + [ + "140", + "three 9s outside stage2; stage2=9" + ], + [ + "73", + [ + 8, + 8, + 9, + 8, + 8 + ] + ], + [ + "6", + [ + 9, + 9, + 0, + 9, + 9 + ] + ] + ], + "global_zero_positions": [ + 0, + 1, + 3, + 4 + ], + "original_pair_bins": [ + [ + 56, + [ + 9, + 7 + ] + ], + [ + 56, + [ + 7, + 9 + ] + ], + [ + 18, + [ + 9, + 6 + ] + ], + [ + 18, + [ + 6, + 9 + ] + ], + [ + 28, + [ + 9, + 5 + ] + ], + [ + 28, + [ + 5, + 9 + ] + ], + [ + 20, + [ + 8, + 8 + ] + ] + ] + }, + "total_fields_per_controller": 667 + } + ], + "historical_identity_profile": { + "1": 2487, + "2": 713, + "3": 224, + "4": 102, + "5": 20, + "8": 8 + }, + "interface_lemma": "If a cleaned physical native word is diag(DataOp,T_i), independent allowed monomial pre/post adapters on each helper change only T_i. When those adapters reduce T_i to its width-r Fourier, productive helper paths can be packed without separately preparing canonical native entrance frames. The internal word and ADDs are unchanged. Exact normal-form phases remain a native proof obligation.", + "p1_synchronous_entrance_debt_without_whole_block_adapters": 371, + "p2_fixed_original_assignment_global_field_minimum": { + "middle_stage_nines": 213, + "minimum": 219, + "ordinary_stage_nines_total": 444, + "ordinary_stage_nonzero_roles_total": 736, + "proof": "The four other stages require736 nonzero global roles. Of213 hosts carrying middle width9, each can fit at most three other roles unless all four are width8. There are only292 other width8 roles, so at most73 hosts have four such roles. The q-213 middle-unused hosts fit at most four other roles. Thus736<=3*213+73+4*(q-213)=4q-140, so q>=219. The219-host construction attains it. Scope fixes the displayed original-bin assignments and width-only whole-helper interface." + }, + "p2_global_field_lower_bound_in_this_unstaggered_middle_window": 213, + "scope": "Exact helper-width inventory, chronological eligibility, integer replica lifts, strict recursive child ranks and outward-rational complete invoices. Native whole-block helper adapters and changed data-controller endpoint closure are not emitted by this certificate.", + "source_profile": { + "1": 2453, + "2": 716, + "3": 224, + "4": 74, + "5": 48 + }, + "source_sha256": "fc8ce7bd7ec4803d2cff816af54feee9ebb5198afe6e58bd33f8f85fde1014f4", + "status": "PASS" +} diff --git a/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-NOTE.md b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-NOTE.md new file mode 100644 index 0000000000..5d5cd5d258 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-NOTE.md @@ -0,0 +1,294 @@ +# Productive helper-only forests + +This is a new recursive-contract candidate. It does not establish a new kappa. +The executor reset during the round. This note and its three standalone +packing certificates were recreated and rerun afterward. The earlier base +repository, frozen cumulative trees, and native receipts were unavailable in +the reset executor when these certificates were written; their previously +reported results must not be confused with a fresh replay of those sources. +The exact packing and outward-rational invoice in +`PRODUCTIVE-HELPER-FOREST.json` pass. A native emission of all helper boundary +adapters, and the changed main-data controller closure, remain required. + +## Why these fields can receive productive credit + +The previously tested controller families required every borrowed local helper +to fit inside an idle segment of an existing main-data field. The new proposal +adds fields whose entire job is a complete Fourier transform of the ambient +rank. Their arbitrary contents must be transformed correctly, rather than +erased or treated as zero. Each field can host several already-cleaned native +helper roles and then finish its remaining transform. + +There is a useful whole-block interface lemma. Suppose the exact physical +operator of a cleaned native word is + +\[ +\operatorname{diag}(D,T_1,\ldots,T_h),\qquad +T_i=F_{\mathrm{end},i}F_{\mathrm{start},i}^{-1}. +\] + +Here `D` is the data operation. The absence of helper/data and helper/helper +cross terms is essential and follows from full arbitrary-input restoration, +not merely from testing zero helper inputs. If allowed monomial adapters +`L_i,R_i` satisfy + +\[ +L_iT_iR_i=F_{r_i}, +\] + +then applying those adapters only at the boundaries gives exactly +`diag(D,F_r1,...,F_rh)`. The internal word and every internal ADD remain +unchanged. A later call can normalize its own complete helper block onto a +disjoint set of unfinished axes of the same field. These transforms compose +to the field's complete ambient Fourier job. + +This is different from changing each internal helper frame independently. +That latter shortcut can invalidate same-frame contacts. In particular, a +literal synchronous preparation of all native first-call entrance frames +would add 371 rank units in the p=1 case. The boundary-only block lemma avoids +that preparation if, and only if, the actual native relative transforms have +the stated allowed monomial normal forms with correct phases. The complex +agent identifies the existing `ce.lower_child(T_i)` normal-form routine for +that check; its fresh all-408-role adapter replay is not part of this receipt. + +## Exact p=1 packing + +Use three calls, ambient rank 25, one224-field main controller, and 408 new +productive helper fields:632 productive fields in total. The helper width +inventory per call is + +| Width | Count | +|---:|---:| +|5|28| +|6|18| +|7|56| +|8|93| +|9|213| + +Three replicas realize every cyclic rotation of each row below. Each call +then receives precisely its native inventory. + +| Hosts per controller | Three local widths | Remaining rank | +|---:|---|---:| +|168|9,9,7|0| +|54|9,9,6|1| +|9|9,9,5|2| +|75|9,8,5|3| +|102|9,8,8|0| + +The remaining profile is `54H1+9H2+75H3`, rank 297. Every host has at least +one width 9 role; this is useful for a potential staggered realization, though +no staggered frame compiler is presumed here. The receipt also preserves an +earlier, slightly weaker packing with remainder `129H1+84H2`. + +For the pruned THREE native profile +`2453H1+716H2+224H3+74H4+48H5`, the complete conditional invoice is + +\[ +632H_{25}-3P_{\rm THREE}-(54H_1+9H_2+75H_3). +\] + +Using the native same-width involution completion costs another 24H1. The +remaining allowance is: + +| C | After three involution completions | +|---:|---:| +|0.001745|125.1728310792885820514| +|0.004|29.1726149353073625021| + +These are allowances for all still-missing main-controller closure and +boundary work. They are not additional independent savings, and they are not +a proof that the altered controller computes the required complete job. + +## Exact p=2 packing + +Use five calls, ambient rank 41,448 main fields, and 224 new productive helper +fields:672 in total. Original idle windows provide224 rank 16 bins for stages +0/1,224 rank 8 bins for stage 2, and 224 rank 16 bins for stages 3/4. Twelve replicas +make all listed allocations integral. + +Each two-stage rank 16 window uses the following224 pairs: + +| Count | Widths | +|---:|---| +|56|9,7| +|56|7,9| +|18|9,6| +|18|6,9| +|28|9,5| +|28|5,9| +|20|8,8| + +The middle rank 8 window supplies93 width 8,56 width 7,18 width 6,28 width 5 +roles, leaving29 bins unused. New global fields supply the remaining roles. +The receipt records exact templates, expands every field and stage, and +checks all five histograms. Its complete remainder is + +\[ +128H_1+130H_2+28H_3+29H_8+ +\frac{433}{3}H_5+\frac{20}{3}H_{32}. +\] + +The corresponding allowance, after five 8H1 involution completions, is: + +| C | Remaining allowance | +|---:|---:| +|0.001745|256.6925912500534969694| +|0.004|62.7790886212506527393| + +A simple temporal obstruction rules out185 global helper fields in this +particular unstaggered allocation: the middle call has213 width 9 roles, +while all original middle idle bins have rank 8. At least213 additional fields +are therefore needed in that window. This is not a lower bound for other +orders, staggered forests, split helper interfaces, or shared native calls. + +## Improved p=2 allocation with 219 new hosts + +Keeping all original-bin assignments above, the new global fields can be +reduced from 224 to 219. No replica lift is needed. Use 140 fields with stage 2 +width 9 and three other width 9 roles plus one unused other stage, assigning35 +fields to each of the four possible unused stages. Use 73 fields with widths +`[8,8,9,8,8]` and 6 with widths `[9,9,0,9,9]`. + +Each outside stage receives111 width 9 and73 width 8 global roles; the middle +stage receives213 width 9 roles. All unused global progress is146H5. The +complete remaining profile is + +\[ +128H_1+130H_2+28H_3+29H_8+146H_5. +\] + +There are 667 productive fields. After five 8H1 involution completions, the +allowances are 256.7538204220371311753 at C=0.001745 and +62.9181652461339883969 at C=0.004. + +For these fixed original-bin assignments,219 is minimal. The four outside +stages require736 nonzero global roles. A host with middle width 9 can fit +only three outside roles unless all four are width 8. The 292 available +outside width 8 roles permit at most73 such exceptions. The remaining +`q-213` hosts fit at most four outside roles. Thus +`736 <= 3*213+73+4*(q-213) = 4q-140`, forcing `q>=219`. +This is a scoped minimum for the displayed assignment, not a restriction on +changing the original idle-bin allocation. + +## Changed original allocation:213 new hosts + +The separate `productive_helper_refinement.py` improves the p=2 construction +by changing the original idle-bin allocation. It preserves the earlier 219 +certificate and its deliberately narrower minimum claim. + +In each rank 16 two-stage window, use the following224 pairs: + +| Count | Widths | +|---:|---| +|56|9,7| +|56|7,9| +|18|9,6| +|18|6,9| +|16|9,5| +|16|5,9| +|44|8,8| + +The middle original rank 8 assignment stays as before. Each outside stage now +needs 123 width 9,49 width 8, and 12 width 5 global helper roles. Use these 213 +global fields, all with middle width 9: + +| Count | Widths in the four outside stages | Residual rank | +|---:|---|---:| +|48|9,9,9,5;12 of each position of5|0| +|116|9,9,9,0;29 of each unused position|5| +|49|8,8,8,8|0| + +No replica lift is needed. The complete residual profile is + +\[ +128H_1+82H_2+28H_3+116H_5+29H_8. +\] + +With 661 productive fields, its allowance after the five involution +completions is 257.8060663457233916759 at C=0.001745 and +65.3169531797331110453 at C=0.004. It attains the213-host lower bound for the +same middle window: all 213 width 9 roles there must go to new hosts because +the original idle bins have rank 8. This minimum does not cover staggered +controllers or changed native helper interfaces. + +## Further p=1 residual improvement + +A small numerical MILP proposed the following integral packing. The separate +`productive_helper_p1_refinement.py` replays its exact inventory, three-replica +cyclic realization, and rational invoice without using an optimizer. + +| Count | Three local widths | Residual rank | +|---:|---|---:| +|18|5,5,5|10| +|1|5,5,8|7| +|28|5,9,9|2| +|54|6,9,9|1| +|168|7,9,9|0| +|139|8,8,9|0| + +Its remaining profile is `54H1+28H2+H7+18H10`. After three involution +completions the allowance is 125.5333563352732444235 at C=0.001745 and +29.9957695582853966376 at C=0.004. This exact candidate improves the earlier +packings. The numerical solver's optimal-status flag is not claimed as a +rational optimality certificate. + +## Identity fallback and a boundary priority + +Substituting the historical repaired-identity profile +`2487H1+713H2+224H3+102H4+20H5+8H8`, even granting these same helper packings, +does not meet either target. For the improved p=1 packing its complete margin +is -42.5195663639361573338 at 0.001745 and -138.1239824801766720710 at 0.004. +For the 219-host p=2 packing these margins are -22.7335086500041011334 and +-215.9094971130060692249. For the latest213-host p=2 packing they remain +-21.6812627263178406327 and -213.5107091794069465765. The latest p=1 packing +also still fails: -42.1590411079514949617 and -137.3008278571986379355. +The changed-kernel controller closure remains +necessary; the productive fields do not by themselves establish the target. + +The latest p=1 allowance at 0.001745 can pay one 112H1 normalization and leave +13.5333563352732444235. This makes the exact source-head and terminal-boundary +contract worth checking: it might allow a final common-frame gauge decoder. +It cannot pay two such normalizations. No entrance obligation has been +deleted, and no one-pass boundary change is admitted by this invoice. + +## What is checked and what remains + +The standalone script checks every expanded host, stage eligibility, +per-call width inventory, nonnegative remaining path, integer replica lift, +strict child-rank decrease, the total rank identity, and the complete tariff +at two exact rational exponents. Its logarithm and exponential calculations +use rational series bounds, and the JSON intervals are rounded outward. + +It does not reconstruct the earlier native source profile from its original +word. That profile was authenticated earlier in this round; the executor +reset interrupted access to those predecessor artifacts. Nor does it verify +fresh monomial normal forms for all408 relative helper transforms, emit a +whole enlarged controller, or admit its outer bilinear use. The scientific +meaning of these receipts is a concrete, fully invoiced productive-helper +interface candidate with explicit remaining obligations. + +The native involution completion also uses one sequentially reusable owned +clear field. Its recursive calls are included in the 8H1-per-call invoice, +but this scratch field receives no productive output credit. A complete +controller must supply that actual scratch capacity and retain its clearing +proof. The productive field counts in this note are not the total allocated +register count after adding such owned scratch. + +The historical repaired-identity kernel cannot be replaced by the pruned +kernel plus free historical query assembly. The separate query-contract +audit rules out the tested deletions. This note uses the actual changed +involution kernel and keeps its main-data closure outstanding. + +## Reproduction + +From the checkpoint-tree directory: + +```bash +python3 -B research/kappa-accounted-composition/recursion/productive_helper_forest.py --output research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-FOREST.json +python3 -B research/kappa-accounted-composition/recursion/productive_helper_refinement.py --output research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-REFINEMENT.json +python3 -B research/kappa-accounted-composition/recursion/productive_helper_p1_refinement.py --output research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-P1-REFINEMENT.json +``` + +No external package is needed. Repeating the command should reproduce the +same receipt bytes for the same script. diff --git a/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-P1-REFINEMENT.json b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-P1-REFINEMENT.json new file mode 100644 index 0000000000..e08b26b74d --- /dev/null +++ b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-P1-REFINEMENT.json @@ -0,0 +1,126 @@ +{ + "case": { + "ambient_rank": 25, + "calls": 3, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-42.159041107951494961653775155515228344688876662", + "upper": "-42.159041107951494961653775155515228344688876661" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-137.300827857198637935501731722781218172953988313", + "upper": "-137.300827857198637935501731722781218172953988312" + } + } + ], + "host_assignment_sha256": "ba92ef1f9bf31d412d77fa30f70bcc2acd8e22401d4c63487de93f424de269a3", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "125.533356335273244423535502683498224487734853654", + "upper": "125.533356335273244423535502683498224487734853655" + }, + "raw_allowance": { + "lower": "149.533356335273244423535502683498224487734853654", + "upper": "149.533356335273244423535502683498224487734853655" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "29.995769558285396637573618084170527237992303486", + "upper": "29.995769558285396637573618084170527237992303487" + }, + "raw_allowance": { + "lower": "53.995769558285396637573618084170527237992303486", + "upper": "53.995769558285396637573618084170527237992303487" + } + } + ], + "lift": 3, + "name": "p1_concentrated_residual", + "optimality_scope": "A scipy/HiGHS MILP discovery run reported an optimum at C=.004, but this certificate verifies only this exact integer pattern and complete invoice. It is not a rational optimality proof or a native owner emission.", + "productive_fields_per_controller": 408, + "rank_bill": "15576", + "rank_credit": 15800, + "rank_margin": "224", + "residual_profile_per_controller": { + "1": "54", + "10": "18", + "2": "28", + "7": "1" + }, + "role_match": true, + "strict_child_rank": true, + "templates": [ + [ + 18, + [ + 5, + 5, + 5 + ] + ], + [ + 1, + [ + 5, + 5, + 8 + ] + ], + [ + 28, + [ + 5, + 9, + 9 + ] + ], + [ + 54, + [ + 6, + 9, + 9 + ] + ], + [ + 168, + [ + 7, + 9, + 9 + ] + ], + [ + 139, + [ + 8, + 8, + 9 + ] + ] + ], + "total_fields_per_controller": 632 + }, + "sources": { + "PRODUCTIVE-HELPER-FOREST.json": "63a4197062a6ea4a96555b4136210efc716317c1251d88c40a47a22b9926b89d", + "productive_helper_forest.py": "fc8ce7bd7ec4803d2cff816af54feee9ebb5198afe6e58bd33f8f85fde1014f4", + "productive_helper_p1_refinement.py": "2982ef4fab8e7844a460579640ad13bf9eb0381b0e7b66fb805c21f73cea1b8d" + }, + "status": "PASS" +} diff --git a/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-REFINEMENT.json b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-REFINEMENT.json new file mode 100644 index 0000000000..4b91f95801 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-REFINEMENT.json @@ -0,0 +1,169 @@ +{ + "case": { + "ambient_rank": 41, + "calls": 5, + "helper_histogram_per_call": { + "5": 28, + "6": 18, + "7": 56, + "8": 93, + "9": 213 + }, + "historical_identity_fallback_same_helper_packing_favorable_screen": [ + { + "C": "349/200000", + "complete_margin": { + "lower": "-21.681262726317840632739489904400586413276967019", + "upper": "-21.681262726317840632739489904400586413276967018" + } + }, + { + "C": "1/250", + "complete_margin": { + "lower": "-213.510709179406946576527880277692889204555019621", + "upper": "-213.510709179406946576527880277692889204555019620" + } + } + ], + "host_assignment_sha256": "5ee500e785abcde1821024c1d2c8618e9c938e21ba480df9dd37cc20aaa05c86", + "invoices": [ + { + "C": "349/200000", + "after_8_H1_per_call": { + "lower": "257.806066345723391675909306493955168307429250173", + "upper": "257.806066345723391675909306493955168307429250174" + }, + "raw_allowance": { + "lower": "297.806066345723391675909306493955168307429250173", + "upper": "297.806066345723391675909306493955168307429250174" + } + }, + { + "C": "1/250", + "after_8_H1_per_call": { + "lower": "65.316953179733111045264369400560019813688800044", + "upper": "65.316953179733111045264369400560019813688800045" + }, + "raw_allowance": { + "lower": "105.316953179733111045264369400560019813688800044", + "upper": "105.316953179733111045264369400560019813688800045" + } + } + ], + "lift": 1, + "minimum_in_fixed_middle_window": { + "attained": true, + "bound": 213, + "proof": "The middle call needs213 width9 helper roles. Every original idle field available there has rank8. A productive field can host at most one simultaneous native helper role. Therefore at least213 new productive fields are required for this same fixed middle window. The displayed213-field word-width packing attains the bound. This does not cover staggered controllers, changed helper interfaces, or different original idle windows." + }, + "name": "p2_changed_original_assignment_213_hosts", + "productive_fields_per_controller": 213, + "rank_bill": "26653", + "rank_credit": 27101, + "rank_margin": "448", + "residual_profile_per_controller": { + "1": "128", + "2": "82", + "3": "28", + "5": "116", + "8": "29" + }, + "role_match": true, + "strict_child_rank": true, + "templates": { + "global_templates": [ + { + "count": 48, + "each_5_position_count": 12, + "middle_width": 9, + "other_widths": [ + 9, + 9, + 9, + 5 + ] + }, + { + "count": 116, + "each_0_position_count": 29, + "middle_width": 9, + "other_widths": [ + 9, + 9, + 9, + 0 + ] + }, + { + "count": 49, + "middle_width": 9, + "other_widths": [ + 8, + 8, + 8, + 8 + ] + } + ], + "original_pair_bins": [ + [ + 56, + [ + 9, + 7 + ] + ], + [ + 56, + [ + 7, + 9 + ] + ], + [ + 18, + [ + 9, + 6 + ] + ], + [ + 18, + [ + 6, + 9 + ] + ], + [ + 16, + [ + 9, + 5 + ] + ], + [ + 16, + [ + 5, + 9 + ] + ], + [ + 44, + [ + 8, + 8 + ] + ] + ] + }, + "total_fields_per_controller": 661 + }, + "scope": "Exact changed idle-bin allocation, complete tariff and fixed-middle-window host minimum. Native whole-block adapters and main-controller closure remain unverified here.", + "sources": { + "PRODUCTIVE-HELPER-FOREST.json": "63a4197062a6ea4a96555b4136210efc716317c1251d88c40a47a22b9926b89d", + "productive_helper_forest.py": "fc8ce7bd7ec4803d2cff816af54feee9ebb5198afe6e58bd33f8f85fde1014f4", + "productive_helper_refinement.py": "d70e2feb935f93e9c9ce30340ce099dbefdc3ffe9ef54b3b338333efe815df71" + }, + "status": "PASS" +} diff --git a/research/kappa-accounted-composition/recursion/productive_helper_forest.py b/research/kappa-accounted-composition/recursion/productive_helper_forest.py new file mode 100644 index 0000000000..5e380981d1 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/productive_helper_forest.py @@ -0,0 +1,161 @@ +#!/usr/bin/env python3 +"""Exact finite packings for productive helper-only fields. + +This is a width/chronology certificate, not a native owner emission. The full +physical cleaned-block contract permits independent helper endpoint adapters +only if each relative helper transform is certified as a rank-width Fourier +with the stated allowed monomial adapters. That adapter obligation and the +changed data-controller endpoint closure remain separate. +""" +import argparse +from collections import Counter +from fractions import Fraction as Q +import hashlib +import json +from pathlib import Path +import sys +sys.dont_write_bytecode = True + +WIDTHS = Counter({5: 28, 6: 18, 7: 56, 8: 93, 9: 213}) +PROFILE = Counter({1: 2453, 2: 716, 3: 224, 4: 74, 5: 48}) +HISTORICAL_IDENTITY_PROFILE = Counter({1: 2487, 2: 713, 3: 224, 4: 102, 5: 20, 8: 8}) + +def log_interval(r): + e = r.bit_length() - 1 + def odd_series(x): + z = (x - 1) / (x + 1) + n = 80 + s = 2 * sum((z ** (2*j+1) / (2*j+1) for j in range(n)), Q(0)) + tail = 2*z**(2*n+1) / ((2*n+1)*(1-z*z)) + return s, s + tail + a, b = odd_series(Q(2)) + c, d = odd_series(Q(r, 2**e)) + return e*a+c, e*b+d + +def exp_interval(x): + assert 0 <= x < 1 + term, s = Q(1), Q(1) + n = 48 + for j in range(1, n+1): + term *= x/j + s += term + following = term*x/(n+1) + return s, s + following/(1-x/(n+2)) + +def power_interval(r, c): + if r == 1: + return Q(1), Q(1) + lo, hi = log_interval(r) + elo, _ = exp_interval(c*lo) + _, ehi = exp_interval(c*hi) + return Q(r)/ehi, Q(r)/elo + +def encode_interval(bounds): + scale = 10**45 + lo, hi = bounds + a = lo.numerator*scale//lo.denominator + b = -((-hi.numerator*scale)//hi.denominator) + def fixed(n): + sign = '-' if n < 0 else '' + s = str(abs(n)).rjust(46, '0') + return sign+s[:-45]+'.'+s[-45:] + return {'lower': fixed(a), 'upper': fixed(b)} + +def invoice(width, rank, calls, residual, c, source_profile=PROFILE): + profile = Counter({r: Q(n*calls) for r, n in source_profile.items()}) + profile.update(residual) + credit = power_interval(rank, c) + lo, hi = width*credit[0], width*credit[1] + for r, n in sorted(profile.items()): + a, b = power_interval(r, c) + lo -= n*b + hi -= n*a + return {'C': str(c), 'raw_allowance': encode_interval((lo, hi)), + 'after_8_H1_per_call': encode_interval((lo-8*calls, hi-8*calls))} + +def verify_case(name, lift, width, rank, calls, hosts, templates): + counts = [Counter() for _ in range(calls)] + residual = Counter() + new_hosts = 0 + for host in hosts: + cap = host['capacity'] + used = sum(host['widths']) + assert used <= cap + assert all(0 <= r < rank for r in host['widths']) + assert 0 <= cap-used < rank + if host['kind'] == 'productive': + new_hosts += 1 + for stage, r in zip(host['stages'], host['widths']): + if r: + counts[stage][r] += 1 + if cap > used: + residual[cap-used] += 1 + for row in counts: + assert row == Counter({r: n*lift for r, n in WIDTHS.items()}) + per = {r: Q(n, lift) for r, n in residual.items()} + total_rank = calls*sum(r*n for r, n in PROFILE.items()) + sum(r*n for r, n in per.items()) + assert width*rank-total_rank == 224*((calls-1)//2) + return {'name': name, 'lift': lift, 'productive_fields_per_controller': new_hosts//lift, + 'total_fields_per_controller': width, 'ambient_rank': rank, 'calls': calls, + 'helper_histogram_per_call': dict(sorted(WIDTHS.items())), + 'role_match': True, 'strict_child_rank': True, + 'residual_profile_per_controller': {str(r): str(n) for r, n in sorted(per.items())}, + 'rank_bill': str(total_rank), 'rank_credit': width*rank, + 'rank_margin': str(width*rank-total_rank), 'templates': templates, + 'host_assignment_sha256': hashlib.sha256(json.dumps(hosts, sort_keys=True, separators=(',', ':')).encode()).hexdigest(), + 'invoices': [invoice(width, rank, calls, per, c) for c in (Q(349, 200000), Q(1, 250))], + 'historical_identity_fallback_same_helper_packing_favorable_screen': [ + {'C': str(c), 'complete_margin':invoice(width, rank, calls, per, c, HISTORICAL_IDENTITY_PROFILE)['raw_allowance']} + for c in (Q(349, 200000), Q(1, 250))]} + +def p1(all_have_nine=False): + rows = ([(168,[9,9,7]),(54,[9,9,6]),(9,[9,9,5]),(75,[9,8,5]),(102,[9,8,8])] + if all_have_nine else [(168,[9,9,7]),(54,[9,9,6]),(84,[9,9,5]),(27,[9,8,8]),(75,[8,8,8])]) + hosts = [] + for count, row in rows: + for shift in range(3): + v = [row[(s+shift)%3] for s in range(3)] + hosts.extend({'kind':'productive','capacity':25,'stages':[0,1,2],'widths':v} for _ in range(count)) + return verify_case('p1_every_host_has_width9' if all_have_nine else 'p1',3,632,25,3,hosts,rows) + +def p2(compact=False): + lift = 1 if compact else 12 + hosts = [] + pairs = [(56,[9,7]),(56,[7,9]),(18,[9,6]),(18,[6,9]),(28,[9,5]),(28,[5,9]),(20,[8,8])] + for stages in ([0,1],[3,4]): + for count, row in pairs: + hosts.extend({'kind':'original_idle','capacity':16,'stages':stages,'widths':row} for _ in range(count*lift)) + for r, count in [(8,93),(7,56),(6,18),(5,28),(0,29)]: + hosts.extend({'kind':'original_idle','capacity':8,'stages':[2],'widths':[r]} for _ in range(count*lift)) + for zero in [0,1,3,4]: + row = [9,9,9,9,9] + row[zero] = 0 + hosts.extend({'kind':'productive','capacity':41,'stages':[0,1,2,3,4],'widths':row} for _ in range(35 if compact else 100*lift//3)) + extra = [(73,[8,8,9,8,8]),(6,[9,9,0,9,9])] if compact else [(73*lift,[8,8,9,8,8]),(20*lift//3,[0,0,9,0,0]),(11*lift,[9,9,0,9,9])] + for count, row in extra: + hosts.extend({'kind':'productive','capacity':41,'stages':[0,1,2,3,4],'widths':row} for _ in range(count)) + templates = [['140','three 9s outside stage2; stage2=9'],['73',[8,8,9,8,8]],['6',[9,9,0,9,9]]] if compact else [['400/3','three 9s outside stage2; stage2=9'],['73',[8,8,9,8,8]],['20/3',[0,0,9,0,0]],['11',[9,9,0,9,9]]] + return verify_case('p2_219_hosts' if compact else 'p2',lift,667 if compact else 672,41,5,hosts,{'original_pair_bins':pairs,'global_zero_positions':[0,1,3,4], + 'global_templates':templates}) + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument('--output', type=Path, required=True) + args = parser.parse_args() + result = {'status':'PASS', 'scope':'Exact helper-width inventory, chronological eligibility, integer replica lifts, strict recursive child ranks and outward-rational complete invoices. Native whole-block helper adapters and changed data-controller endpoint closure are not emitted by this certificate.', + 'interface_lemma':'If a cleaned physical native word is diag(DataOp,T_i), independent allowed monomial pre/post adapters on each helper change only T_i. When those adapters reduce T_i to its width-r Fourier, productive helper paths can be packed without separately preparing canonical native entrance frames. The internal word and ADDs are unchanged. Exact normal-form phases remain a native proof obligation.', + 'source_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest(), + 'source_profile':dict(sorted(PROFILE.items())), + 'historical_identity_profile':dict(sorted(HISTORICAL_IDENTITY_PROFILE.items())), + 'p1_synchronous_entrance_debt_without_whole_block_adapters':371, + 'p2_global_field_lower_bound_in_this_unstaggered_middle_window':213, + 'p2_fixed_original_assignment_global_field_minimum': {'minimum':219, + 'ordinary_stage_nines_total':444, 'ordinary_stage_nonzero_roles_total':736, + 'middle_stage_nines':213, + 'proof':'The four other stages require736 nonzero global roles. Of213 hosts carrying middle width9, each can fit at most three other roles unless all four are width8. There are only292 other width8 roles, so at most73 hosts have four such roles. The q-213 middle-unused hosts fit at most four other roles. Thus736<=3*213+73+4*(q-213)=4q-140, so q>=219. The219-host construction attains it. Scope fixes the displayed original-bin assignments and width-only whole-helper interface.'}, + 'cases':[p1(),p1(True),p2(),p2(True)]} + args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'cases':[{'name':c['name'],'rank_margin':c['rank_margin']} for c in result['cases']]})) + +if __name__ == '__main__': + main() diff --git a/research/kappa-accounted-composition/recursion/productive_helper_p1_refinement.py b/research/kappa-accounted-composition/recursion/productive_helper_p1_refinement.py new file mode 100644 index 0000000000..085ed5bd43 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/productive_helper_p1_refinement.py @@ -0,0 +1,38 @@ +#!/usr/bin/env python3 +"""Exact certificate for a smaller p=1 residual invoice. + +A numerical MILP proposed the integer pattern. This replay uses no optimizer +and does not assert an exact optimum over other allocations. +""" +import argparse +import hashlib +import json +from pathlib import Path +import sys +sys.dont_write_bytecode = True +import productive_helper_forest as B + +ROWS=[(18,[5,5,5]),(1,[5,5,8]),(28,[5,9,9]),(54,[6,9,9]),(168,[7,9,9]),(139,[8,8,9])] + +def construct(): + hosts=[] + for count,row in ROWS: + for shift in range(3): + v=[row[(s+shift)%3] for s in range(3)] + hosts.extend({'kind':'productive','capacity':25,'stages':[0,1,2],'widths':v} for _ in range(count)) + r=B.verify_case('p1_concentrated_residual',3,632,25,3,hosts,ROWS) + assert r['residual_profile_per_controller']=={'1':'54','2':'28','7':'1','10':'18'} + r['optimality_scope']='A scipy/HiGHS MILP discovery run reported an optimum at C=.004, but this certificate verifies only this exact integer pattern and complete invoice. It is not a rational optimality proof or a native owner emission.' + return r + +def main(): + p=argparse.ArgumentParser() + p.add_argument('--output',type=Path,required=True) + args=p.parse_args() + here=Path(__file__).resolve().parent + r={'status':'PASS','sources':{name:hashlib.sha256((here/name).read_bytes()).hexdigest() for name in ['productive_helper_p1_refinement.py','productive_helper_forest.py','PRODUCTIVE-HELPER-FOREST.json']},'case':construct()} + args.output.write_text(json.dumps(r,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':r['status'],'name':r['case']['name']})) + +if __name__=='__main__': + main() diff --git a/research/kappa-accounted-composition/recursion/productive_helper_refinement.py b/research/kappa-accounted-composition/recursion/productive_helper_refinement.py new file mode 100644 index 0000000000..2ac6c91956 --- /dev/null +++ b/research/kappa-accounted-composition/recursion/productive_helper_refinement.py @@ -0,0 +1,50 @@ +#!/usr/bin/env python3 +"""Change original idle-bin allocation and attain213 productive helpers.""" +import argparse +import hashlib +import json +from pathlib import Path +import sys +sys.dont_write_bytecode = True +import productive_helper_forest as B + +def construct(): + hosts = [] + pairs = [(56,[9,7]),(56,[7,9]),(18,[9,6]),(18,[6,9]),(16,[9,5]),(16,[5,9]),(44,[8,8])] + for stages in ([0,1],[3,4]): + for count, row in pairs: + hosts.extend({'kind':'original_idle','capacity':16,'stages':stages,'widths':row} for _ in range(count)) + for r, count in [(8,93),(7,56),(6,18),(5,28),(0,29)]: + hosts.extend({'kind':'original_idle','capacity':8,'stages':[2],'widths':[r]} for _ in range(count)) + for special in [0,1,3,4]: + for value, count in [(5,12),(0,29)]: + row = [9,9,9,9,9] + row[special] = value + hosts.extend({'kind':'productive','capacity':41,'stages':[0,1,2,3,4],'widths':row} for _ in range(count)) + hosts.extend({'kind':'productive','capacity':41,'stages':[0,1,2,3,4],'widths':[8,8,9,8,8]} for _ in range(49)) + result = B.verify_case('p2_changed_original_assignment_213_hosts',1,661,41,5,hosts, + {'original_pair_bins':pairs,'global_templates':[ + {'count':48,'middle_width':9,'other_widths':[9,9,9,5],'each_5_position_count':12}, + {'count':116,'middle_width':9,'other_widths':[9,9,9,0],'each_0_position_count':29}, + {'count':49,'middle_width':9,'other_widths':[8,8,8,8]}]}) + assert result['productive_fields_per_controller'] == 213 + assert result['residual_profile_per_controller'] == {'1':'128','2':'82','3':'28','5':'116','8':'29'} + result['minimum_in_fixed_middle_window'] = {'bound':213,'attained':True, + 'proof':'The middle call needs213 width9 helper roles. Every original idle field available there has rank8. A productive field can host at most one simultaneous native helper role. Therefore at least213 new productive fields are required for this same fixed middle window. The displayed213-field word-width packing attains the bound. This does not cover staggered controllers, changed helper interfaces, or different original idle windows.'} + return result + +def main(): + p = argparse.ArgumentParser() + p.add_argument('--output',type=Path,required=True) + args=p.parse_args() + here=Path(__file__).resolve().parent + base_receipt=here/'PRODUCTIVE-HELPER-FOREST.json' + assert base_receipt.exists(), 'Reproduce the base certificate first.' + r={'status':'PASS','scope':'Exact changed idle-bin allocation, complete tariff and fixed-middle-window host minimum. Native whole-block adapters and main-controller closure remain unverified here.', + 'sources':{name:hashlib.sha256((here/name).read_bytes()).hexdigest() for name in ['productive_helper_refinement.py','productive_helper_forest.py','PRODUCTIVE-HELPER-FOREST.json']}, + 'case':construct()} + args.output.write_text(json.dumps(r,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':r['status'],'productive_fields':r['case']['productive_fields_per_controller'],'total_fields':r['case']['total_fields_per_controller']})) + +if __name__=='__main__': + main() diff --git a/research/kappa-causal-continuation/README.md b/research/kappa-causal-continuation/README.md new file mode 100644 index 0000000000..1557b27592 --- /dev/null +++ b/research/kappa-causal-continuation/README.md @@ -0,0 +1,75 @@ +# Causal continuation: output accumulation and native cleanup reuse + +Research date: 20 October 2026. Apache-2.0; substantial OpenAI assistance. + +**This package contains verified local improvements and complete paid small +blocks. It does not claim a new multiplication kappa.** All recoverable prior +work remains unmerged; no selected result, public branch, PR, or discussion +was changed. + +Read [REPORT.md](REPORT.md) for the research result and next synthesis targets. + +## Main results + +| Construction | Verified improvement | Remaining boundary | +|---|---|---| +| BIT paired output accumulation | 2 side helpers to 0; h19 unit ADDs 10,512 to 5,254 | A pair's sources must be supplied at its actual frames | +| BIT recursive output blocks | Fully paid 4/8/12-output routes save 994/2,198/2,430 helpers at h19 | Blocks do not yet compose over all 11,628 targets | +| Conditional .004 acquisition screen | A full 11,628-helper source bank now fits several explicit path budgets | Counts do not construct those paths globally | +| Complex compensation | 125,676 to 55,114 conservative real scalar primitives | Native child profile unchanged | +| Complex cleanup | 91 separate fields to 36, with no added operations | Caller must host the remaining 36 | +| Integrated complex word | THREE 148,646; FOUR 148,624 scalar allowance, both directions; no C workspace | Global repair/controller, rounding, hosting, outer admission | + +## Reproduce the scoped evidence + +Python 3 with NumPy is required for the bounded complex coefficient search. +The other new verification code uses the standard library. Keep assertions +enabled. From the recovered repository root: + +```sh +python3 -B research/kappa-causal-continuation/validate.py +``` + +The validator checks this package's exact inventory, reconstructs the two +predecessor packages from their pinned closed inventories, and reruns the +new constructions in an isolated tree. It compares fresh artifacts with +the frozen results and then runs the 41-stage predecessor replay. It does +not rewrite predecessor sources. `--new-only` omits those inherited replay +stages while retaining source authentication; `--integrity-only` checks +inventories without running the probes. + +`VALIDATION.json` is the definitive replay receipt, including exact stage +counts, comparison hashes, scope, and any failures. `INTEGRITY.json` binds +all other continuation files, excluding only itself and `VALIDATION.json`. + +## Where the new work lives + +- `bit/`: paired output consumer, recursive dirty-source router, exact frame + certificate, cost screens, proof, and scoped replay. +- `complex/`: dyadic output-basis compensation, bounded coefficient search, + exact one-shot capture exclusion, and independent integration review. +- `buffer-borrowing/`: independent literal native interpreter, the first + 53 lifetime aliases, source pins, and proof. +- `literature-audit/`: primary literature and partial public overlap review, + independent BIT/budget/lifetime audits, and two additional fully decoupled + buffer aliases. +- `integrated-complex/`: four usable canonical/native clean-J word pairs, + with optimized compensation, 55 borrowed fields, and empty centre + bookkeeping removed. +- `recovery/`: exact base/overlay provenance, unmerged state, and the original + attached context with its historical status explained. +- `delivery/`: byte-preserving reconstruction of the available older + cumulative inventory for the new cumulative overlay. + +## Carry-forward policy + +The supplied compact base matches the predecessor's required SHA256 exactly, +despite its changed download suffix. Every one of the 2,963 retained overlay +paths is preserved with its original bytes and mode. The 508 earlier +supplement files remain in the cumulative unmerged supplement, alongside +this continuation. They must not be dropped merely because a later turn +focuses on another direction. No user declaration of a merge has occurred. + +The older reported unavailable archive and the last interrupted session's +unprovided word artifacts are not claimed recovered. See +[recovery/RECOVERY.md](recovery/RECOVERY.md). diff --git a/research/kappa-causal-continuation/buffer-borrowing/SOURCES.json b/research/kappa-causal-continuation/buffer-borrowing/SOURCES.json new file mode 100644 index 0000000000..2e9f9b3ed2 --- /dev/null +++ b/research/kappa-causal-continuation/buffer-borrowing/SOURCES.json @@ -0,0 +1,20 @@ +{ + "research/kappa-native-resume/native-source/rebased-core/sources/bw_emit.py": "99b59a6fa3feda86785e4225157dcb1a6169d256cec4f5e0acc9cea7a701df76", + "research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz": "a5ed85fac5ebf3c091482a113ab0d351eb5a7892fe2841a35e16017ec1ff49f7", + "research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-CANONICAL.json.gz": "f07fae1011160b53e2df1f3135465db1277a093319de2a1da52d648f00a39542", + "research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-PHASES.json": "9d5316a1465a3eb2c62fac76c291394c0401c34a96fd8b94863d662b16a64552", + "research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-CANONICAL.json.gz": "4526f0c379c0a4e4ae7731b7086a15382c1d1e3960bdf73245f7121299a7ac73", + "research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-NATIVE-WORD.json.gz": "bde29d4ac306aa528a1237c8b7197291f29f1733f50058ca13f72c2ae328dccf", + "research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-CANONICAL.json.gz": "882271d70602354cbada8d77f479703ccc6c47e689e8ccd91f9e4c87de0f4379", + "research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-PHASES.json": "81c01d4b007bb720311dbdbbc63df673fdaf7baba707b360bbecc4b9cb6ee7b0", + "research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-CANONICAL.json.gz": "6d214cd24e00a2af5b237df8acadd6bd8d3e77bf307fafd0e3e8451c66d93203", + "research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-NATIVE-WORD.json.gz": "bde29d4ac306aa528a1237c8b7197291f29f1733f50058ca13f72c2ae328dccf", + "research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-CANONICAL.json.gz": "bff67890d0d350646d11f45aa1b098d6a415b1850b4982e8a05b4c146d247eda", + "research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-PHASES.json": "e273538de7957b959c40f593c4aad5e8e3d81d83d662fe2895c17770c5fad8bd", + "research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-CANONICAL.json.gz": "71f03fe085db07399cdfeffadbb65f1910bbdb5b7b0873562fb228e4cb292c8b", + "research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-NATIVE-WORD.json.gz": "57d7c53bacc42c6d403d3f2ced05c7e277ade034a2cb2db9b07e20962cc0b0d6", + "research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-CANONICAL.json.gz": "e0c473d62127a09cb3cbf451564dbdd19c0c10f1674dab0575e65251aa5e1e3c", + "research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-PHASES.json": "c8f744c62670ad7d76faf4dee65e2f708f4036953041e72383929c7d0b32ec38", + "research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-CANONICAL.json.gz": "d67623412d0de8aab251702d708e9c48421b2721938657193370ee7065322e25", + "research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-NATIVE-WORD.json.gz": "57d7c53bacc42c6d403d3f2ced05c7e277ade034a2cb2db9b07e20962cc0b0d6" +} diff --git a/research/kappa-causal-continuation/buffer-borrowing/verify_buffers.py b/research/kappa-causal-continuation/buffer-borrowing/verify_buffers.py new file mode 100644 index 0000000000..5a55034ec9 --- /dev/null +++ b/research/kappa-causal-continuation/buffer-borrowing/verify_buffers.py @@ -0,0 +1,351 @@ +#!/usr/bin/env python3 +"""Independent literal native replay with arbitrary dirty cleanup buffers. + +The retained emitter supplies words; this module imports no retained evaluator. +Free warp symbols distinguish primal and inverse-transpose operators. Physical +phase receipts are authenticated inherited evidence, not regenerated here. +Apache-2.0. Substantial OpenAI assistance. +""" +from collections import Counter, defaultdict +from fractions import Fraction +from hashlib import sha256 +from pathlib import Path +import argparse +import ast +import gzip +import json + + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +CORE = ROOT / "research/kappa-native-resume/native-source/rebased-core" +PACKET = ROOT / "research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz" +TAGS = ("THREE-base", "THREE-extra", "FOUR-base", "FOUR-extra") +Q = Fraction + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def read(path): + raw = path.read_bytes() + return json.loads(gzip.decompress(raw) if path.suffix == ".gz" else raw) + + +def digest(path): + return sha256(path.read_bytes()).hexdigest() + + +def authenticate(): + pins = read(HERE / "SOURCES.json") + for relative, expected in pins.items(): + need(digest(ROOT / relative) == expected, "Changed retained source: " + relative) + return pins + + +def emit_original(canonical): + """Reuse only the pinned source's pure writer; no source verifier executes.""" + source = CORE / "sources/bw_emit.py" + names = {"basis", "reg", "dual", "scalar", "lregs", "expand", "emit", "primitive_bound"} + tree = ast.parse(source.read_text()) + functions = [n for n in tree.body if isinstance(n, ast.FunctionDef) and n.name in names] + need({n.name for n in functions} == names, "Exact writer function inventory") + context = dict(c=canonical, frames=canonical["frames"], v=112, R=408, h=9, n=632, Q=Q) + exec(compile(ast.Module(body=functions, type_ignores=[]), str(source), "exec"), context) + return {"forward": context["emit"](False), "backward": context["emit"](True)} + + +def span(basis): + points = {0} + for vector in basis: + need(isinstance(vector, int) and 0 <= vector < 512, "Nine-dimensional frame") + points |= {v ^ vector for v in tuple(points)} + return frozenset(points) + + +def identity(role): + return {(role, ()): Q(1)} + + +def add(destination, source, coefficient): + for key, value in tuple(source.items()): + value = destination.get(key, Q(0)) + coefficient * value + if value: + destination[key] = value + else: + destination.pop(key, None) + + +def warp(row, operator, backward): + symbol = (1 if operator > 0 else -1) * (abs(operator) + (10**6 if backward else 0)) + out = {} + for (variable, word), coefficient in row.items(): + moved = word[1:] if word and word[0] == -symbol else (symbol,) + word + key = variable, moved + out[key] = out.get(key, Q(0)) + coefficient + return {k: v for k, v in out.items() if v} + + +def expected(role, R, backward, clean): + out = identity(role) + name, i = role + if name != "L": + return out + target = i < 112 if backward else 112 <= i < 224 + if not target: + return out + port = i if backward else i - 112 + source_offset = 112 if backward else 0 + sign = -1 if backward else 1 + add(out, identity(("L", source_offset + port)), Q(sign)) + if clean: + for query in R: + if port in query: + for j, value in query.items(): + add(out, identity(("L", source_offset + j)), -sign * query[port] * value / 4) + return out + + +def replay(program, canonical, R, backward, clean, omit_cleanup_last=False): + """All L inputs and all Z inputs are independent, including dirty outputs.""" + n = 2 * canonical["v"] + canonical["R"] + need(n == 632, "Retained compact width") + rows = {("L", i): identity(("L", i)) for i in range(n)} + current = {("L", i): frame for i, frame in enumerate(canonical["start"])} + center_indices = set() + for event in program: + if event[0] == "scalar": + op = event[1] + for register in op[1:3] if op[0] in ("add", "copy") else op[1:2]: + if register[0] == "C": + center_indices.add(register[1]) + elif event[0] == "center": + center_indices.add(event[1]) + need(center_indices <= set(range(8)), "Known center workspace indices") + for i in center_indices: + rows["C", i] = identity(("C", i)) + current["C", i] = None + spaces = [span(vectors) for vectors in canonical["frames"]] + dimensions = [len(s).bit_length() - 1 for s in spaces] + need(dimensions[1] == 9, "Cleanup uses the full native frame") + live = set() + profile = Counter() + counts = Counter() + scalar_primitives = 0 + cleanup_start = next(i for i, event in enumerate(program) if event == ["phase", "full_frame_cleanup"]) + last_scalar = max(i for i, event in enumerate(program) if event[0] == "scalar" and i > cleanup_start) + for position, event in enumerate(program): + kind = event[0] + counts[kind] += 1 + if kind in ("climb", "center"): + if kind == "climb": + _, part, index, old, new, rank = event + role = "L", index + need(spaces[old] < spaces[new], "Non-nested or empty paid climb") + else: + _, index, old, new, rank = event + role = "C", index + need(spaces[old] < spaces[new] or spaces[new] < spaces[old], "Center residual is not nested") + need(current[role] == old and rank == abs(dimensions[new] - dimensions[old]), "Paid edge source/rank") + current[role] = new + profile[rank] += 1 + elif kind == "zero_buffer": + _, action, index = event + role = "Z", index + if action == "alloc": + need(role not in rows and role not in live, "Cleanup buffer duplicate") + rows[role] = identity(role) + current[role] = 1 + live.add(role) + else: + need(action == "free" and role in live and current[role] == 1, "Cleanup buffer retirement") + live.remove(role) + elif kind == "warp": + _, role, operator, old, new = event + role = tuple(role) + need(current[role] == old, "Warp entrance frame") + rows[role] = warp(rows[role], operator, backward) + current[role] = new + elif kind == "scalar": + op, frame = event[1:] + target = tuple(op[1]) + if op[0] == "clear": + need(target[0] == "C", "A borrowed input was cleared") + rows[target] = {} + current[target] = frame + scalar_primitives += 1 + continue + need(current[target] == frame, "Scalar target frame") + if op[0] in ("add", "copy"): + source = tuple(op[2]) + need(current[source] == frame, "Scalar source frame") + if op[0] == "copy": + need(target[0] == "C", "Copy overwrites borrowed state") + rows[target] = dict(rows[source]) + scalar_primitives += 1 + continue + coefficient = Q(op[3], op[4]) + need(target != source, "Self-ADD is not a reversible shear") + numerator, denominator = abs(coefficient.numerator), coefficient.denominator + need(numerator and denominator & (denominator - 1) == 0, "Non-dyadic ADD") + scalar_primitives += 3 + numerator.bit_length() - 1 + numerator.bit_count() + denominator.bit_length() - 1 + if not (omit_cleanup_last and position == last_scalar): + add(rows[target], rows[source], coefficient) + else: + need(op[0] == "scale", "Unknown scalar event") + coefficient = Q(op[2], op[3]) + numerator, denominator = abs(coefficient.numerator), coefficient.denominator + need(numerator and numerator & (numerator - 1) == 0 and denominator & (denominator - 1) == 0, + "Nonunit dyadic SCALE") + scalar_primitives += abs(numerator.bit_length() - denominator.bit_length()) + int(coefficient < 0) + if not (omit_cleanup_last and position == last_scalar): + rows[target] = {k: coefficient * v for k, v in rows[target].items()} + else: + need(kind in ("phase", "adapter", "residual_adapter"), "Unknown native event") + need(not live and [current["L", i] for i in range(n)] == canonical["final"], "Retired fields/final frames") + need(dict(profile) == {int(k): v for k, v in canonical["local_profile"].items()}, "Complete paid profile") + mismatches = [] + for role, actual in rows.items(): + wanted = {} if role[0] == "C" else expected(role, R, backward, clean) + if actual != wanted: + mismatches.append({"role": list(role), "unequal_coefficients": sum(actual.get(k, 0) != wanted.get(k, 0) for k in set(actual) | set(wanted))}) + if not omit_cleanup_last: + need(not mismatches, "Native dirty-buffer operator mismatch: " + repr(mismatches[:3])) + else: + need(mismatches, "Missing-cleanup negative control unexpectedly accepted") + buffer_count = sum(role[0] == "Z" for role in rows) + return dict(direction="backward" if backward else "forward", clean_J=clean, + independent_borrowed_input_fields=632+buffer_count, original_data_fields=224, + expected_restored_helpers=408, expected_restored_arbitrary_cleanup_buffers=buffer_count, + explicitly_owned_C_fields_cleared=len(center_indices), event_counts=dict(counts), + paid_profile=dict(sorted(profile.items())), paid_rank_mass=sum(k*v for k,v in profile.items()), + conservative_real_scalar_primitive_allowance=2*scalar_primitives, + all_scalar_frames_checked=True, dyadic_coefficients_and_unit_scales_checked=True, + free_warp_operator_endpoint_matches=not mismatches, + negative_control_omitted_last_cleanup_scalar=omit_cleanup_last, + mismatch_examples=mismatches[:3]) + + +def liveness(canonical): + times = defaultdict(list) + for i, event in enumerate(canonical["cleanup"]): + for role in event[1:3] if event[0] == "add" else event[1:2]: + role = tuple(role) if isinstance(role, list) else ("L", role) + times[role].append(i) + full = [i for i, f in enumerate(canonical["final"]) if f == 1] + candidates = [] + for z in canonical["zero_buffers"]: + begin, end = min(times["Z", z]), max(times["Z", z]) + possible = [i for i in range(112) if all(t < begin or t > end for t in times["L", i])] + candidates.append(dict(buffer=z, first_use=begin, last_use=end, + noninterfering_same_call_X_candidates=possible, + noninterfering_full_frame_L_candidates=[i for i in full if all(t < begin or t > end for t in times["L", i])])) + return dict(full_frame_live_roles=len(full), full_frame_roles_unused_in_cleanup=[i for i in full if not times["L", i]], + same_call_X_outputs_touched=sum(bool(times["L", i]) for i in range(112)), + buffers_with_no_noninterfering_same_call_X=sum(not row["noninterfering_same_call_X_candidates"] for row in candidates), + buffers_with_no_noninterfering_full_frame_L=sum(not row["noninterfering_full_frame_L_candidates"] for row in candidates), + intervals=candidates, + conclusion="Arbitrary-buffer restoration does not establish a legal same-call completed-output alias.", + external_same_frame_bank_requirement=91, + external_bank_hosting_constructed=False) + + +def find_aliases(canonical): + """Injective lifetime-safe matching; no native L access during its Z span.""" + options = {row["buffer"]: row["noninterfering_full_frame_L_candidates"] + for row in liveness(canonical)["intervals"]} + owner = {} + + def augment(z, seen): + for role in options[z]: + if role in seen: + continue + seen.add(role) + if role not in owner or augment(owner[role], seen): + owner[role] = z + return True + return False + + for z in sorted(options, key=lambda z: (len(options[z]), z)): + augment(z, set()) + bindings = {z: role for role, z in owner.items()} + need(len(set(bindings.values())) == len(bindings), "Alias must be injective") + need(all(role in options[z] for z, role in bindings.items()), "Interfering alias") + return bindings + + +def alias_program(program, bindings): + """Replace only Z register operands, retaining every native/scalar event.""" + out = [] + + def mapped(role): + return ["L", bindings[role[1]]] if role[0] == "Z" and role[1] in bindings else list(role) + + for event in program: + if event[0] == "zero_buffer" and event[2] in bindings: + continue + if event[0] == "scalar": + op = event[1].copy() + op[1] = mapped(op[1]) + if op[0] in ("add", "copy"): + op[2] = mapped(op[2]) + out.append(["scalar", op, event[2]]) + elif event[0] == "warp": + out.append([event[0], mapped(event[1]), *event[2:]]) + else: + out.append(event) + return out + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--output", type=Path, default=HERE / "RESULT.json") + args = parser.parse_args() + pins = authenticate() + R = [{int(k): Q(v) for k, v in row.items()} for row in read(PACKET)["R_rows"]] + providers = {} + for tag in TAGS: + providers[tag] = {} + for clean in (False, True): + folder = "converted" if clean else "sources" + canonical = read(CORE / folder / (tag + "-CANONICAL.json.gz")) + programs = read(CORE / folder / (tag + "-NATIVE-WORD.json.gz")) if clean else emit_original(canonical) + checks = [replay(programs[direction], canonical, R, direction == "backward", clean) + for direction in ("forward", "backward")] + bindings = find_aliases(canonical) + aliases = {direction: alias_program(program, bindings) for direction, program in programs.items()} + aliased_checks = [replay(aliases[direction], canonical, R, direction == "backward", clean) + for direction in ("forward", "backward")] + need(len(bindings) == 53, "Expected source-bound 53-field lifetime match") + for original, aliased in zip(checks, aliased_checks): + need(original["paid_profile"] == aliased["paid_profile"] and + original["conservative_real_scalar_primitive_allowance"] == aliased["conservative_real_scalar_primitive_allowance"], + "Aliasing changes the charged operation list") + providers[tag]["clean_J" if clean else "ordinary"] = dict(checks=checks, liveness=liveness(canonical), + lifetime_alias_bindings={str(z): role for z, role in sorted(bindings.items())}, + aliased_checks=aliased_checks, + eliminated_cleanup_fields=len(bindings), remaining_cleanup_fields=91-len(bindings), + aliased_word_sha256={direction: sha256(json.dumps(program, sort_keys=True, separators=(",", ":")).encode()).hexdigest() + for direction, program in aliases.items()}) + print("PASS all arbitrary cleanup inputs", tag, flush=True) + canonical = read(CORE / "converted/THREE-base-CANONICAL.json.gz") + programs = read(CORE / "converted/THREE-base-NATIVE-WORD.json.gz") + controls = [replay(programs[direction], canonical, R, direction == "backward", True, True) + for direction in ("forward", "backward")] + result = dict(schema="literal-native-dirty-buffers/1", status="PASS_NATIVE_ARBITRARY_DIRTY_BUFFER_REPLAY", + new_multiplication_kappa=False, physical_phase_generation_rerun=False, + phase_evidence="Pinned inherited exact native phase receipts; unchanged native edges.", + source_hashes=pins, verifier_sha256=digest(Path(__file__)), providers=providers, + negative_controls=controls, + lifetime_safe_local_full_frame_aliases_per_word=53, + remaining_external_bank_fields_per_word=38, + all_91_local_output_alias_claim=False, external_bank_placement_and_cost_closed=False) + args.output.parent.mkdir(parents=True, exist_ok=True) + args.output.write_text(json.dumps(result, indent=2) + "\n") + print(result["status"], flush=True) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-function-composition/README.md b/research/kappa-function-composition/README.md new file mode 100644 index 0000000000..0a8a6c1662 --- /dev/null +++ b/research/kappa-function-composition/README.md @@ -0,0 +1,49 @@ +# Function composition and the complete recursive bill + +Research continuation, 20 October 2026. **No new integer-multiplication κ +is established. All earlier unmerged work is retained.** + +This round makes two constructive advances: a complete, independently +checked BIT continuation through ten outputs, and a cheaper complex clean-J +word. It also derives exact budget limits that require the next complex +advance to change query acquisition, consumer computation, endpoint ownership, +or the recursive interface. Optimizing the present role paths alone cannot +fund the known repair at either target. + +Start with [REPORT.md](REPORT.md). Detailed proofs and reproducible receipts: + +| Scope | Main result | Evidence | +|---|---|---| +| BIT source functions | Complete ninth output; exact optimum within the declared forest family; staged 8→9→10 with restored originals | [bit/PROOF.md](bit/PROOF.md), [COMPLETE-NINTH.json](bit/COMPLETE-NINTH.json), [STAGED-CONTINUATION.json](bit/STAGED-CONTINUATION.json) | +| Deferred cross-pair sharing | Two future outputs read the same twelve functions at different times; 153 helpers and 21,782 ADDs | [DEFERRED-CONTINUATION.json](bit/DEFERRED-CONTINUATION.json), [DEFERRED-INDEPENDENT.json](literature/DEFERRED-INDEPENDENT.json) | +| Independent BIT audits | Both signs, arbitrary inputs and dirt, every cut, separate Bellman enumeration | [NINTH-INDEPENDENT.json](literature/NINTH-INDEPENDENT.json), [STAGED-INDEPENDENT.json](literature/STAGED-INDEPENDENT.json) | +| Complex native pruning | 28 matched compute/restore deletions; 31 fewer paid child calls; 448 fewer scalar primitives | [pruned-native/RESULT.json](complex/pruned-native/RESULT.json), [PRUNED-NATIVE-AUDIT.json](invoice/PRUNED-NATIVE-AUDIT.json) | +| Complex mixed producer | Exact rank-six query; complete partial repair emitted and rejected on recursive cost | [mixed-contact/RESULT.json](complex/mixed-contact/RESULT.json), [MIXED-NATIVE-AUDIT.json](invoice/MIXED-NATIVE-AUDIT.json) | +| Actual timed consumer obstruction | All nearby staggered contacts fail; a simple physical output-coordinate witness is independently checked at every cut | [staggered-consumer/RESULT.json](complex/staggered-consumer/RESULT.json), [CONSUMER-WITNESS-AUDIT.json](invoice/CONSUMER-WITNESS-AUDIT.json) | +| Affordable two-frame routes | Complete finite tree dictionary rejected by physical projected coordinate witnesses | [two-frame-consumer/RESULT.json](complex/two-frame-consumer/RESULT.json), [its proof](complex/two-frame-consumer/PROOF.md) | +| Complete budget | Perfect fixed-role coalescing cannot fund 896-H1 repair; optimistic mixed-job extension still fails | [invoice/PROOF.md](invoice/PROOF.md), [ENDPOINT-BUDGET.json](invoice/ENDPOINT-BUDGET.json), [MIXED-JOB-SCREEN.json](invoice/MIXED-JOB-SCREEN.json) | +| Global BIT alternative | Exact F3 rank and rectangle screens; independent persistent pair banks fail .004 under stated stock relaxation | [literature/PROOF.md](literature/PROOF.md), [FIXED-PAIR-SCREEN.json](literature/FIXED-PAIR-SCREEN.json) | +| Literature and application | Classic and recent primary sources, with explicit model mismatches and finite gates | [literature/LITERATURE.md](literature/LITERATURE.md), [bit/LITERATURE.md](bit/LITERATURE.md), [UNCOMPUTATION-LITERATURE.md](research-notes/UNCOMPUTATION-LITERATURE.md) | + +From the repository root, run: + +```sh +python3 -B research/kappa-function-composition/validate.py +``` + +The validator authenticates the unchanged 74-stage predecessor coverage, +reconstructs a private isolated tree, removes every declared new generated +artifact, runs its generator, and compares complete proof payloads. It does +not rerun unchanged predecessor stages or claim a full selected cold replay. +The final [VALIDATION.json](VALIDATION.json) records the actual completed +stages, comparisons, and environment. [INTEGRITY.json](INTEGRITY.json) binds +all new source and evidence bytes and modes. + +The cumulative delivery also tests application against the attached base ZIP +and tests the complete unmerged supplement review patch. Its package receipt +is separate from the mathematical replay. The frozen predecessor is +`research/kappa-global-routing/`; it is not edited by this scope. + +Apache-2.0; substantial OpenAI assistance. Computational checks are not +represented as independent human review. See [UNMERGED-STATE.json](UNMERGED-STATE.json) +for continuity and admission status. diff --git a/research/kappa-function-composition/complex/README.md b/research/kappa-function-composition/complex/README.md new file mode 100644 index 0000000000..36cd18251f --- /dev/null +++ b/research/kappa-function-composition/complex/README.md @@ -0,0 +1,168 @@ +# Complex function composition: an actual paid improvement and a changed query producer + +Apache-2.0; substantial OpenAI assistance. All predecessor packages remain +unchanged. This directory contains new finite constructions and explicitly +scoped negative certificates. It does not establish a new multiplication κ. + +## Retain the native pruning + +The strongest immediately reusable result is `pruned-native/RESULT.json`. +Deleting 28 exactly unconsumed centre-assembly operations together with their +scheduled inverses preserves the complete clean-J operator in all four +providers and both orientations. The independent literal replay treats all +632 persistent inputs as arbitrary, including all 408 helper inputs. It +checks their restoration, legal scalar frames and dyadic coefficients. + +Write H_r(C)=r^(1-C). The paid profiles change as follows; base and extra +variants agree. + +| Provider | Old profile | New profile | Rank mass | +|---|---|---|---:| +| THREE | 2487 H1 + 713 H2 + 224 H3 + 102 H4 + 20 H5 | 2453 H1 + 716 H2 + 224 H3 + 74 H4 + 48 H5 | 5093 | +| FOUR | 2489 H1 + 713 H2 + 227 H3 + 99 H4 + 20 H5 | 2455 H1 + 716 H2 + 227 H3 + 71 H4 + 48 H5 | 5092 | + +The exact saving per core call is + +\[ +34H_1-3H_2+28H_4-28H_5. +\] + +There are 31 fewer recursive calls, but no rank-mass reduction. At the two +diagnostic primitive values C=.001745 and C=.004, the saving is approximately +0.1292770134 and 0.2956617046 H1 per call, respectively. The overall invoice +and its much larger remaining repair gap are handled in +[`../invoice/PROOF.md`](../invoice/PROOF.md). This local gain must not be +reported as a 31-unit recursive saving or as a new κ. + +The deletion also removes 56 ADDs, reducing the conservative scalar primitive +allowance by 448 to 115382 for THREE and 115360 for FOUR, in either direction. +The word uses the same 632 persistent fields, zero external cleanup fields, +and zero owned centre fields. Exact physical recipes are emitted and decoded +for every newly coalesced child edge, including the global phase and both +orientations. + +The prefix bounds do not increase: forward formal row L1 is at most 457, +with maximum coefficient magnitude 32 and denominator 32; backward L1 is at +most 258, with maximum coefficient magnitude 6 and denominator 64. These are +exact logical-prefix bounds, not a rounded recursive-error proof. + +**Interface limitation:** the complete endpoint operator is preserved, but +the old eight final centre queries are not. A caller using only the clean-J +endpoint can use this replacement. A caller reading historical intermediate +centre queries must supply and price a new producer. This distinction is +recorded in the machine receipt. + +## The 24+32 contact witness now has a native word + +The previous geometric witness has been turned into an actual chronological +construction for query R3, on THREE-base. It forms a 24-coordinate partial +query in an existing rank-seven frame and a 32-coordinate partial query in a +new rank-seven frame. Five unit-coefficient preparation ADDs and five inverse +ADDs are explicitly emitted. The two partials sum to the complete 56-coordinate +query plus its exact dirty-helper nuisance. The nuisance has 45 nonzero helper +coefficients after cancellation. + +Both the reference and candidates implement the same **partial repair**: + +\[ +Y\gets Y+\left(I-\frac14\sum_{k\ne3}R_k^T R_k\right)X, +\] + +with X and every helper restored. The backward word is its inverse transpose. +This comparison tests one complete query acquisition and its actual consumer; +it is not a completed eight-query ordinary-response controller. + +| Candidate | Additional owned fields | Candidate minus the one-H8 reference | Scalar allowance, forward / backward | Decision | +|---|---:|---|---:|---| +| One-centre reference | 1 | 0 | 127628 / 127634 | Exact reference | +| Two independent rank-seven descents | 1, recycled | 5(H3+H1−H4)+2H7−H8 | 127978 / 127990 | Exact, costs more | +| Merge at the rank-six intersection | 2 | 5(H3+H1−H4)+2H1+H6−H8 | 127722 / 127728 | Exact, still costs more | + +All fields marked owned are cleared; no external cleanup fields are used. +Both candidates include the 990 zero-frame nuisance-compensation ADDs required +by this partial repair. Every changed frame transition is physically lowered +and phase checked. An omitted-compensation negative control fails. + +The second candidate is a real change of premise: the two rank-seven frames +intersect in + +```text +G6 = span([257, 129, 65, 39, 20, 10]) over F2. +``` + +Two rank-one descents produce a complete R3 query at G6, after which the two +carriers can be added. `mixed-contact/RESULT.json` verifies the actual live +merged row, including all dirty-helper coefficients. This escapes the old +fixed-history one-shot-support restrictions. Its existing consumer requires a +further rank-six descent, so the complete bill still loses to H8. + +## The consumer has now been tested across time + +`mixed_contact.py` transports the original demanded consumer function through +every cut of the unchanged B, tail and dirty cleanup. At each cut, it prices a +restricted delivery graph carrying scalar copies of the already formed query. +There are 10646 inspected post-event cuts: 2674 require a frame at distance at +least six from G6; the other 7972 require at least 28 distinct frames. The +necessary delivery costs exceed H8 once the producer's 2H1 formation is +included. Exact Clifford tableaus verify the distance formula on all 1155 +catalogued frames. See the precise free-warp and scalar-copy qualifications +in [`PROOF.md`](PROOF.md). + +`staggered_consumer.py` then removes the one-cut restriction. It grants scalar +injections at every register and every chronological cut, all subsequent +unchanged operations, unlimited copies and their distribution for free. + +| Maximum distance from G6 | Existing frames with contacts | Unique projected impulse columns | Exact projected rank | Desired function in span? | +|---:|---:|---:|---:|---| +| 0 | 0 | 0 | 0 | No | +| 1 | 1 | 25 | 9 | No | +| 2 | 14 | 145 | 58 | No | +| 3 | 80 | 5286 | 533 | No | +| 4 | 351 | 6738 | 632 | Yes, necessary projection only | +| 5 | 726 | 8591 | 632 | Yes, necessary projection only | + +This projection is physical: every actual warp and its inverse fixes the +512-dimensional basis vector |0>, including phase. It therefore respects all +identities among the actual operators. The distance-at-most-three rejection +has a one-coordinate witness: every admitted impulse has zero final value +at Y[3], while the demanded R3 function has coefficient 1/4 there. All 726 +individual frames within distance five also have exact separating witnesses. + +Thus staggered scalar delivery inside these nearby frames fails for a specific +output reason. A successful candidate in this model must use multiple frames +and at least one at distance four or greater. Distance-four feasibility is not +a native circuit, and the projected calculation does not guarantee dyadic +coefficients, chronological distribution, restoration or a lower bill. + +## Next discriminating construction + +The next bounded test is a small dyadic set of distance-four contact frames +whose complete impulse operators realize the demand, followed by an explicit +chronological delivery tree. It must charge the two rank-one formation calls, +every frame transition, source preparation and return, dirty compensation, +and owned-field hosting. The current construction has less than H8 available +for this complete acquisition. If that menu fails, the consumer trajectory +must change enough to create a function-spanning G6 or nearer contact, or +several queries must share a different demanded function. Merely moving the +old complete scatter to another single cut cannot use this producer cheaply. + +## Reproduce + +From the repository root, using standard-library Python 3: + +```sh +python3 -B research/kappa-function-composition/complex/prune_native.py +python3 -B research/kappa-function-composition/complex/contact_native.py +python3 -B research/kappa-function-composition/complex/mixed_contact.py +python3 -B research/kappa-function-composition/complex/staggered_consumer.py +``` + +Each program writes only this new directory, authenticates frozen inputs and +records artifact hashes. The second, third and fourth commands are sequential +dependencies. For a read-only replay of preserved receipts, give `--out` a new +directory and use the cumulative root validation procedure. The native +checker and Clifford compiler are reused from attributed, pinned predecessors; +the new construction and search code do not replace those predecessors. + +Literature selection, exact model differences and the associated probes are +recorded in [`PRIMARY-LITERATURE.md`](PRIMARY-LITERATURE.md). diff --git a/research/kappa-function-composition/complex/prune_native.py b/research/kappa-function-composition/complex/prune_native.py new file mode 100644 index 0000000000..8c3e11b240 --- /dev/null +++ b/research/kappa-function-composition/complex/prune_native.py @@ -0,0 +1,253 @@ +#!/usr/bin/env python3 +"""Delete exact unconsumed centre-assembly/inverse pairs, including dirty state. + +Apache-2.0; substantial OpenAI assistance. The native writer and final verifier +are reused with attribution from the frozen causal-continuation package. The +new search checks literal rational perturbations, never numerical similarity. +""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import importlib.util +import json +import sys +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +RESEARCH = HERE.parents[1] +BASE = RESEARCH / 'kappa-global-routing/integrated-complex' +CHECKER = RESEARCH / 'kappa-causal-continuation/buffer-borrowing/verify_buffers.py' +CHECKER_SHA = '106b4d5e052427230df452f0c901d5390a0583782960b80a8af94d9a984022c7' +CLIFFORD = RESEARCH / 't74-native-fast/vendor/native/new/code/borrow/vendor/clifford/clifford_exact.py' +TAGS = ('THREE-base', 'THREE-extra', 'FOUR-base', 'FOUR-extra') + + +def need(ok, why): + if not ok: raise ValueError(why) + + +def digest(path): return sha256(path.read_bytes()).hexdigest() + + +def read(path): + data = path.read_bytes() + return json.loads(gzip.decompress(data) if path.suffix == '.gz' else data) + + +def save(path, value): + data = json.dumps(value, sort_keys=True, separators=(',', ':')).encode() + path.write_bytes(gzip.compress(data, mtime=0)) + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + obj = importlib.util.module_from_spec(spec) + sys.modules[name] = obj + spec.loader.exec_module(obj) + return obj + + +def apply(v, rows, event, backward=False): + if event[0] == 'warp': + r = tuple(event[1]); rows[r] = v.warp(rows[r], event[2], backward) + elif event[0] == 'scalar': + g = event[1]; r = tuple(g[1]) + if g[0] == 'add': v.add(rows[r], rows[tuple(g[2])], Q(g[3], g[4])) + else: + need(g[0] == 'scale', 'No owned field in the clean-J baseline') + rows[r] = {k: Q(g[2], g[3])*x for k, x in rows[r].items()} + + +def index_sections(program): + phase = None; result = {'A': [], 'full_frame_cleanup': []} + for i, e in enumerate(program): + if e[0] == 'phase': phase = e[1] + if phase in result and e[0] in ('scalar', 'warp'): result[phase].append(i) + return result + + +def exact_candidate_screen(v, canonical, program, R): + """Test the final 48 assembly gates, with their actual scheduled inverses.""" + indices = index_sections(program) + need(len(indices['A']) == len(canonical['A']), 'Native A gate correspondence') + need(len(indices['full_frame_cleanup']) == len(canonical['cleanup']), 'Cleanup gate correspondence') + options = []; unmatched = [] + for j in range(len(canonical['A']) - 48, len(canonical['A'])): + g = canonical['A'][j] + need(g[0] == 'add' and len(canonical['frames'][g[-1]]) in (7, 8), 'Final centre-assembly grammar') + inverse = g[:]; inverse[3] *= -1; inverse[-1] = 1 + hits = [i for i, h in enumerate(canonical['cleanup']) if h == inverse] + if len(hits) == 1: + cj = hits[0] + options.append((j, cj, indices['A'][j], indices['full_frame_cleanup'][cj])) + else: unmatched.append({'A_index': j, 'gate': g, 'matching_inverse_occurrences': hits}) + sites = {k for option in options for k in option[2:]} + effects = {}; rows = {('L', i): v.identity(('L', i)) for i in range(632)} + for i, e in enumerate(program): + if i in sites: + g = e[1] + effects[i] = (tuple(g[1]), {k: -Q(g[3], g[4])*x for k, x in rows[tuple(g[2])].items()}) + apply(v, rows, e) + need(all(row == v.expected(r, R, False, True) for r, row in rows.items()), 'Pinned baseline endpoint') + accepted = []; rejected = [] + for aj, cj, ai, ci in options: + delta = {('L', i): {} for i in range(632)} + live_data_effect = False + for i in range(ai, len(program)): + if i in (ai, ci): + target, effect = effects[i]; v.add(delta[target], effect, Q(1)) + else: apply(v, delta, program[i]) + if i < indices['full_frame_cleanup'][0]: + e = program[i] + if e[0] == 'scalar' and e[1][0] in ('add', 'scale'): + r = tuple(e[1][1]) + if r[0] == 'L' and 112 <= r[1] < 224 and delta[r]: live_data_effect = True + bad = [(r, row) for r, row in delta.items() if row] + item = {'A_index': aj, 'cleanup_index': cj, 'A_gate': canonical['A'][aj], + 'cleanup_gate': canonical['cleanup'][cj], + 'native_scalar_sites': [ai, ci], 'Y_difference_before_cleanup': live_data_effect} + if not bad: + item['all_632_arbitrary_input_columns_exactly_preserved'] = True + accepted.append(item) + else: + r, row = bad[0]; key, value = next(iter(sorted(row.items()))) + item['first_exact_nonzero_residual'] = {'output': list(r), 'input': list(key[0]), + 'operator_word': list(key[1]), 'coefficient': str(value)} + item['changed_output_fields'] = len(bad) + rejected.append(item) + need(all(not a['Y_difference_before_cleanup'] for a in accepted), 'No changed data consumer before cleanup') + return accepted, rejected, unmatched + + +def prefix_growth(v, program, backward): + rows = {('L', i): v.identity(('L', i)) for i in range(632)} + maximum_l1 = Q(1); max_coefficient = Q(1); max_numerator = 1; max_denominator = 1 + witness = {'event': -1, 'register': ['L', 0]} + for i, event in enumerate(program): + apply(v, rows, event, backward) + if event[0] not in ('scalar', 'warp'): continue + role = tuple(event[1] if event[0] == 'warp' else event[1][1]) + values = [abs(x) for x in rows[role].values()] + norm = sum(values, Q(0)) + if norm > maximum_l1: + maximum_l1 = norm; witness = {'event': i, 'register': list(role)} + if values: + max_coefficient = max(max_coefficient, max(values)) + max_numerator = max(max_numerator, max(x.numerator for x in values)) + max_denominator = max(max_denominator, max(x.denominator for x in values)) + return {'max_formal_row_l1': str(maximum_l1), 'max_abs_coefficient': str(max_coefficient), + 'max_abs_numerator': max_numerator, 'max_denominator': max_denominator, + 'l1_witness': witness, 'independent_input_columns': 632, + 'scope': 'Exact logical scalar/monomial prefixes over arbitrary inputs; no rounded child error bound.'} + + +def monomial(ce, recipe, n): + result = ce.diag(n, tuple(recipe['quadratic_rows']), tuple(recipe['linear_Z4'])) + result = result.then(ce.perm(n, tuple(recipe['matrix_rows']))).then(ce.xgate(n, recipe['offset'])) + return result.then(ce.Op(n, ce.identity(n), [('phase', recipe['phase'])])) + + +def conjugate(ce, op): + n = op.n; result = ce.ident(n) + for g in op.circuit: + if g[0] == 'P': part = ce.perm(n, g[1]) + elif g[0] == 'X': part = ce.xgate(n, g[1]) + elif g[0] == 'C': part = ce.cgate(n, g[1], -g[2]) + elif g[0] == 'D': part = ce.diag(n, g[1], tuple(-x % 4 for x in g[2])) + else: + need(g[0] == 'phase', 'Known Clifford primitive') + part = ce.Op(n, ce.identity(n), [('phase', -g[1] % 4)]) + result = result.then(part) + return result + + +def check_new_edges(canonical, old_program, new_program, cache): + ce = load(CLIFFORD, 'function_composition_clifford') + old = {(e[3], e[4]) for e in old_program if e[0] == 'climb'} + new = {(e[3], e[4]) for e in new_program if e[0] == 'climb'} + reports = [] + for a, b in sorted(new - old): + A, B = canonical['frames'][a], canonical['frames'][b] + key = tuple(A), tuple(B) + if key not in cache: + source = ce.canonical_frame(9, A).adjoint().then(ce.canonical_frame(9, B)) + records = {} + for name, op in (('forward', source), ('inverse_transpose', conjugate(ce, source))): + recipe, _ = ce.lower_child(op) + need(recipe['rank'] == len(B)-len(A), 'Exact new physical child rank') + decoded = monomial(ce, recipe['right'], 9).then(ce.fourier(9, range(recipe['rank']))).then(monomial(ce, recipe['left'], 9)) + need(decoded.tab == op.tab and decoded.zero_state() == op.zero_state(), 'Decoded Clifford equality including global phase') + records[name] = recipe + cache[key] = records + reports.append({'old_frame': a, 'new_frame': b, 'old_basis': A, 'new_basis': B, + 'rank': len(B)-len(A), 'exact_physical_recipes': cache[key]}) + return reports + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out', type=Path, default=HERE/'pruned-native') + args = parser.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + need(digest(CHECKER) == CHECKER_SHA, 'Frozen independent checker') + v = load(CHECKER, 'function_composition_independent'); v.authenticate() + base_receipt = read(BASE/'RESULT.json') + R = [{int(k): Q(x) for k, x in row.items()} for row in v.read(v.PACKET)['R_rows']] + providers = {}; phase_cache = {} + for tag in TAGS: + path = BASE/(tag+'-CANONICAL.json.gz') + need(digest(path) == base_receipt['providers'][tag]['artifacts'][path.name], 'Pinned baseline canonical') + old = read(path); old_native = v.emit_original(old) + yes, no, unmatched = exact_candidate_screen(v, old, old_native['forward'], R) + candidate = deepcopy(old) + a_drop = {r['A_index'] for r in yes}; c_drop = {r['cleanup_index'] for r in yes} + candidate['A'] = [g for i, g in enumerate(old['A']) if i not in a_drop] + candidate['cleanup'] = [g for i, g in enumerate(old['cleanup']) if i not in c_drop] + native = v.emit_original(candidate) + histogram = Counter(e[5] for e in native['forward'] if e[0] == 'climb') + candidate['local_profile'] = {str(r): n for r, n in sorted(histogram.items())} + candidate['edges'] = [list(e) for e in sorted({(e[3], e[4]) for e in native['forward'] if e[0] == 'climb'})] + checks = {} + for name in ('forward', 'backward'): + checks[name] = {'independent_literal_replay': v.replay(native[name], candidate, R, name == 'backward', True), + 'baseline_prefix_growth': prefix_growth(v, old_native[name], name == 'backward'), + 'new_prefix_growth': prefix_growth(v, native[name], name == 'backward')} + phases = check_new_edges(candidate, old_native['forward'], native['forward'], phase_cache) + # Removing the computation while retaining its inverse must be rejected. + negative = deepcopy(candidate) + first = yes[0] + negative['cleanup'].insert(0, first['cleanup_gate']) + bad_program = v.emit_original(negative) + rejected_control = False + try: v.replay(bad_program['forward'], negative, R, False, True) + except ValueError as error: + rejected_control = 'operator mismatch' in str(error) + need(rejected_control, 'Unpaired deletion negative control') + cp = args.out/(tag+'-CANONICAL.json.gz'); np = args.out/(tag+'-NATIVE-WORD.json.gz') + save(cp, candidate); save(np, native) + providers[tag] = {'baseline_canonical_sha256': digest(path), 'candidate_pairs_tested': len(yes)+len(no), + 'accepted_pairs': yes, 'rejected_pairs': no, 'unmatched_pairs': unmatched, + 'old_paid_profile': old['local_profile'], 'new_paid_profile': candidate['local_profile'], + 'old_paid_rank_mass': sum(int(r)*n for r, n in old['local_profile'].items()), + 'new_paid_rank_mass': sum(int(r)*n for r, n in candidate['local_profile'].items()), + 'recursive_calls_removed': sum(old['local_profile'].values())-sum(histogram.values()), + 'scalar_ADDs_removed': 2*len(yes), 'checks': checks, + 'new_physical_edges': phases, 'unpaired_deletion_negative_control_rejected': True, + 'artifacts': {cp.name: digest(cp), np.name: digest(np)}} + print('PASS pruned native', tag, len(yes), 'pairs;', dict(sorted(histogram.items())), flush=True) + result = {'schema': 'consumer-driven-clean-J-native-pruning/1', + 'status': 'PASS_ALL_FOUR_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS', + 'generator_sha256': digest(Path(__file__)), 'baseline_receipt_sha256': digest(BASE/'RESULT.json'), + 'independent_interpreter_sha256': CHECKER_SHA, 'phase_compiler_path': str(CLIFFORD.relative_to(RESEARCH.parent)), + 'phase_compiler_sha256': digest(CLIFFORD), 'providers': providers, + 'native_word_new_children_phase_lowered': True, 'local_data_and_helper_endpoints_unchanged': True, + 'same_632_field_inventory': True, 'zero_owned_fields': True, 'zero_external_cleanup_fields': True, + 'no_new_kappa_claimed': True, 'query_producer_contract_preserved': False, + 'scope': 'Exact ideal clean-J local endpoints only. Final historical query assemblies are partially removed because this local word has no centre consumers. A caller that reads those historical queries cannot substitute this word without reconstructing and pricing its producer interface. The full caller and rounded induction remain open.'} + (args.out/'RESULT.json').write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(result['status'], flush=True) + + +if __name__ == '__main__': main() diff --git a/research/kappa-function-composition/complex/pruned-native/FOUR-base-CANONICAL.json.gz b/research/kappa-function-composition/complex/pruned-native/FOUR-base-CANONICAL.json.gz new file mode 100644 index 0000000000..1d23783dbb Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/FOUR-base-CANONICAL.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/FOUR-base-NATIVE-WORD.json.gz b/research/kappa-function-composition/complex/pruned-native/FOUR-base-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..8efa57f8e0 Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/FOUR-base-NATIVE-WORD.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/FOUR-extra-CANONICAL.json.gz b/research/kappa-function-composition/complex/pruned-native/FOUR-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..dd2ad305b6 Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/FOUR-extra-CANONICAL.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/FOUR-extra-NATIVE-WORD.json.gz b/research/kappa-function-composition/complex/pruned-native/FOUR-extra-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..8efa57f8e0 Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/FOUR-extra-NATIVE-WORD.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/RESULT.json b/research/kappa-function-composition/complex/pruned-native/RESULT.json new file mode 100644 index 0000000000..8ad4ed8445 --- /dev/null +++ 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Final historical query assemblies are partially removed because this local word has no centre consumers. A caller that reads those historical queries cannot substitute this word without reconstructing and pricing its producer interface. The full caller and rounded induction remain open.", + "status": "PASS_ALL_FOUR_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS", + "zero_external_cleanup_fields": true, + "zero_owned_fields": true +} diff --git a/research/kappa-function-composition/complex/pruned-native/THREE-base-CANONICAL.json.gz b/research/kappa-function-composition/complex/pruned-native/THREE-base-CANONICAL.json.gz new file mode 100644 index 0000000000..ef65766aaa Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/THREE-base-CANONICAL.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/THREE-base-NATIVE-WORD.json.gz b/research/kappa-function-composition/complex/pruned-native/THREE-base-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..5db603d2dc Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/THREE-base-NATIVE-WORD.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/THREE-extra-CANONICAL.json.gz b/research/kappa-function-composition/complex/pruned-native/THREE-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..bc62bd984c Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/THREE-extra-CANONICAL.json.gz differ diff --git a/research/kappa-function-composition/complex/pruned-native/THREE-extra-NATIVE-WORD.json.gz b/research/kappa-function-composition/complex/pruned-native/THREE-extra-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..5db603d2dc Binary files /dev/null and b/research/kappa-function-composition/complex/pruned-native/THREE-extra-NATIVE-WORD.json.gz differ diff --git a/research/kappa-global-routing/README.md b/research/kappa-global-routing/README.md new file mode 100644 index 0000000000..d7f5e5ffa9 --- /dev/null +++ b/research/kappa-global-routing/README.md @@ -0,0 +1,70 @@ +# Mutable function routing and scheduled complex cleanup + +**Verified local improvements and explicit next interfaces; no new multiplication κ.** + +The complete account is [REPORT.md](REPORT.md). This scope carries forward all +earlier unmerged research without changing frozen predecessor files. + +The main positive results are: + +- An h19 eight-output BIT producer with **76 dirty helpers and 16,875 ADDs**, + compared with 10,228 helpers and 50,500 ADDs for its uncoded joined-tree + comparator. It groups sources by the function needed at the consumer. +- A complete local complex word with **all 91 cleanup roles borrowed from + existing registers**, no external cleanup bank, and all 408 persistent + helpers restored. A paid reversible helper basis reduces the full scalar + allowance to **115,830 THREE / 115,808 FOUR**, both directions. The recursive + child profiles remain unchanged. +- Explicit source/contact probes that distinguish real ways to change the + prior model from retuning a rejected fixed-history route. Their native + construction and whole-supplier costs remain separate obligations. + +## Evidence map + +| Directory or file | Scope | +|---|---| +| [bit/README.md](bit/README.md) | Joined output trees, mutable source-pattern sums, bounded scaling, and source-interface probes | +| [complex/README.md](complex/README.md) | Paid helper basis, stateful observation cuts and changed-contact geometry | +| [cleanup/PROOF.md](cleanup/PROOF.md) | Joint instruction scheduling and borrowing, with exact interval certificates | +| [integrated-complex/RESULT.json](integrated-complex/RESULT.json) | Combined full native words, four providers and both directions, 632 arbitrary inputs | +| [literature-audit/LITERATURE-AND-AUDIT.md](literature-audit/LITERATURE-AND-AUDIT.md) | Primary literature, independently reconstructed checks, limitations and next tests | +| [SOURCE-BASELINE.json](SOURCE-BASELINE.json) | Public baseline, exact guidance hashes and scope change | +| [VALIDATION.json](VALIDATION.json) | Fresh isolated replay and source-bound reuse of unchanged prior coverage | + +## Reproduce + +Apply the cumulative overlay to the supplied compact base first. From the +resulting repository root: + +```sh +python3 -B research/kappa-global-routing/validate.py \ + --output kappa-global-routing-validation.json +``` + +Python 3 and NumPy are required. The driver authenticates this complete scope, +the preceding causal checkpoint, and the two older closed dependency +inventories. It runs all new stages in an isolated copy, removing their outputs +before regeneration and comparing complete canonical proof payloads afterward. +It records compression-container byte identity separately from mathematical +payload equality. + +The prior 55-stage receipt is authenticated and reused with unchanged source +bytes. Those stages are not counted as newly run. This is a research-module +replay, not a selected-result cold replay or a complete multiplication theorem. +Individual generators and audits are documented in their respective folders. + +## Open next questions + +The immediate κ-relevant tests are a BIT source producer that can legally +supply a second output block, and a complex contact at one of the new joins +that eliminates more recursive cost than its split source and carrier paths +add. Every proposed word must restore arbitrary borrowed contents and price +all source, output and helper transitions. The report gives the exact existing +allowances and stop conditions. + +Keep the combined complex word as the implementation baseline for caller +ownership and precision work. Further scalar reductions need a concrete +caller purpose; they do not change the recursive histogram by themselves. + +Apache-2.0. Substantial OpenAI assistance. All predecessor attribution, +source-specific notices and immutable archives remain preserved. diff --git a/research/kappa-global-routing/integrated-complex/FOUR-base-CANONICAL.json.gz b/research/kappa-global-routing/integrated-complex/FOUR-base-CANONICAL.json.gz new file mode 100644 index 0000000000..fdb63a12fb Binary files /dev/null and b/research/kappa-global-routing/integrated-complex/FOUR-base-CANONICAL.json.gz differ diff --git a/research/kappa-global-routing/integrated-complex/FOUR-extra-CANONICAL.json.gz b/research/kappa-global-routing/integrated-complex/FOUR-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..9b198b231c Binary files /dev/null and b/research/kappa-global-routing/integrated-complex/FOUR-extra-CANONICAL.json.gz differ diff --git a/research/kappa-global-routing/integrated-complex/RESULT.json b/research/kappa-global-routing/integrated-complex/RESULT.json new file mode 100644 index 0000000000..c89da2e5fa --- /dev/null +++ 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compensation and scheduled cleanup composed and checked together. Existing phase kernels are inherited. This is not a whole-supplier or all-size multiplication admission.", + "status": "PASS_ALL_FOUR_PROVIDERS_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS" +} diff --git a/research/kappa-global-routing/integrated-complex/THREE-base-CANONICAL.json.gz b/research/kappa-global-routing/integrated-complex/THREE-base-CANONICAL.json.gz new file mode 100644 index 0000000000..019d216da3 Binary files /dev/null and b/research/kappa-global-routing/integrated-complex/THREE-base-CANONICAL.json.gz differ diff --git a/research/kappa-global-routing/integrated-complex/THREE-extra-CANONICAL.json.gz b/research/kappa-global-routing/integrated-complex/THREE-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..e45526ea8f Binary files /dev/null and b/research/kappa-global-routing/integrated-complex/THREE-extra-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/README.md b/research/kappa-native-resume/README.md new file mode 100644 index 0000000000..c92a4497b2 --- /dev/null +++ b/research/kappa-native-resume/README.md @@ -0,0 +1,32 @@ +# Kappa native continuation + +This cumulative research continuation adds an exact cheaper clean-J core +rebase, a padded shared-helper cover with full-macro connector fusion, +causal scalar simplification, changed QQ16 geometry probes, and exact +native-cost/minrank target gates. **No new multiplication κ is admitted.** + +Read [REPORT.md](REPORT.md) for the findings, quantitative limits and next +implementation targets. [RECOVERY.md](RECOVERY.md) states exactly which +prior archive bytes were available and the known later-archive recovery gap. + +## Replay + +```sh +python3 research/kappa-native-resume/validate.py \ + --output /absolute/path/to/fresh-native-validation.json +``` + +The scoped replay needs Python 3.12, NumPy and Node.js. It authenticates +the delivered files and runs from an isolated copy. It does not invoke +network access or the historical full release engine. The stage-level +claims are enumerated in `REPLAY-PLAN.json`. + +`INTEGRITY.json` authenticates source and reference results. The generated +`VALIDATION.json` binds the actual fresh run to that manifest. The enclosing +cumulative overlay has its own full archive/path/blob manifest and safe +apply procedure. + +The literature links and their scoped use are in `LITERATURE.json`. +Retained sources are unmodified and pinned in their adjacent provenance +records. New work includes OpenAI assistance and follows the repository's +Apache-2.0 context. diff --git a/research/kappa-native-resume/causal-controller/README.md b/research/kappa-native-resume/causal-controller/README.md new file mode 100644 index 0000000000..9010d96a64 --- /dev/null +++ b/research/kappa-native-resume/causal-controller/README.md @@ -0,0 +1,363 @@ +# QQ16 causal-controller continuation + +This component provides two concrete native query acquisitions, a simpler +104-port producer contract, and an exact reason the current pair's joint +records cannot be compressed to eleven fields. It establishes no new κ. + +The most useful integration development is the **compatible cheaper clean-J +core** proved in the sibling [rebased-core component](../native-source/rebased-core/PROOF.md). +That removes an inherited source mismatch: the original QQ16 packet uses a +5522-mass,480-helper core, whereas the retained mixed supplier can now provide +the same clean-J data boundary at mass5093 or5092 with408 helpers. Its final +source receipt is pinned at `f255c40fccad74eaaa8022da5524f8a3329626e41f20e596ee06e60327cddf15`. +The remaining +join is the changed caller, observations, records, and global hosting. + +All new probe code used OpenAI assistance and is Apache-2.0. Frozen source +documents and code retain their notices; the mixed supplier's credits are in +[its copied notice](sources/baseline-mixed/NOTICE.md). This directory preserves +the old packet and reports conservative comparisons explicitly rather than +silently assigning the cheaper supplier's invoice to a different native word. + +## Replay and files + +Python3.12 and its standard library are sufficient. From this directory run: + +```sh +python replay.py --output REPLAY-RESULT.json +``` + +The runner regenerates each receipt and literal word in a temporary directory +and compares their bytes with the saved artifacts. Each program also has a +standalone CLI: + +| Program | Main receipt | What it checks | +|---|---|---| +| `producer_query_probe.py` | `RESULT.json` | Exact selected-X snapshot and a/query acquisition | +| `core_frame_tap_probe.py` | `CORE-TAP-RESULT.json` | Completed-core observations cannot expose the clean Y query | +| `macro_commutator_probe.py` | `MACRO-COMMUTATOR-RESULT.json` | Whole-macro phase cancellation failures | +| `delete_A_geometry_probe.py` | `DELETE-A-RESULT.json` | Changed104-port scalar map and current-output inverse | +| `deleted_A_native_probe.py` | `DELETE-A-NATIVE-RESULT.json` | Literal native104 wrappers and conservative core restriction | +| `deleted_A_c_record_probe.py` | `DELETE-A-C-RECORD-RESULT.json` | Producer capture of the later c query | +| `joint_record_minrank_probe.py` | `JOINT-RECORD-MINRANK-RESULT.json` | Full-owner coded-side-information lower bound | +| `core_budget_probe.py` | `CORE-BUDGET-RESULT.json` | Rational target gates and exact supplier provenance | + +Every program accepts `--output PATH`. The acquisition, native104 wrapper, and +c-record programs additionally accept `--word-output PATH`; their saved words +are `ACQUISITION-WORD.json`, `DELETE-A-NATIVE-WORD.json`, and +`DELETE-A-C-RECORD-WORD.json` respectively. + +The public QQ16 snapshot is pinned in [sources/MANIFEST.json](sources/MANIFEST.json) +to commit `12fa8c5c844be21bfefb38a19503fbd867a11244`. The original local core +frame members have independent archive/member pins in +[sources/qq16-core/MEMBERS.json](sources/qq16-core/MEMBERS.json). The attached +retained mixed supplier's exact profile, proof, and rational enclosure engine +are independently copied in [sources/baseline-mixed](sources/baseline-mixed/MANIFEST.json). +The huge owner PLAN is represented here only by a source-hashed dimension +summary; this component does not pretend to replay its owner fill. + +## 1. A native producer acquisition that costs8H8 + +Write \(F_s\) for the completed phase on the first \(9+8s\) axes. Before a +backward producer \(B_s\), its selected eight \(X_T\) fields already share +\(F_{s-1}\). Copy them into eight explicitly cleared, newly owned full-cover +records. The producer runs while these records remain unchanged. Each record +then crosses the eight fresh axes through one canonical rank8 child. + +After promotion to \(F_s\), two legal query views can be formed: + +\[ +a=R_T D_s X_{T,\mathrm{pre}},\qquad +q=R_T X_{T,\mathrm{pre}}. +\] + +The actual backward source identity identifies a with its current Y query, +\(a=R_sY_{\mathrm{post}}\). At the first F0/B1 pair, q is exactly the old +forward query f. At later stages q is the stated selected-X query; identifying +or converting it to a differently routed prior f is a further consumer task. +The word does not silently count that later identification as free. + +The exact per-invocation bill is: + +| Item | Count | +|---|---:| +| New rank8 native children | 8 | +| Explicit query-alignment cover maps | 8 | +| Canonical child wrapper maps | 16 | +| Real scalar primitive allowance | 650 | +| Newly owned live output fields | 16 | +| Eventual retirement CLEAR allowance after last use | 32 | + +Only eight records live across the backward producer. The second eight output +fields are allocated after promotion. All sixteen start with independent dirty +symbols and are explicitly cleared; the records are **live outputs**, not +returned borrowed scratch or the source's owner-serial C0. + +The canonical C8 kernel is checked on all256 Fourier characters, using65536 +exact Gaussian integer terms. Its multiplier is \(i^{|w|}\); the wrapper +coordinate permutations and signed phases are literal. The sibling +[independent acquisition audit](../native-source/QQ16-ACQUISITION-AUDIT.json) +replays complete owner products and complete Z4 phase polynomials together, +without importing this emitter. It passes all four stages and rejects all20 +mutations, including missing translation, promotion, exit wrapper, and CLEAR. + +This is one directional acquisition. Four independent invocations cost32H8; +one8H8 invoice does not replace every record of a whole nine-call controller. + +## 2. What the cheap-tap and commutator probes exclude + +The unchanged old core provides514 independent entries into its completed local +frame:112 data outputs and402 borrowed helper outputs. Granting free retention +of every completed-frame value over the whole core yields7194 observations, +but their joint span still has dimension514. Their projection onto independent +X/helper/private nuisance columns also has rank514. Adding the eight pure Y +queries raises the rank to522. + +This is an exact rational obstruction:514 arrival rows give the rational span +upper bound, and a nonzero514-by514 minor modulo65521 gives the matching +nuisance lower bound. It rules out acquiring the pure Y query by merely copying +and combining those existing completed-frame observations. Earlier paid side +information and a changed core are outside this observer class. + +The literal mid-B normalized X_T and Y fields have rank8s, not completed rank +9+8s. A tap there needs rank9 promotion to the completed output, costing8H9. +The pre-B snapshot's8H8 acquisition is cheaper. + +The whole-macro probe also checks the possibility of cancellation across the +full query-factor correction. The selected Y output block at the all-ones +character produces i where the required physical operator produces1. An +independent whole ordinary-Y block has a nonzero owner-path coefficient−1/2 +whose uniform fresh-C8 bias phases are0 and3. These full-block witnesses rule +out those two uncompensated fusions; they do not exclude a redesigned macro. + +## 3. Deleting A changes the data geometry honestly + +The old A ports + +`[0,7,20,27,36,46,57,111]` + +are arbitrary data, not scratch. The changed construction removes them from +both data banks. It has104 ports,208 independent data fields, the same96 U +ports and eight T ports, and + +\[ +RK=13I_8,\qquad R K_U=12I_8,\qquad R P_T=I_8. +\] + +The future A logger disappears, together with all A data restoration +obligations. The new forward wrapper consists of two matched completed-X +shears. The new backward wrapper keeps the selected-X native normalization, +three current-chart sweeps, and its inverse normalization; it has no future +logger or A-copy block. + +For \(f=R_sx\), the new forward endpoint is + +\[ +X_U=x_U-K_{s,U}f,\quad X_T=P_{s,T}f, +\] +\[ +Y_U=y_U+X_U,\quad Y_T=y_T+x_T-X_T. +\] + +For a backward input, let \(a=\ell_{s,T}x\), \(b=R_sy\). Its endpoint is + +\[ +Y_U=y_U+K_{s,U}a,\quad Y_T=y_T-P_{s,T}(b+11a), +\] +\[ +X_U=x_U-y_U,\quad X_T=P_{s,T}(b+12a). +\] + +All four inverse identities hold on208 independent rational input columns. +The first pair's repair to \((-y,x+y)\) uses a/f followed by +\(c=R_{0,U}(X_U+Y_U^*)\): + +\[ +Y_U^*=Y_U-K_{1,U}a+K_{0,U}f,\quad Y_T^*=X_T+Y_T, +\] +\[ +X_U^*=X_U,\quad X_T^*=P_{0,T}(f-c)-Y_T^*. +\] + +After updating Y_U, a is dead and its eight slots can be reused for c. Thus the +logical repair uses16 query fields plus one selected delta field, compared with +the original33-field template. The scalar checker uses the same-chart +specialization for its complete pair inverse; native output correction is not +inferred from that scalar equality. + +The **literal native wrappers are also emitted**. Five forward and four +backward stage cases check every73D phase polynomial with the actual source +maps. The exact per-call wrapper invoice is: + +| Wrapper | New rank1 children | Explicit maps | Child wrapper maps | Real scalar allowance | +|---|---:|---:|---:|---:| +| F0 | 0 | 0 | 0 | 7680 | +| Later F | 0 | 208 | 0 | 7680 | +| B | 16 | 224 | 32 | 7888 | + +The conservative old-core implementation creates16 owned internal fields in +the former A positions, clears them before the core, runs its unchanged native +word, projects the104 outputs, then clears those16 internal outputs. Zero +insertion commutes with the native frames. Both complete core orientations are +replayed on all712 columns:208 retained data,480 arbitrary borrowed helpers, +16 owned internal A fields, and eight original owned private fields. Every +borrowed helper is restored. Omitting either initial A-clear family fails. +This costs64 extra real scalar primitives per call and saves no old core child. + +The native104 word is a valid local restriction at the inherited boundaries; +it is not a global104-port owner construction. In particular, W is **not** +automatically scaled by104/112. New data families, internal fields, helper +windows, records, and idle complements must be assembled and counted together. + +## 4. Capture c before its cheap source disappears + +The new c probe uses the real F_s cut immediately after its X_U query +subtraction and before its existing X unalignment. Every source at that cut +already shares the completed frame. The identity + +\[ +c=R_{s,U}x_{\mathrm{pre},U} + =R_U X_{U,\mathrm{aligned}}+12R_T X_{T,\mathrm{aligned}} +\] + +holds with the actual owner paths, because \(R_UK_U=12I\). It supplies the +later repair query since \(X_U+Y_U^*=x_{\mathrm{pre},U}\). + +Eight explicitly cleared records capture c using384 unit ADDs and32 +coefficient12 ADDs. They cross the next B through8H8 with16 canonical wrapper +maps. The scalar allowance is3600, with eight owned live output fields and an +eventual16-scalar CLEAR allowance. No new source-alignment map is required: +the existing aligned producer cut supplies the reads. All four stage pairs +pass the exact routed identity and phase checks. Missing CLEAR, coefficient11 +in place of12, and missing promotion are rejected. + +This turns a post-consumer input-acquisition problem into a specific cheap +producer record. It still leaves the native writes into hole-framed Y and +completed X, record lifetime, and global storage to implement. + +## 5. Why sixteen fixed query records do not compress to eleven + +This check retains every owner rather than replacing owner maps by scalar +identities. Work on payload Fourier character zero, where all payload phases +and translations disappear but right owner multiplication remains exact. + +Grant the receiver every completed-X output and even free normalization of +the eight selected Y_T fields. Put \(u=x_{\mathrm{pre},U}\), +\(A=R_{s,U}/2\), and \(B=R_{s+1,U}/2\). The selected section makes f and u +independent. Eliminating arbitrary old-Y nuisance from the receiver leaves +only + +\[ +h=Bu-Zf, +\] + +where \(Z=(R_{s+1,U}K_{s,U}-11\ell_{s+1,T}P_{s,T})/2\). Recovering the exact +joint targets f and c therefore has minimum extra linear record rank + +\[ +\operatorname{rank}([f;Au;h])-\operatorname{rank}(h) +=\operatorname{rank}([A;B]). +\] + +This allows arbitrary coding with every source input and arbitrary cross-owner +mixing; it does not assume that eight records remain separately designated f. + +Both diagonal Gram blocks are12I. For \(C=AB^*\), all96 relative ordinary-port +owner routes are distinct. Collecting every identical73D owner path gives + +\[ +\frac{\operatorname{tr}(CC^*)}{|G|}=96,\qquad +\frac{\operatorname{tr}((CC^*)^2)}{|G|}=2208. +\] + +The block Gram spectrum lies in[0,24]. Applying +\(q(t)=1-(1-t/12)^4\), which is zero at0 and lies in[0,1] on that interval, +proves + +\[ +\frac{\operatorname{rank}([A;B])}{|G|} +\ge16-\frac{2\cdot2208}{12^4} +=\frac{1705}{108}>15. +\] + +One full-cover record has only \(|G|\) coordinates on this specialization. +Thus **at least16 full-cover records** are needed for these exact joint +queries, even under the generous receiver. All four actual stage pairs give +the same bound. Collapsing every route to identity reduces the bound to8, +which explains why a scalar specialization can badly underestimate this cost. + +This rules out transferring the historical eleven-block endpoint result to +the present fixed f/c interface. That earlier result changed the source +encoding. A changed producer, use of other completed banks, paid access to +ordinary Y holes, different record frames, or a decoder that computes final +corrections without reconstructing both queries can change this bound's target. + +## 6. Target budgets and the now-closed local rebase + +The retained outer uses +\(\kappa=C-10^{-46}-10^{-70}\). Values C=.001745 and C=.004 below are relaxed +screens; a strict final improvement needs a positive margin beyond the outer +backoff and both applicable supplier transfers. + +Under the retained mixed W/idle invoice, removing all nine eight-rank8 centre +groups leaves the following per-controller native allowances: + +| Primitive target C | Replacement allowance | After64 B lines and four a/query captures | Excess from adding four independent c captures | +|---|---:|---:|---:| +| .001745 | 571.332681182781… | 252.259925978816… | 2.812829225148… | +| .004 | 155.819896738978… | −162.059586330025… | 415.939069399028… | + +The last column tests the total64H1+64H8 before consumers, retirement, +precision, or changed incidence. It is a rejected implementation strategy, +not an impossibility result for another architecture. The exact rational +intervals are in the budget and c-record receipts. + +The original5522-mass QQ16 core would add +\(429+109/324\) rank mass per call compared with the mixed noncentre profile. +At C=.004, its nine-call weighted excess is3828.15672911048…. Those negative +old-core comparisons remain in `CORE-BUDGET-RESULT.json` as provenance history. +They are **locally avoidable now**: the sibling rebase emits all four mixed +clean-J providers at mass5093/5092 without a new native child, field, or phase +kernel. Its exact correction uses5636 common-frame scalar ADDs, based on367 +nuisance coefficients from106 immutable entrance-frame helpers. All723 +independent input columns,408 borrowed helpers,91 arbitrary dirty buffers, +forward/dual native operators, and matching data boundaries are checked. +The final eight-column transport check also proves that the prefix fixes the +raw scatter and the post-cut word transports it to Rᵀ/4 on Y only. The added +and removed scalar work has a net allowance of61,136 per call; it introduces +no new native child or helper width. + +The [rebase receipt](../native-source/rebased-core/RESULT.json) and +[rebase proof](../native-source/rebased-core/PROOF.md) are part of the cumulative +continuation. A source-hashed receipt snapshot is included here so the budget +checker distinguishes the new positive local result from its conservative +historical comparison. The newly introduced scalar observations still change +the hosting obligations; the old fill is not silently reused. + +For a changed104 geometry, the executable gate keeps W_new and I_new symbolic. +Let S be the new core saving relative to nine old QQ16 cores, D_W the cover +weight loss per controller, D_I the changed idle cost, and Q all new native +record/consumer/bridge/return costs beyond the64 B lines. Then + +\[ +S>\Delta_{\mathrm{core}}-A_{\mathrm{mixed}}+64+Q+D_I+D_WH_{73}(C). +\] + +The rebase supplies the old-core/mixed difference locally. A valid complete +invoice must still instantiate the changed caller's W_new, I_new, and Q. + +## Precise next task + +The promising search target is now the **final correction operator**, with +producer output encoding allowed to change, rather than minrank of the fixed +f/c query list. Use the compatible mixed clean-J boundary, retain every data +and dirty/helper input, and specify which completed and hole-framed outputs +the receiver can actually read. Find a producer/consumer factorization that +reduces the native acquisition and restoration bill of those final outputs. +The exact16-field bound above should be a rejecting control whenever the +candidate secretly reduces back to the same f/c receiver. + +After a positive factorization, emit its literal common-frame word and native +children, bind the mixed caller and new scalar observation windows, construct +the changed owner/private/record fill, and only then recompute the complete +complex/BIT/outer admission. No frozen baseline package is changed here. diff --git a/research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz b/research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz new file mode 100644 index 0000000000..c6e876d8b4 Binary files /dev/null and b/research/kappa-native-resume/causal-controller/sources/research/joint-query/certificate.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-CANONICAL.json.gz new file mode 100644 index 0000000000..c3eb094570 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-NATIVE-WORD.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..c3c3c77ece Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-base-NATIVE-WORD.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..b84ba97bdd Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-NATIVE-WORD.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..c3c3c77ece Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/FOUR-extra-NATIVE-WORD.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-CANONICAL.json.gz new file mode 100644 index 0000000000..fbdc18bfae Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-NATIVE-WORD.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..33c89dbc17 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-base-NATIVE-WORD.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..052bcf3d67 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-NATIVE-WORD.json.gz b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-NATIVE-WORD.json.gz new file mode 100644 index 0000000000..33c89dbc17 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/converted/THREE-extra-NATIVE-WORD.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-CANONICAL.json.gz new file mode 100644 index 0000000000..77ee265be3 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-PHASES.json b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-PHASES.json new file mode 100644 index 0000000000..22fd191b74 --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-base-PHASES.json @@ -0,0 +1,23 @@ +{ + "status": "PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES", + "canonical_word_sha256": "bff67890d0d350646d11f45aa1b098d6a415b1850b4982e8a05b4c146d247eda", + "frames": 1156, + "directed_edges": 3071, + "ranks": { + "1": 2072, + "2": 667, + "8": 16, + "3": 209, + "4": 87, + "5": 20, + "warp": 180 + }, + "endpoint_adapters": 224, + "warp_recipes": 180, + "warp_recipes_rank_zero": true, + "warp_recipes_equal_exact_512_column_product": true, + "all_child_equalities_include_global_phase": true, + "recipe_sha256": "f99fef8a175ff4952bc546d4b09c78d45d0bda5540c391eba82d539e477a78ba", + "seconds": 55.57171139499951, + "scope": "Every local frame, centre transition and warp. Expanded scalar-interface integration and global emitted program are separate obligations." +} diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..bc6433bbf5 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-PHASES.json b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-PHASES.json new file mode 100644 index 0000000000..f99019e597 --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/FOUR-extra-PHASES.json @@ -0,0 +1,23 @@ +{ + "status": "PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES", + "canonical_word_sha256": "e0c473d62127a09cb3cbf451564dbdd19c0c10f1674dab0575e65251aa5e1e3c", + "frames": 1156, + "directed_edges": 3071, + "ranks": { + "1": 2072, + "2": 667, + "8": 16, + "3": 209, + "4": 87, + "5": 20, + "warp": 180 + }, + "endpoint_adapters": 224, + "warp_recipes": 180, + "warp_recipes_rank_zero": true, + "warp_recipes_equal_exact_512_column_product": true, + "all_child_equalities_include_global_phase": true, + "recipe_sha256": "d259754558d9616665069423644a71535b5c331a565c77211b9719f0a319d29d", + "seconds": 77.28612599100052, + "scope": "Every local frame, centre transition and warp. Expanded scalar-interface integration and global emitted program are separate obligations." +} diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/NOTICE.md b/research/kappa-native-resume/native-source/rebased-core/sources/NOTICE.md new file mode 100644 index 0000000000..56c36d6f59 --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/NOTICE.md @@ -0,0 +1,39 @@ +# Attribution and source boundaries + +New work: Cinna Davis with substantial OpenAI Codex assistance, Apache-2.0. +No independent human or formal-kernel review is claimed. + +- Rohan Arun, PR4076: three staged entries, exact source words, T360 literal + fill, replay integration and predecessor exports. The present result starts + from his exact head 9744f9df268590295a9872b1952c695a678192b9, then + composes the later receiver-sharing improvement below. +- Dugongue / David Ganz, PR4085: receiver sharing composed with the staged + supplier, inherited native receipts, exact all-character Z4 operator proof + and event-level owner admission. The shared-covector rewrite follows Nick + Petalas' adaptive-continuation construction. All exports are pinned at + c5854c77620723aae6914c32122870519a6f820d. +- Dugongue, PR3945: terminal-width-five words, native wrapper, binary-basis + ledger, rational root enclosures and owner audit. The unmodified functions + and source-specific notices remain in their files. +- Ross Richardson, PR3605, and dataisfire, PR3734: staged-entry mechanism and + literal producer-return source construction. +- ChaloupeMusic / Erwan Secousses, PR3949: general staged-entry construction, + candidate/commuting-range machinery; PR3815: T360 clock, pattern/Hall model + and original EDF fill builder. Discovery adapted those tools to nonuniform + per-call widths and call-wide augmenting matching. +- Marius-Paul Dumitrean: exact frame solver in + research/rank7-skip-restore-optimum/frames_exact.py. Discovery used the + repository's integer-surrogate solver; explicit resulting frames, not an + asserted new optimality theorem, are certified here. +- Sagar Thakrar and roger-sato, PR3687/3693: retained exact point-set and + arbitrary-dirty semantics interpreters. Their reuse is not an independently + invented semantics engine. +- zxv, PR3682, and the complete older geometry, native, compiler, frame, owner, + packing and outer-transfer lineage retained by the predecessor packages. + +vendor/PR4076-NOTICE.md preserves the predecessor notice, and native/ preserves +its complete source-specific native and accelerator notices and licenses. +EXPORTED-SOURCES.json records byte-identical exports and source commits. +The mixed selector, four-entry source binding, new literal integer fill, +complete mixed-profile certificate and their integration are the contribution +of this package. Unsuccessful bounded searches are not claimed as barriers. diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-CANONICAL.json.gz new file mode 100644 index 0000000000..40d74e5638 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-PHASES.json b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-PHASES.json new file mode 100644 index 0000000000..494dd79f94 --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-base-PHASES.json @@ -0,0 +1,23 @@ +{ + "status": "PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES", + "canonical_word_sha256": "f07fae1011160b53e2df1f3135465db1277a093319de2a1da52d648f00a39542", + "frames": 1155, + "directed_edges": 3069, + "ranks": { + "1": 2070, + "2": 667, + "8": 16, + "3": 207, + "4": 89, + "5": 20, + "warp": 180 + }, + "endpoint_adapters": 224, + "warp_recipes": 180, + "warp_recipes_rank_zero": true, + "warp_recipes_equal_exact_512_column_product": true, + "all_child_equalities_include_global_phase": true, + "recipe_sha256": "bf88fc775e883227bbea9532ebf97df9e29576054dc0a893f8dc7fa1dc9744e2", + "seconds": 25.168864687999303, + "scope": "Every local frame, centre transition and warp. Expanded scalar-interface integration and global emitted program are separate obligations." +} diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-CANONICAL.json.gz b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-CANONICAL.json.gz new file mode 100644 index 0000000000..2af11b77b0 Binary files /dev/null and b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-CANONICAL.json.gz differ diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-PHASES.json b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-PHASES.json new file mode 100644 index 0000000000..00a86cdddc --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/THREE-extra-PHASES.json @@ -0,0 +1,23 @@ +{ + "status": "PASS_NEW_WORD_EXACT_LOCAL_PHASE_RECIPES", + "canonical_word_sha256": "882271d70602354cbada8d77f479703ccc6c47e689e8ccd91f9e4c87de0f4379", + "frames": 1155, + "directed_edges": 3069, + "ranks": { + "1": 2070, + "2": 667, + "8": 16, + "3": 207, + "4": 89, + "5": 20, + "warp": 180 + }, + "endpoint_adapters": 224, + "warp_recipes": 180, + "warp_recipes_rank_zero": true, + "warp_recipes_equal_exact_512_column_product": true, + "all_child_equalities_include_global_phase": true, + "recipe_sha256": "3d90c6657966224c7b92e85dddec28633ef7fd56af9eafeeb44fef468605ff06", + "seconds": 25.273792084997694, + "scope": "Every local frame, centre transition and warp. Expanded scalar-interface integration and global emitted program are separate obligations." +} diff --git a/research/kappa-native-resume/native-source/rebased-core/sources/bw_emit.py b/research/kappa-native-resume/native-source/rebased-core/sources/bw_emit.py new file mode 100644 index 0000000000..6d47f63ab7 --- /dev/null +++ b/research/kappa-native-resume/native-source/rebased-core/sources/bw_emit.py @@ -0,0 +1,192 @@ +# Warp-aware copy of hpe_emit_local.py (GK / PR #433 lineage: icekylinx with Codex assistance; DreamingOfClouds with Claude Opus 5.5). +# BW: every change for anticipatory dirty borrowing (PR2226, Cinna Davis @1337isnot1337, with OpenAI Codex assistance) is marked "BW:". +# On a word without warps / zero buffers / twisted additions the behaviour equals hpe_emit_local.py. +"""Emit both local programs of a BORROW canonical word: scalar macros, climbs, centres, wrappers, monomial warps, zero buffers. + +A warp event ['warp', r, op, f0, f1] is the rank-zero logical reference change K_G^{+-1} of register r moving it from +frame f0 to frame f1 (G(f0) = f1 for op > 0, G^-1(f0) = f1 for op < 0). The same event stream serves both orientations: +the inverse-transpose of a monomial K_G is the monomial conj(K_G) (same permutation, conjugated phases), a different +formal symbol of the free operator algebra, and the exact column check of the physical product is run for both. +""" +if not __debug__: + raise RuntimeError('assertions must remain enabled') # every check of this stage is an assert +import sys as _sys, os as _os; _sys.path.insert(0, _os.path.dirname(_os.path.abspath(__file__))) +import gzip,hashlib,json +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path + +OUT=Path(_sys.argv[1]);cp=OUT/'CANONICAL-WORD.json.gz' +c=json.load(gzip.open(cp)) +frames=c['frames'];v,R,h=c['v'],c['R'],c['h'];n=2*v+R +def digest(x):return hashlib.sha256(json.dumps(x,sort_keys=True,separators=(',',':')).encode()).hexdigest() +def basis(vs): + b={} + for x in vs: + for p in sorted(b,reverse=True): + if x>>p&1:x^=b[p] + if x: + p=x.bit_length()-1 + for q in b: + if b[q]>>p&1:b[q]^=x + b[p]=x + return tuple(b[p] for p in sorted(b,reverse=True)) +def reg(x):return list(x) if isinstance(x,list) else ['L',x] # BW: ['Z', c] is a private zero buffer +def dual(e): + e=list(e) + if e[0]=='add':e[1],e[2]=e[2],e[1];e[3]*=-1 + else:e[2],e[3]=e[3],e[2] + return e +def scalar(e,back): + e=dual(e) if back else list(e) + if e[0]=='add':return ['add',reg(e[1]),reg(e[2]),e[3],e[4]] + return ['scale',reg(e[1]),e[2],e[3]] +def primitive_bound(e): + if e[0] in ('clear','copy'):return 1 + q=Q(e[3],e[4]) if e[0]=='add' else Q(e[2],e[3]) + a,b=abs(q.numerator),q.denominator + assert b&(b-1)==0 + if e[0]=='scale': + assert a&(a-1)==0 + return abs(a.bit_length()-b.bit_length())+int(q<0) + return 3+(a.bit_length()-1)+a.bit_count()+(b.bit_length()-1) +def lregs(e):return [r for r in ([e[1]]+([e[2]] if e[0]=='add' else [])) if not isinstance(r,list)] + +def expand(e): + """BW: a twisted addition t += c*W(s) at frame zero: warp s by W, add, unwarp (all rank zero at frame 0).""" + if e[0]!='tadd':return [e] + _,t,s,word,a,b,f=e + return [['warp',s,op,0,0] for op in reversed(word)]+[['add',t,s,a,b,0]]+[['warp',s,-op,0,0] for op in word] + +def emit(back): + out=[];cur=c['start'].copy() + def event(e):out.append(e) + def scal(e,f):event(['scalar',e,f]) + def warp(e): + _,r,op,f0,f1=e + if isinstance(r,list):assert r[0]=='Z' and f0==f1==1 + else: + assert cur[r]==f0,('warp frame mismatch',r,cur[r],f0);cur[r]=f1 + event(['warp',reg(r),op,f0,f1]) + event(['phase','zero_frame_precondition']) + for e0 in c['precondition']+c['compensation']: + for e in expand(e0): + if e[0]=='warp':warp(e);continue + regs=lregs(e) + assert all(c['start'][r]==0 for r in regs) + scal(scalar(e,back),0) + event(['phase','entry_adapters']) + entering=[r for r,f in enumerate(c['start']) if r>=2*v and f!=0] + hpe={r for r in range(2*v,n) if c['final'][r]!=1} + wrapped=sorted(set(entering)|hpe) + for r,f in enumerate(c['start']): + if r in wrapped:event(['residual_adapter','pre',r,f]+([c['final'][r]] if c['final'][r]!=1 else [])) + else:event(['adapter','old_to_new',r,f]) + def climb(r,f,part): + old=cur[r] + if old==f:return + assert basis(frames[old]+frames[f])==tuple(frames[f]) + rank=len(frames[f])-len(frames[old]);assert rank>0 + event(['climb',part,r,old,f,rank]);cur[r]=f + for part in ('A','centers','B')+(('cleanup_tail',) if c.get('tail') else ())+('final',): + event(['phase',part]) + if part in ('A','B','cleanup_tail'): + for e in (c['tail'] if part=='cleanup_tail' else c[part]): + if e[0]=='warp':warp(e);continue # BW: monomial reference change, no climb + for r in lregs(e):climb(r,e[-1],part) + scal(scalar(e,back),e[-1]) + elif part=='centers': + lv=lambda r:c.get('target_levels',{}).get(str(r-v),0) + assert all(cur[r]==lv(r) for r in range(v,2*v)) + for k,r in enumerate(c['centers']): + cf=c['center_frames'][k];assert cur[r]==cf + terms=[e for e in c['cut'] if e[2]==r] + path=[cf]+sorted({lv(e[1]) for e in terms}-{cf},key=lambda x:-len(frames[x])) + C=['C',k] + if not back: + scal(['clear',C],cf);scal(['copy',C,['L',r]],cf) + for a in range(len(path)): + if a>0:event(['center',k,path[a-1],path[a],len(frames[path[a-1]])-len(frames[path[a]])]) + for e in terms: + if lv(e[1])==path[a]:scal(['add',['L',e[1]],C,e[3],e[4]],path[a]) + scal(['clear',C],path[-1]) + else: + scal(['clear',C],path[-1]) + for a in range(len(path)-1,-1,-1): + for e in terms: + if lv(e[1])==path[a]:scal(['add',C,['L',e[1]],e[3],e[4]],path[a]) + if a>0:event(['center',k,path[a],path[a-1],len(frames[path[a-1]])-len(frames[path[a]])]) + scal(['add',['L',r],C,-1,1],cf);scal(['clear',C],cf) + else: + for r,f in enumerate(c['final']):climb(r,f,part) + event(['phase','exit_adapters']) + for r,f in enumerate(c['final']): + if r not in hpe:event(['adapter','new_to_old',r,f]) + event(['phase','full_frame_cleanup']) + zb=c.get('zero_buffers',[]) + for z in zb:event(['zero_buffer','alloc',z]) # BW: private FULL-frame zero buffers stand in for the restored (hyperplane-ended) helpers + for e in c['cleanup']: + if e[0]=='warp':warp(e);continue + regs=lregs(e) + assert all(c['final'][r]==1 for r in regs) + assert all(r[0]=='Z' and r[1] in zb for r in [e[1]]+([e[2]] if e[0]=='add' else []) if isinstance(r,list)) + scal(scalar(e,back),1) + for z in zb:event(['zero_buffer','free',z]) + for r in wrapped:event(['residual_adapter','post',r,c['start'][r]]+([c['final'][r]] if r in hpe else [])) + event(['phase','done']) + return out + +def add_row(dst,src,q): + for j,x in src.items(): + z=dst.get(j,Q(0))+q*x + if z:dst[j]=z + else:dst.pop(j,None) +def warp_row(row,op): + out={} + for (var,word),x in row.items(): + word=word[1:] if word and word[0]==-op else (op,)+word + out[var,word]=x + return out + +def check_local(events,back): + # BW: literal scalar replay of the emitted stream in the free warp algebra (rows over (register, warp word)), centres and zero buffers. + state={('L',r):{(r,()):Q(1)} for r in range(n)};live=[];paid=Counter();prims=0;macros=0;zlive=set();nwarp=0 + for e in events: + if e[0]=='climb':assert not live;paid[e[5]]+=1 + elif e[0]=='center':assert live==[e[1]];paid[e[4]]+=1 + elif e[0]=='zero_buffer': + z=e[2] + if e[1]=='alloc':assert z not in zlive;zlive.add(z);state['Z',z]={} + else:assert not state['Z',z] and z in zlive;zlive.remove(z) + elif e[0]=='warp': + t=tuple(e[1]);nwarp+=1;state[t]=warp_row(state[t],e[2]) + elif e[0]=='scalar': + op=e[1];tag=op[0];t=tuple(op[1]);macros+=1;prims+=2*primitive_bound(op) + if tag=='clear': + state[t]={} + if t[0]=='C' and t[1] in live:live.remove(t[1]) + elif tag=='copy':state[t]={(var,w):x for (var,w),x in state.get(tuple(op[2]),{}).items()};live.append(t[1]) + elif tag=='scale':state[t]={j:Q(op[2],op[3])*x for j,x in state[t].items()} + else: + s=tuple(op[2]);q=Q(op[3],op[4]);row=state.setdefault(t,{}) + if t[0]=='C' and t[1] not in live:live.append(t[1]) + add_row(row,state.get(s,{}),q) + assert not live and not zlive and all(not row for r,row in state.items() if r[0] in 'CZ') + for r in range(n): + expected={(r,()):Q(1)} + if back and r 0) are not reference changes. + - Dropping one is invisible to a frame-only label check, but the algebraic replays reject it. +- **`new/SPEC.json` `width_spot_helpers`:** now [278, 281, 284, 289] (registers 506, 510, 513, 518, all width 8). The old spot helpers 482 and 494 are pre-life borrowers. + +**Endpoints.** +- The strict discovery assembly certified both endpoints with the fine engines at 1e-3000. Each is the 1e-3000 floor of its exact moment root. +- During discovery only, a scratch copy of `code/assemble.py` read these endpoints in place of its 1e-24 discovery values. The shipped strict run uses the unmodified code. +- Afterwards the SPEC was unpinned and refrozen, and `outer/B.json`, `LEDGER.json`, `INDEPENDENT.json` and `OUTER-CLAIM.json` were regenerated. +- The LF-pin, line-ending-insensitive and receipts-to-scratch portability fixes are already in PR2588 and are kept. + +**Dependencies:** Python 3.12 with numpy, gmpy2, sympy 1.14, mpmath and psutil. The search tools also need scipy and ortools. + +**Reproduce:** `python3 -B verify_release.py --main --work [--native-cache ]` ends with `PASS_RELEASE `. + +**Credits.** +- **zxv:** PR2530 — the source-partner words. +- **jeremykothe2:** PR2588 (helper-label swap and evacuated fill), PR2575, PR2526, PR2474, PR2462, PR2447, PR2435 — the package lineage, exact pricing and the gmx layout. +- **rohanarun:** PR2571 — repacked fill. +- **tekuuu:** PR2436. +- **Vlado Vojdanovski:** PR2322 and PR1673. +- **mdumitrean:** PR2478 and PR1625. +- **ywz65:** PR2508. +- **Cinna Davis:** PR2226. +- All inherited credits below. + +New here, prepared with substantial Claude Code assistance: the pre-life hosting, the refit alias, the refill, the controls and the fixes above. Earlier versions of this line are PR2559 and philippeVV's evacuation work. + +## Previous package text (PR2588), retained; its kappa figures are superseded by the section above + +**Credit first: this construction is PR2530's (zxv).** Our change is the swap of two helper labels (see below). PR2530 adds eight dirty aliases to the PR2462 supplier by restoring the second member of a retired source pair before the first member's handoff: 656 borrows 49, 657 borrows 83, 658 borrows 98, 660 borrows 101, 632 borrows 68, 596 borrows 80, 600 borrows 88, 633 borrows 31 (422 helpers per call, local profile {1:2367, 2:632, 3:318, 4:120, 5:26, 7:8}, width inventory {4:33, 5:9, 7:62, 8:78, 9:240}, all eight centres with 7+1 paths, twenty lifted targets (twelve on line 30, eight on line 45), a diagonal call order). Its words, order and dispatch are reproduced here exactly (`new/borrow/inputs/alias8_43base.json`, `alias8_43extra.json`, applied by `apply_lowfour` in `new/borrow/word_patch.py`; `new/borrow-fill/ORDER.json`, `DISPATCH.json`; its `independent_core.py` with the twenty-target pins). The aliases extend PR2436's (tekuuu) lifted-target mechanism on our PR2462 supplier, which rests on PR2322's handoff construction (Vlado Vojdanovski) and the PR2312 words. + +**What this package adds.** (0) The helper-label swap: the aliases of lenders 80 and 88 use registers 596 and 600 instead of 648 and 649 (same lenders and partners; `new/borrow/inputs/alias8_43*.json`); the local profile becomes {1:2367, 2:632, 3:318, 4:120, 5:26, 7:8}, with twenty lifted targets, so the checker pins in `independent_core.py` accept targets on lines 30 and 45 (twenty offsets). (1) An owner fill obtained by the evacuation method of PR2559 (philippeVV), run by Jeremy Kothe (jeremykothe2), with AI assistance on PR2530's owner map (`new/borrow-fill/FILL.json.gz`; words, order, windows and plan unchanged; `reconstruct.py` checks every window, bin, helper and family rule literally); leftover idle {1:22090, 3:6661, 5:422, 7:170, 14:1, 23:1, 32:77, 48:26489}. (2) Hardening stages in `new/rebuild_borrow.py`: the cut-centre guard of PR2478 (mdumitrean; log line `CUT-CENTRE-GUARD`) and the helper-index guard of PR2508 (ywz65; log line `HELPER-INDEX-GUARD`), each with a negative control. (3) Portability and determinism, after philippeVV's audit of PR2474 and with the fixes of PR2559: every generated artifact and every pinned artifact hash is LF-only (the package has `.gitattributes` `* -text`); `outer/run_outer.py` compares regenerated artifacts line-ending-insensitively; verification writes its receipts to its work directory and never into the package; the alias controls re-derive the inverse cleanup so each is rejected for its own reason; the four-alias control picks its warps by lender; the predecessor gates of `verify_release.py` are de-duplicated; `requirements.txt` declares the dependencies. (4) The private cleanup-buffer stock accounting (PR2309, PR2315; physical stock 66814 = 66304 + 422 + 8 + 78 + 2), the pooled-slot time binding (PR2301, PR1959) and the 3000-digit endpoints with the fixed-grid-maximum outer kappa C - 1/F - B (PR2180 / PR2161). + +**Entrance frames and hosting (answering the open question of PR2446).** The raised entrance frames reach the hosted helpers through the widths. (1) `new/rebuild_borrow.py`, stage `ENTRANCE-WIDTH-BINDING` (log line `ENTRANCE-WIDTH-BINDING base raised 24 bound True dim1 tamper rejected True`, and the same for extra): for each word, width = |helper_final| - |start_extra| of every remaining helper must equal the ordered widths of `new/BORROW-PROVIDERS.json`; a raised buffer with its start frame lowered to dimension 1 changes a width and is rejected. (2) `code/bind_providers_portable.py` re-derives the same widths from the native canonical word (|final| - |start|) and asserts they equal the plan registry widths. (3) `reconstruct.py` and `code/emit_allcold.py` use exactly those widths for every hosted role (`w == registry widths`), and `code/audit_allcold.py` rejects a wrong per-call residual width, a wrong provider width registry and an entering helper width silently restored to nine. A buffer delivered at dimension 1 would be width 8, not 7, and fails (1), (2) and the audit. **Scope of these stages:** the width stages bind dimensions only (|final| - |start_extra|); the frame contents of `start_extra` are established by the independent audit_local forward and dual replay of each word and by the native chain, which consume the same frames. + +**Private cleanup stock (charged as physical stock).** The borrowed words clear their private zero buffers at the end of each local program; these 78 buffers per call (`zero_buffers` of the canonical words) are private physical arrays outside the helper inventory, not paid rank. This package charges them exactly as PR2309 (Dugongue) does and as the audit PR2315 (thakrarsagar) requires: physical stock = data + helpers + 8 centres + 78 + 2 = 66830 (a disjoint private bank `zero_buffers` in `private_namespace`), a FULL-frame CLEAR on every allocation and release (156 extra scalar macros and 312 extra real primitives per orientation), a monomial routing reserve for the 84 warp events per orientation and the 78 buffers in the local adapter bound (355712), with the matching identities in `code/emit_allcold.py`, `code/invoice_allcold.py`, `code/audit_allcold.py`, `code/assemble.py` and `independent_core.py`, and two new controls in `code/audit_allcold.py` (`omit private zero buffers`, `private buffer aliases data`). The profile and kappa are unchanged by this accounting (it only adds guards; stock bits 2644, row coefficient 19545 < 20161). `evidence/INDEPENDENT.json` is regenerated for this fill (291708 roles, 438 helpers per call). + +**Pooled-slot time binding.** `code/audit_allcold.py` now binds all eleven fields of each pooled slot of `OWNERS.json.gz`, including `time`, to the base-derived call owners, with tamper controls (equal-width slots exchanging times; a shifted time). The gap was found by PR1959 (Josie Elliston) and executed by PR2301 (navinor / Aaron Wang); our owners satisfy it exactly. + +**Precision of this package** (`new/SPEC.json:outer_refinement`, `CLAIM.json:outer_refinement`): endpoint_digits 3000 (the native and conservative complex endpoints are certified to 1e-3000: accept at the endpoint, reject at endpoint + 1e-3000, by finer-constant copies of both fine engines (`code/fine_mercator.py`, `code/fine_moment.py`) in directed-rounding integer fixed point, floor for lower and ceiling for upper bounds), grid_digits 2995 (the final kappa grid), margin_digits 3005, bootstrap_steps 1116; kappa decimals are printed with 3120 significant digits. Every free slack of the ledger is the fixed-grid-maximum point of PR2180 / PR2161 (dataisfire) at this package's own endpoint: r0, layout slack, q gap and endpoint gap 1e-3080, epsilon = 1 - (1025/1024) 1e-3080, beta = 1 - 3 (1025/1024) 1e-3080, zeta 1e-3082, stop 1e-3086, backoff 1e-3022, kappa floor grid 1e-3046. With C the certified complex endpoint, F the floor denominator and B the backoff, the ledger kappa is exactly C - 1/F - B, the maximum of the declared floor-grid/backoff family (every point has epsilon q < C and C F is an integer, so floor(F epsilon q)/F <= C - 1/F; attained since 0 < C - epsilon q < 1/F, checked exactly; `outer/PROOF-GRID-MAX.md`). The main-binding family condition (Gaussian saving minus main saving = delta (1 - 2q) > 0) is an explicit 63rd strict row `gaussian_above_main`; the 62 earlier strict inequalities and all 17 rejecting controls are retained. The single 3000-digit rejection moment of the strict assembly uses the exact-integer coarse exponential of PR2168 (dataisfire; derivation as PR2184): `code/exact_coarse.py`, `EXACT-ARITHMETIC-DERIVATION.json`, intervals identical to the retained engine. Margins are asymptotic: the all-size statements still need p beyond an astronomically large cutoff. + +`new/CHAIN.json` lists every kernel step; `new/rebuild_chain.py` re-verifies each step against its own base words (exact B + early-cleanup map over all rows, remaining cleanup, interfaces). Pivot reads: every inserted read of the pivot q must see the row of q equal to s*I with s = +1 or -1 (`new/pivotcheck.py`); a pivot restored by a sign-flipping cleanup gate is read with that sign. Tamper controls (signs unaccounted; each new coefficient moved one dyadic step) must all fail the map check. + +Repairs retained: the PR1956 carried bootstrap cutoffs (every stage cutoff carries the previous constant; exact check at proof time, with the control that the uncarried cutoffs fail) and the PR1964 Lemma G extended-label stage `lemmag-labels` with the X/Y/Z adapter mutants. + +Credits: PR2530 (zxv): the eight source-partner aliases, the order, the words and the starting owner map (new/borrow/inputs/alias8_43*.json, new/borrow-fill/ORDER.json, DISPATCH.json, independent_core.py). PR2559 (philippeVV): the owner-fill evacuation method, and the portability fixes (line endings, receipts outside the package, re-derived controls, dependency declaration) reused here; philippeVV's audit of PR2474. Jeremy Kothe (jeremykothe2), with AI assistance: running the evacuation on PR2530's owner map (new/borrow-fill/FILL.json.gz) and the integration. PR2436 (tekuuu): the lifted-target aliases and the A-phase prelude mechanism that PR2530 extends. PR2322 (Vlado Vojdanovski): the handoff construction; PR2435/PR2447/PR2462/PR2474/PR2526 and PR2312 (ours): the entrance raise, the supplier chain and the borrowed words. PR2478 (mdumitrean) and PR2508 (ywz65): the cut-centre and helper-index guards. PR2309 (Dugongue) and PR2315 (thakrarsagar): private-stock accounting; PR2301 (navinor / Aaron Wang) and PR1959 (Josie Elliston): pooled-slot time binding; PR2161 / PR2180 (dataisfire): fixed-grid maximum; PR2168 / PR2184: exact-integer coarse exponential. PR2226 (Cinna Davis / 1337isnot1337 with OpenAI Codex): anticipatory dirty borrowing. PR1786 (chafreaky / Chafik Boukhalfa): fine source-prime padding, its proof and checks, vendored under vendor/pr1786; Tim Trudgian: effective prime number theorem (arXiv:1401.2689). PR1673 (Vlado Vojdanovski, with AI assistance): capped phase-cell fusion outer construction, its proof and exact evidence, vendored under vendor/pr1673. PR1758 (IdeaOuter and contributors): fractional-shift gauge-free interiors, accretive shallow-circuit lemma, per-visit tag accounting. Jeremy Kothe and contributors (PR1731 ledger lineage, radix-union schedule), LeverRank (independent schedule simulation), Gohberg, Semencul, Kailath, Kung, Morf, Bitmead, Anderson, Pan (structured Toeplitz-like algorithms of the retained alternative), Francois Le Gall et al. and Chafik Boukhalfa (PR1699). Supplier: PR1625 (mdumitrean), PR1661 (midhatn), pivot-554 kernel (PR1672 lineage) and the PR1319/PR1297/PR1274/PR1286 lineage as in earlier packages. PR1956 (Josie Elliston: the bootstrap-cutoff carry finding and repair), PR1897 (M6LI), PR1876, PR1875 and PR1828 (rohanarun): the endpoint and slack refinements that this package exceeds; rohanarun also PR1319 and PR1297; chafreaky PR1786; Vlado Vojdanovski PR1673; mdumitrean PR1625; midhatn; philippeVV PR1519 (odd-term Mercator finding); Kothe and the existing lineage. PR1964 (thakrarsagar / Sagar Thakrar): Lemma G label and adapter-rule checks and the X/Y/Z adapter mutants (code/lemmag_checks.py), Lemma G by DreamingOfClouds (PR509 / PR516). PR2161 / PR2180 (dataisfire, OpenAI Codex-assisted): the fixed-grid maximum point and its proof, re-derived at this endpoint; PR2044 (Kothe) free-epsilon refinement they stack on; PR1786 (chafreaky / Chafik Boukhalfa) and PR1673 (Vlado Vojdanovski) ledger formulas. PR2168 / PR2184 (dataisfire): exact-integer coarse exponential and its one-site derivation. PR2309, PR2302 and PR2265 are the records exceeded. PR2286 (thakrarsagar) side-condition audit and philippeVV (PR1897 / PR2044 comments): the text defects (minimum-margin claim, xi aside, HALO K0^2 branch, p^10 umbrella, vacuous or-True guards, stale xi digits) corrected in outer/FINE-PADDING-COMPOSITION.md section 4, outer/check_ledger.py, outer/struct_check.py and outer/check_gaugefree.py. Jeremy Kothe (jeremykothe2), with AI assistance: change of the lender of alias 651 to 530 and dropping of alias 653, the exact owner fill for the new width inventory, the exact frame re-solve of both borrowed words and the exact kernel search and pricing, new work in this package (new/borrow/inputs/refit_43*.json, kernel_43*.json, new/borrow-fill/FILL.json.gz). PR1625 (mdumitrean): the helper-donor kernel shape reused here at pivots 518 and 506. PR2309 (Dugongue): concurrent independent work, a 44th dirty alias with owner repacking that also borrows from lender 530; the leader this package exceeds. PR2301 (navinor / Aaron Wang) and PR1959 (Josie Elliston): the unbound pooled-slot time field, bound in code/audit_allcold.py here. PR2315 (thakrarsagar): the audit finding that the private cleanup buffers must be charged. PR2309 (Dugongue): the private-stock accounting (physical stock, CLEAR charges, warp routing reserve, adapter bound, controls) ported here. + +## Lineage and construction notes (text of earlier packages; their kappa and precision figures are superseded by the section above and were removed) + +Adds the exact helper-donor kernel k554-signed-001 (pivot 554, found by LeverRank/kernel566) to the previous record's local words: re-solved frames, fresh natives, PLAN rebound, SPEC refrozen, certified fine endpoints and exact outer witness of the base package. + +Credits: pivot-554 helper-donor kernel generalising PR1625 (kernel566.py; sign-aware pivot-read check pivotcheck.py). PR1625 (mdumitrean): pivot-566 helper-donor kernel, words, edits, rebuild.py, rebind.py. PR1319, PR1297, PR1424, PR1485, PR1574 (rohanarun); PR1274, PR1378... (see README) and the 1e-200/1e-300/1e-400 refinement PR1479, PR1510, PR1513, PR1536, PR1578 (jeremykothe2); PR1534, PR1549 (midhatn); PR1519 (philippeVV, fine Mercator engine soundness finding); PR1286, PR1193 (jeremydiviney), PR1148 (Thomas DiFiore) and the retained lineage. Research direction, prioritisation and release strategy: jeremykothe2. Implementation assistance: oh-my-pi. + +--- +# PR1625's pivot-566 helper-donor kernel on the T74 exact fill, with certified 1e-3000 endpoints and a 1e-2995 outer witness + +4. Stage `lemmag-labels` (beside the frame-only `local-replay`; `code/lemmag_stage.py` with `code/lemmag_checks.py`, vendored verbatim from PR1964 by thakrarsagar / Sagar Thakrar; Lemma G by DreamingOfClouds, PR509 / PR516): runs the Lemma G extended-label check (`label_check`, strict and permissive readings) and the adapter rule (`adapter_rule`) on each provider's emitted LOCAL-PROGRAMS in both orientations, asserting 0 violations, and requires the three physically wrong adapter mutants that the frame-only replay accepts (X exit straddle, Y entry straddle, Z exit adapters moved before a phase-B add) to be REJECTED on every provider. + +The top-level PLAN.json/SPEC.json are the unchanged PR1424/PR1319 files used by the owner reconstruction and parent lift checks; the strict stage uses `new/`. All 47 strict outer inequalities, the rejection controls and next-grid rejection hold (checked offline with the package's own outer witness and engine). The remaining gap to the exact root is the 1e-3000 endpoint bracket; the displayed digits are set by the 1e-2995 grid. + +The three witness settings are frozen in `SPEC.json:outer_refinement`: 12 bootstrap iterations, eta = beta = 10^-40 with target backoff 10^-43, and a 10^-36 final κ grid. The inherited complex and bit endpoint grids and their native/surcharge models remain unchanged. Every bootstrap cutoff is recomputed from the exact gap. The next final κ grid point rejects for this witness. No optimality over all outer parameters is claimed. + +The recurrence is a_(j+1) = (1-s)s + s a_j. Its closed form is a_j = s - (s-a_0)s^j; 0 < a_0 < s < 1 gives a finite increasing chain below s. This only changes witnesses to the inherited conditional bootstrap and outer lemmas, whose assumptions remain in force. Root endpoints, arithmetic engines and finite construction are inherited unchanged. + +Reproduce complete admission with `python3 -B verify.py --work /absolute/new/work --base-native-from /bound/base/native --extra-native-from /bound/extra/native`, or omit the two reuse flags for fresh native supplier runs. The output directory must not exist and assertions must be enabled. `evidence/OUTER-DISCOVERY.json` is discovery evidence, not a completed new-package replay. `evidence/pr1377/` contains predecessor receipts; inherited `PRICE.json` and input/parent receipts are not new verification. + +Conditional finite construction only. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, exact recovery, analytic transfer, PR559 recurrence correction and PR589 bit H1-H5 assumptions remain. No unconditional theorem, full Lean certification, or global packing claim. + +Credit Rohan Arun PR1319 (74-replica exact fill) and PR1297 (four cold kernels); Jeremy Kothe PR1377 (finer final grid), PR1019 and PR1274; Jeremy Diviney PR1193; Thomas DiFiore PR1148; Hao Yan and retained owner/controller lineage; eumemic and icekylinx/Zhihao Chen for the exact outer assembly; the PR547/548/559 recurrence and PR589 bit suppliers. Prepared by Rohan Arun with substantial OpenAI Codex assistance. Apache-2.0; predecessor notices and bytes are retained. No independent human or proof-assistant verification is claimed. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/CLASSICAL-BOUNDARY.md b/research/kappa-observation-integration/release-work/current/package/outer/CLASSICAL-BOUNDARY.md new file mode 100644 index 0000000000..33272b5545 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/CLASSICAL-BOUNDARY.md @@ -0,0 +1,30 @@ +# Classical-matrix boundary (tau = 2.372864 here; PR1699 used 12/5) + +PR1673 (Vlado Vojdanovski) charges the batched boundary algebra W^(tau-1) where tau is a fixed target exponent for W-square complex matrix +products; its stability lemma only requires omega_C < tau (it is stated there for the supplied omega_C <= 9/4 premise, tau = 19/8). +Chafik Boukhalfa (PR1699, `research/t74-signed554-phase-fusion/CLASSICAL-MATRIX-TRANSFER.md`) observed that Francois Le Gall's published +theorem omega < 2.3728639 (ISSAC 2014, arXiv 1401.7714, Table 1 and Section 6.3) already gives omega_C < 12/5, so the matrix branch needs no +unproven premise. This package adopts that specialisation: the boundary exponent is epsilon - xi + (tau-1)/2 = epsilon - xi + 7/10, the boundary +saving is 3/10 + xi - epsilon, and the ledger constant is c = (3 - tau)/2 = 3/10. The checks `matrix_exponent_above_published_bound` +(12/5 > 2.3728639) and `matrix_exponent_below_five_halves` are in the ledger; the exponent is an external published theorem and is not +re-proved here (neither in PR1699 nor in this package). The boundary row is far above the binding rows, so the final kappa is set by the +main transform = fused Gaussian balance with the sparse-union row carrying the layout slack, exactly as in PR1673's matrix corollary (this paragraph describes the PR1699 configuration; in this package the union row is relaxed by the new radix schedule and the boundary row binds, see the end of this file). + +What changes here relative to PR1699 is purely numerical and uses the freedom already in PR1673's proof: every strict slack (r0, zeta, layout slack, +q gap, stop exponent, decimal floor, backoff) is only required to be positive; they are set to 1e-3080 or below, and b is the certified complex +endpoint minus 1e-3080 instead of a 22-digit truncation minus 1e-30. The native moment test of the supplier is re-enclosed on a 2^-10413 grid +(directed-rounding integer fixed point, atanh/exponential term counts derived from the grid in check_ledger.py): moment(b) < 1 and moment(endpoint + 1e-3000) > 1 are both proven by exact rational enclosures. +Resulting kappa exceeds PR1699's exact kappa 0.001677641731343354564251058287... by about 6.7e-25. + + +## Change in this package + +The target exponent is tau = 2.372864 (just above Le Gall's published 2.3728639; PR1699's lemma only needs omega_C < tau), so c = (3 - tau)/2 = 0.313568. With the union row relaxed by the radix-distributed schedule (outer/UNION-SCHEDULE.md, new in this package) the boundary row binds: boundary saving c + xi - eps = kappa + slack, and kappa is set by the balance epsilon = (1 + q + q(c - s - r - zeta))/(1 + q)^2, xi = epsilon (1 + q) - c + s. The checks `matrix_exponent_above_published_bound` (tau - 2.3728639 = 1e-7 > 0) and `matrix_exponent_below_five_halves` are in the ledger. + +## Superseded in this package + +The square exponent tau = 2.372864 above is replaced by the rectangular-batching effective exponent tau_eff = 2.1567081 (k = 5, outer/RECT-BOUNDARY.md). Le Gall's bound is no longer used in the ledger. + +## Superseded again in pkg7 + +Historical (not used for kappa): the structured solves of outer/STRUCTURED-BOUNDARY.md give tau_eff = 2 + 1e-3080; the ledger uses the explicit Strassen boundary tau = 45/16 (c = 3/32); the fallback is the k = 5 batching of outer/RECT-BOUNDARY.md. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/FINE-PADDING-COMPOSITION.md b/research/kappa-observation-integration/release-work/current/package/outer/FINE-PADDING-COMPOSITION.md new file mode 100644 index 0000000000..5a85d6854f --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/FINE-PADDING-COMPOSITION.md @@ -0,0 +1,61 @@ +# Fine source-prime padding on this package's stack, and epsilon -> 1 + +**Credit first.** The mechanism of this note is **PR1786 (chafreaky, Chafik Boukhalfa)**: source primes at relative gap Theta(p^-2) from their dyadic targets, which exist by **Tim Trudgian's** effective prime number theorem (arXiv:1401.2689v2, theorem 2), the resulting setup precision `Cp + 64 p^2 ceil(log2 p)`, the theta-dependent conditioning, the polynomial halo p^48 / core p^128 and the dyadic normalizers. Its full text is vendored verbatim in `vendor/pr1786/FINE-PRIME-PADDING.md` and is the proof of every statement about primes, precision, locality and normalization used here. It builds on Vlado Vojdanovski's PR1673 (capped phase-cell fusion), the fractional-shift interiors and per-visit tag accounting of PR1758 (`outer/GAUGEFREE-INTERIORS.md`), the radix-distributed union schedule (`outer/UNION-SCHEDULE.md`), Jeremy Kothe's PR1731 ledger lineage and the PR1699/PR1672 supplier. **New in this package** (only): (a) the observation that epsilon is a free parameter and the resulting exact point epsilon = 1 - 1.0009765625e-3080; (b) the composition analysis below; (c) the exact ledger and independent derivation at that point. + +## 1. What is used from where + +| ingredient | source | role at epsilon -> 1 | +|---|---|---| +| fine primes, theta in (1/(4p^2), 1/(2p^2-1)] | PR1786 sec. 1-2 | chirp range a theta B^2 = O(a B^2/p^2): guard beta + 2 xi - 2 < 1 | +| setup precision, locality, normalizers | PR1786 sec. 3-5 | halo 48 / core 128; online precision stays O(p) | +| fractional-shift interiors, accretive circuit lemma | PR1758 (outer/GAUGEFREE-INTERIORS.md) | removes the linear gauge; **still required**: the old linear guard beta + xi <= 1 is violated at beta ~ 1, xi = 0.908 (ledger control `old_linear_guard`) | +| radix-distributed union schedule, per-visit tag work | outer/UNION-SCHEDULE.md | **still required**: the old g-factor union row 1 - eps - beta - 1/2 + xi is negative (control `old_g_factor_union_row`) and 2 beta < 1 fails (control `old_union_tag_2beta`) | +| boundary | explicit Strassen, tau = 45/16 (7^16 < 2^45), c = 3/32 | **not** the structured boundary of this package series: at xi = 0.908 < eps the boundary row is not binding, so the unproved stability claim of STRUCTURED-BOUNDARY.md is dispensable. Its file is retained as an alternative; it does not enter kappa. | + +## 2. epsilon is a free parameter (answer to "what sets epsilon = 0.9999999 in PR1786") + +Write delta = 1 - epsilon, beta = 1 - 3 delta, xi = epsilon(1+q) - c + s (boundary saving = main + s), c = 3/32. Every constraint of the ledger that involves epsilon or beta: + +* main = epsilon q is the binding saving, kappa = floor(epsilon q) - 1e-3022; +* Gaussian row: 1 - epsilon + q beta >= epsilon q iff **beta >= 1 - delta(1 + 1/q)** (a window of width ~ 594 delta below 1); +* membership: 1 - beta - 2(1-epsilon) > 0 iff **beta < 1 - 2 delta**; header / tag per visit: 1 - beta > 0; group below dimension: epsilon - beta > 0; +* **1 - epsilon - r0 > 0** (`global_gamma_power_slack`, r0 = 1e-3080: the spare power for u <= p^r0) and `source_axis_superpolynomial` 1 - epsilon > 0; +* boundary / union rows: xi = 0.9079 < epsilon (cap_below_cell), xi - 1/2 > 0, guard 1 - (2 xi - 2) - r0 - beta = 0.184 > 0: independent of delta; +* core-face corrections epsilon + beta + 48 - 128 + 2 < 0: independent of delta. + +The window for beta is non-empty for every delta > 0, and beta = 1 - 3 delta lies in it (the Gaussian row exceeds main by delta(1 - 2q)). The only lower bound on delta is **delta > r0 = 1e-3080**, a free strict slack of the proof. So PR1786's 1e-7 is an explicit interior point, not a limit; its kappa is q - 1.68e-10 only because delta = 1e-7. In this package delta = 1.0009765625e-3080 = (1025/1024) 1e-3080 (the smallest ratio that keeps the gamma spare power 1 - epsilon - r0 = r0/1024 > 0 and puts C - epsilon q below 1/F): + + epsilon = 1 - 1.0009765625e-3080, beta = 1 - 3.0029296875e-3080, xi = 0.9079463109..., + kappa = 0.0016797549561255445213356709939207025769983114385025802954603275201628645... + +which is q - 1.70e-3083 (q = b - 1e-3080 = 0.00167975495612554452133567099392070257699831143850258029546032752016286...). Exact sweep (`outer/independent_check.py`): delta in {1e-7, 1e-12, 1e-30, 1e-60, 1.0009765625e-3080, 1.1e-3080 ... } all satisfy every row; delta = r0 is rejected by `global_gamma_power_slack`; beta below the Gaussian window or at the membership edge is rejected (ledger controls `beta_below_gaussian_window`, `beta_above_membership_window`, `epsilon_equals_one`, `epsilon_not_above_r0_slack`). + +**Maximum epsilon.** Within this construction epsilon < 1 strictly (epsilon = 1 would mean d = p axes: source axes of O(1) size, `epsilon_equals_one` control; also kappa = epsilon q <= q), and 1 - epsilon > r0. The hard cap on any outer of this form is kappa < q (the supplier's complex saving); the remaining distance is q delta = 1.70e-3083. + +## 3. Does composing with the rest of the stack relax anything further? (answer: no, nothing is left to relax) + +* The binding rows at this point are main = epsilon q; boundary = main + s and union = main + s + 13/32, all other rows have strictly positive margins; the smallest non-main margin rows above kappa are global_coordinate_router (1.00e-3022), fused_gaussian (1.00e-3022) and group_bit_router (1.00e-3022), and the smallest strict checks overall are stop_positive = 1e-3086, global_gamma_power_slack = 9.77e-3084, lambda_above_tau = 2.5e-3081, lambda_above_internal = 2.5e-3081, lambda_prime_above_lambda = 2.5e-3081. They are positive but NOT all >= 1e-3080 (an earlier version of this sentence claimed so; corrected after the PR2286 audit). Nothing but delta moves kappa. +* Structured boundary / k5 boundary: boundary row non-binding, no effect on kappa (only on c). xi is not free: the ledger fixes xi = epsilon (1 + q) - c + s, which is 0.9079463109 at the explicit Strassen boundary c = 3/32. Using c = 1/2 would give xi = epsilon (1 + q) - 1/2 + s = 5.016963e-1 with the same kappa (xi = 0.5 itself is excluded by the cap xi > 1/2); it would add the unproved structured-stability claim for nothing. +* The router row a_eff = (q+b)/2 > epsilon q by about 1.00e-3022 (global_coordinate_router row); using the true bit endpoint there would not change kappa (main binds). +* Therefore kappa = epsilon q - 1e-3022 with epsilon the only free quantity, and it is already within q delta = 1.70e-3083 of the ceiling q. + +## 4. Obligations (same as PR1786 sec. 8, with sharper asymptotics) + +1. Every "eventually" in vendor/pr1786/FINE-PRIME-PADDING.md is now needed with exponent margin delta = 1.0009765625e-3080 instead of 1e-7: e.g. log x = Theta(p^(1-epsilon)) = p^(1.0009765625e-3080) must dominate log p in the prime-count estimate (true for p beyond an astronomically large cutoff), source axes 2^(p^delta) must exceed every fixed polynomial of p, and the polylog factors of every row are absorbed by p^delta-type slack. This is the same logical status as the 1e-3080 slacks of every row of this package (and of PR1786's 1e-7 version with a smaller cutoff); no practical cutoff is claimed. +2. Trudgian's theorem is a cited mathematical input (vendor/pr1786); the prime and precision lemmas are PR1786's and need their review (PR1786 is a draft). +3. The composition with PR1758's accretive circuit lemma uses theta^-1 = O(p^2) in the symbol bound (PR1786 sec. 3); the Gauge-free note's guard is superseded by `beta + 2 xi - 2 < 1` at fine theta. +4. Fresh native admission of the unchanged supplier: `verify_release.py`. + +## 5. Executable support + +`outer/check_ledger.py` (exact, 62 strict checks (including improvement over the exact kappas of PR1828, PR1839, PR1875, PR1876 and PR1897), 17 rejecting controls, improvement over PR1786's exact kappa and PR1758), `outer/independent_check.py` (independent balance derivation, the epsilon-free sweep), `outer/check_fine_padding.py` (PR1786's finite prime witnesses, underflow control, power relations), `outer/check_gaugefree.py` (PR1758 identities). + +## 4. Audit notes (PR2286 and philippeVV; none changes kappa) + +Credit: the side-condition audit of PR2286 (thakrarsagar) and the related checks of philippeVV (PR1897 / PR2044 comments) found the following text defects, corrected in this package. + +* The minimum-margin sentence of Section 3 and the aside on xi were wrong as written (corrected above, numbers from `outer/LEDGER.json`). +* HALO theorem branch. `vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md` line 205 states `K0^2 = o(p)`; at this point K0 = O(g h) = p^(beta + 1 - epsilon) and the statement needs 2(beta + 1 - epsilon) < 1, which is false here. The branch of the theorem that this package uses is the explicit-metadata branch (the theorem's own metadata term, lines 32 and 44 of that file), which holds: it is dominated by the ledger row `microbox_face_corrections` (margin 7.700e1). The `K0^2 = o(p)` branch is not used. The vendored file is left verbatim. +* The PR1786 p^10 umbrella (`vendor/pr1786/FINE-PRIME-PADDING.md`, `check_fine_padding.py` norm_power = 10) is now tied to THETA_POWER in `outer/check_ledger.py` (`halo_umbrella` in `LEDGER.json`): budget 2 THETA_POWER + 2 + 1/2 = 13/2 < 10, margin 7/2; the check would fail from THETA_POWER = 15/4. +* The two vacuous `or True` guards (`outer/struct_check.py` in the far-block loop and `outer/check_gaugefree.py` line 23) are deleted; nothing depended on them and both scripts still pass. +* Stale digits of xi and of the superseded pkg8-era / PR1673-era numbers are regenerated from the ledger where they are stated for the current point; text marked pkg8-era or superseded is historical. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/GAUGEFREE-INTERIORS.md b/research/kappa-observation-integration/release-work/current/package/outer/GAUGEFREE-INTERIORS.md new file mode 100644 index 0000000000..449c308761 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/GAUGEFREE-INTERIORS.md @@ -0,0 +1,123 @@ +> **Package note (fine primes).** In this package the source primes are chosen at relative gap Theta(p^-2) (PR1786, `vendor/pr1786`); theta >= 1/(4p^2) replaces theta >= 1/(4d) in every bound below, the chirp range becomes O(a B^2/p^2) and the guard of Section 5 becomes beta + 2 xi - 2 < 1 (see `outer/FINE-PADDING-COMPOSITION.md`). The accretive lemma and the shift identities hold for every theta > 0; the symbol bound mu >= (1 - e^{-gamma theta/2}) e^{-pi^2/(3 gamma)} is polynomial for theta^-1 = O(p^2). + +# Gauge-free interiors: fractional-shift Toeplitz kernels, accretive shallow circuits, and the chirp-only guard + +**Status.** New in this package series (not in PR1673 by Vlado Vojdanovski, whose capped phase-cell fusion, SHALLOW-TOEPLITZ.md, CAPPED-GAUGE-AUDIT.md and LOG-A-PHASE-SCHUR-AUDIT.md this note modifies and relies on). Conditional on every hypothesis of PR1673 except the three statements replaced below. Independent review required before release. Executable support: `outer/check_gaugefree.py` (exact identities, 50-digit symbol checks, section/Schur bounds, circuit replay) and `outer/check_ledger.py` / `outer/independent_check.py` (the ledger). + +## 0. Result and what changes + +kappa (this package = structured boundary + this note) = **0.0016797455088022990515465336851999736723...** (exact, 58 strict checks positive, all rejecting controls fire; previous package (structured boundary, linear guard) 0.0016783441717365..., gain +1.4013e-6; rectangular k=5 package 0.0016781238306335..., gain +1.6217e-6). Parameters: eps = 0.9999943758, xi = 0.5016741213, beta = 0.9966461332. + +Three statements of #1673 are replaced (everything else is unchanged): + +| # | #1673 / earlier packages | this note | +|---|---|---| +| R1 | local gauge of a capped interior: diagonals of range O(aB) (narrow), O(b0 B) (broad) | the linear part of the gauge is exactly a fractional centre shift of the Toeplitz kernel; only the quadratic chirp remains, range O(a theta B^2) = O(a B^2/d). (Section 2) | +| R2 | SHALLOW-TOEPLITZ sec. 9 needs mu I <= H <= M I (SPD) or ||H-I||<1/4 | accretive: (H+H^*)/2 >= mu I and ||H|| <= Lambda, with the absolute-scale guard log(2+Lambda+1/mu) (a ratio Lambda/mu alone is false, Remark 4.1a); for the unit-diagonal kernels used here Lambda >= 1 >= mu, so the guard is O(log(2+Lambda/mu)) (Corollary 4.1b). The shifted Gaussian symbol is accretive with explicit mu. (Sections 3, 4) | +| R3 | ledger guard beta + xi <= 1-r-zeta; union tag check 1-2 beta > 0 | guard beta + 2 xi - eps <= 1-r-zeta; tag check 1-beta > 0 (Sections 5, 6) | + +## 1. Setting + +Phase cell: beta_i = theta i - c in [-1/2,1/2), rho = 1+theta, lambda = 1/rho, 1/(4d) <= theta <= 1/(2d-1). An interior is a contiguous block of selected target coordinates (unit q-gaps), length n <= B (dyadic), anchor m at its centre; x = i-m, y = k-m, t = i-m for the narrow operator. B = p^xi, d = p^eps, xi < eps. + +* Broad forward block (FORWARD-SMALL-A-AUDIT, CAPPED-GAUGE-AUDIT sec. 3): K_{ki} = alpha^-1 exp(-b0 (x - lambda y + delta)^2), delta = (1-lambda) m - c, |delta| <= 1/2, b0 = pi/alpha^2. +* Narrow inverse block (LOG-A-PHASE-SCHUR-AUDIT sec. 1): N_{i,i+h} = exp(-a[(rho h + beta_i)^2 - beta_{i+h}^2]), a = pi alpha^2, beta_i = beta0 + theta t_i. + +## 2. Exact gauge <-> shift identities + +**Lemma 2.1 (broad).** For every real s, +(x - lambda y + delta)^2 - lambda (x - y - s)^2 = (1-lambda) x^2 - lambda(1-lambda) y^2 + delta^2 - lambda s^2 + 2(delta + lambda s) x - 2 lambda (delta + s) y. +Hence K = Drow G_s Dcol exactly, with the Toeplitz kernel G_s(y,x) = exp(-b0 lambda (x-y-s)^2) and diagonals separable in (y,x) (proved by expansion; sympy and 600 exact rational mixed-difference checks in `check_gaugefree.py` stage A/B). +Choose **s = -delta (1-theta)**. Then the x-linear coefficient is 2 b0 (delta + lambda s) = 2 b0 delta (1 - lambda(1-theta)) = 2 b0 delta * 2 theta/(1+theta), |.| <= 2 b0 theta, and the y-linear coefficient is -2 b0 lambda (delta + s) = -2 b0 lambda theta delta, |.| <= b0 theta. Over a block with |x|,|y| <= B/2 the linear leftover has logarithmic range <= 2 b0 theta B (total), and the quadratic parts b0 (1-lambda) x^2, b0 lambda (1-lambda) y^2 have range <= b0 theta B^2/4 each. Also |s| <= (1-theta)/2, i.e. 1 - 2|s| >= theta. +(With s = -delta/lambda the linear terms vanish in x and are 2 b0 delta (1-lambda) in y, but |s| can exceed 1/2 by theta/2, which changes the winding number of the symbol; so the shift is shrunk by (1-theta). The price is the O(b0 theta B) linear leftover, dominated by the chirp.) + +**Lemma 2.2 (narrow).** With beta_{i+h} = beta_i + theta h, rho = 1+theta: +(rho h + beta_i)^2 - beta_{i+h}^2 = (1+2theta) h^2 + 2 beta_i h. Writing beta_i = beta0 + theta t_i, j = i+h, t_j = t_i + h, and 2 t_i h = t_j^2 - t_i^2 - h^2: +log N_{ij} = -a rho h^2 - 2 a beta0 h + a theta (t_i^2 - t_j^2) (exact; sympy check, stage A). +So N = D' k D'^{-1}-type: N_{ij} = e^{a theta t_i^2} k(j-i) e^{-a theta t_j^2} with the Toeplitz kernel k(h) = exp(-a rho h^2 - 2 a beta0 h) = exp(a beta0^2/rho) exp(-gamma (h-s)^2), **gamma = a rho, s = -beta0/rho**. Here |s| <= 1/(2 rho) so **1 - 2|s| >= 1 - 1/rho = theta/rho >= theta/2**: no shrinking is needed, and the gauge is the chirp D' = exp(a theta t^2) of logarithmic range a theta B^2/4 (centred). The old weights are the same objects: log w_i = const + a theta (i - c/theta)^2 (the old linear slope 2 a beta0 is exactly the term moved into k). + +**Normalisation.** G'_s(u) := e^{gamma s^2} e^{-gamma (u-s)^2} = exp(-gamma u^2 + 2 gamma s u) has G'_s(0) = 1 (unit diagonal, as the audited N). The scalars e^{gamma s^2} are exact constants (stored in the anchor ratios). + +**Rounding.** The circuit applies the ideal truncated kernel (band W) rounded to Cp bits (the unit diagonal G'_s(0) = 1 is kept exactly by truncation and by rounding, so the circuit kernels are exactly unit-diagonal, Corollary 4.1b and stage F of `check_gaugefree.py`); the audited global rounded matrix differs from it by <= poly(p) 2^{-Cp} times the diagonal range e^{O(g (a theta B^2 + polylog))}, which is covered by the same precision guard (Section 5). **Restated comparison (audited rounded matrix vs ideal shifted kernel).** The audits (BILATERAL-LDU, FORWARD-SMALL-A, CAPPED-GAUGE-AUDIT sec. 2-3) bound the unshifted globally rounded matrix H^. Let K be the ideal block (exact Gaussian entries) and K~ = Drow' G^_s Dcol' the matrix the circuit applies (truncated at W, G^_s rounded to Cp bits, anchored diagonals). Entrywise |H^ - K| <= 4 * 2^{-P} e^{a/(4 theta)} relative to the weights (audit sec. 2), and |K~ - K| <= 2^{-Cp} W-banded; the diagonal conversions contribute the factor e^{O(a theta B^2)} per axis (chirp range of Section 5; the linear leftover is smaller). Hence ||K~ - H^|| <= poly(p) 2^{-Cp} e^{O(g a theta B^2 + g polylog p)} (g axes tensored), and the exponent is o(p) bits by the precision guard beta + 2 xi - eps < 1 of Section 5. For C > 1 the perturbation is <= 2^{-(C-o(1))p} <= 2^{-p}/poly, and since ||K^-1||, ||H^-1|| <= p^{O(1)} (Section 3; audits) every audited uniform bound (row gap, inverse norms, Neumann tail J = Cp^2, halo radius A) holds for K~ with polynomially perturbed constants. No structural change. + +## 3. The shifted Gaussian symbol is accretive + +Let q = e^{-gamma}, 0 <= s' = |s| < 1/2, phi(w) = sum_u e^{-gamma(u-s)^2} e^{-iuw}. + +**Lemma 3.1 (product form).** Jacobi triple product with Z = q^{-2s} e^{-iw}: +phi(w) = e^{-gamma s^2} prod_{m>=1}(1-q^{2m}) (1 + a_m e^{-iw})(1 + b_m e^{iw}), a_m = q^{2m-1-2s}, b_m = q^{2m-1+2s}. +For s < 0 the symbol is the complex conjugate (swap a, b). All a_m, b_m in (0,1) iff |s|<1/2; a_1 = q^{1-2|s|} is the unique factor near 1. At |s| = 1/2: a_1 = 1 and phi(pi) = 0. (Verified at 50 digits, relative error 2.5e-42 on 1025 points.) + +**Lemma 3.2 (phase).** For w in [0,pi] and s >= 0, f(x) := arg(1 + x e^{iw}) = atan(x sin w/(1 + x cos w)) is increasing in x in [0,1) (f'(x) = sin w/|1+x e^{iw}|^2 > 0). arg phi = sum_m [f(b_m) - f(a_m)]. Since 2m+1-2s >= 2m-1+2s iff s <= 1/2, a_{m+1} <= b_m, so f(b_m) >= f(a_{m+1}) and +0 <= -arg phi = sum_m [f(a_m)-f(b_m)] <= sum_m [f(a_m) - f(a_{m+1})] = f(a_1) < w/2 <= pi/2. +For w in [-pi,0] use phi(-w) = conj(phi(w)). + +**Lemma 3.3 (real part).** cos(f(a_1)) = (1 + a_1 cos w)/|1 + a_1 e^{iw}|, so +Re phi(w) >= |phi(w)| cos f(a_1) >= e^{-gamma s^2} prod_{m>=1}(1-q^{2m}) * (1 - a_1) prod_{m>=2}(1-a_m) prod_{m>=1}(1 - b_m) =: m_s(gamma). +(The bound is attained at w = pi up to the other factors: stage C finds min Re phi / m_s = 1.0.) Normalising by e^{gamma s^2}: + +mu(gamma, s) := m_s e^{gamma s^2} >= (1 - e^{-gamma(1-2|s|)}) exp(-pi^2/(3 gamma)). + +Proof of the closed form: -ln prod_k (1 - e^{-k gamma x}) = sum_h (1/h)/(e^{h x gamma}-1) <= pi^2/(6 x gamma); the three infinite products are bounded by prod_{k>=1}(1-q^{2k}) >= e^{-pi^2/(12 gamma)} (twice, the second for m >= 2 since a_m <= q^{2m-2}) and prod_{k>=1}(1-q^{2k-1}) >= prod(1-q^k) >= e^{-pi^2/(6 gamma)}; total e^{-pi^2/(3 gamma)}. +Norm: Lambda := ||G'_s||_2 <= ||symbol||_{l1} = sum_u e^{-gamma(u^2 - 2su)} <= 3 + sqrt(2 pi/gamma) (u^2 - 2su >= u^2 - |u| >= 0 and >= u^2/2 for |u| >= 2). + +Parameter values. Broad: gamma = b0 lambda = pi/(alpha^2 rho) = Theta(1/log p), 1-2|s| >= theta >= 1/(4d): mu >= (gamma theta/2) e^{-pi alpha^2 rho /3} = p^{-O(1)}. Narrow: gamma = a rho >= 4 log(16 p), 1 - 2|s| >= theta/2: mu >= (1 - e^{-a theta/2}) e^{-pi^2/(3 gamma)} >= a theta/3 = p^{-O(1)} (this is the audited row gap). + +## 4. The accretive Toeplitz circuit (replaces SHALLOW-TOEPLITZ sec. 9) + +**Theorem 4.1.** Let N be a power of two and H an N x N complex Toeplitz matrix with ||H||_2 <= Lambda and (H + H^*)/2 >= mu I, mu > 0. There is an N-wire circuit of disjoint invertible 1x1/2x2 gates applying H, with O(log^4 N) rounds, O(N log^4 N) gates, no ancilla, and +log(max prefix / suffix / inverse-prefix operator norm) = O(log N * log(2 + Lambda + 1/mu) + log N). +Rounding every coefficient to p + O(log N * log(2+Lambda+1/mu)) bits gives operator error 2^-p. If Lambda and 1/mu are p^{O(1)} the guard is O(log N log p) bits per axis. (The guard is the absolute-scale one of SHALLOW-TOEPLITZ sec. 9, M = Lambda, 1/mu; it is not a function of Lambda/mu alone.) + +**Remark 4.1a (why the absolute scale is needed).** The statement is not uniform in Lambda/mu: for N = 2 and H = tI, t > 0, Lambda = mu = t, Lambda/mu = 1, but the completed circuit has norm t and its inverse has norm 1/t, so the left side is >= |log t|, unbounded as t varies. (Reviewer observation, Mister-Meeseeks, review of PR 1758.) The proof below is written with Lambda + 1/mu throughout. + +**Corollary 4.1b (unit-diagonal kernels: ratio-only form).** If in addition every diagonal entry d of H satisfies |d - 1| <= 1/2 (in particular d = 1, the normalised kernels G'_s), then Lambda >= |d| >= 1/2 and mu <= Re d <= 3/2 (test the accretivity inequality on a unit vector e_j: Re H_jj >= mu; and ||H|| >= |H_jj|). Hence 1/mu <= 2 Lambda/mu and Lambda = (Lambda mu)/mu <= (3/2) Lambda/mu, so Lambda + 1/mu <= (7/2) Lambda/mu and log(2 + Lambda + 1/mu) <= log(2 + 4 Lambda/mu) <= 2 log(2 + Lambda/mu) + O(1). For d = 1 exactly: Lambda >= 1 >= mu and Lambda + 1/mu <= 2 Lambda/mu. Therefore for unit-diagonal kernels log(max prefix / suffix / inverse-prefix norm) = O(log N log(2 + Lambda/mu) + log N), the ratio-only form originally stated, and the rounding guard is p + O(log N log(2+Lambda/mu)) bits. + +*Proof.* Identical to SHALLOW-TOEPLITZ sec. 4, 5, 6, 9 except for the single place where the hypothesis is used, the uniform control of every recursive matrix, which we prove now. + +(a) *Descendants are Schur complements of sections.* Let T_b be the leading b x b section of H. The recursion of sec. 4 acts on index intervals J = [a,b); the matrix at J is S(a,b) = T_b / T_a (Schur complement of the leading a x a block of T_b, as an (b-a) x (b-a) matrix). Indeed the root is S(0,N); the children of S(a,b) with midpoint m are S(a,m) = T_m/T_a (leading block of S(a,b)) and S(a,b)/S(a,m) = T_b/T_m by the quotient property (T_b/T_a)/(T_m/T_a) = T_b/T_m. All of these are contiguous leading-type sections of the same Toeplitz H. + +(b) *Accretive matrices.* If Re >= mu ||x||^2 then ||Tx|| >= mu||x||, so ||T^-1|| <= mu^-1; every principal submatrix has the same property. Hence ||T_b^-1|| <= 1/mu for all b and ||S(a,b)^-1|| = ||(T_b^-1)[a,b)|| <= 1/mu (absolute bounds, not ratios). Also ||S(a,b)|| <= ||T_b[a,b)|| + ||T[a,b); [0,a)|| ||T_a^-1|| ||T[0,a); [a,b)|| <= Lambda + Lambda^2/mu =: Lambda'. The shear matrices K = A^-1 B, L = C A^-1 have norm <= Lambda/mu (a product of an inverse bound 1/mu and a norm bound Lambda; with the absolute guard Lambda/mu <= (Lambda+1/mu)^2). Writing M := 2 + Lambda + 1/mu, every quantity so far is <= M^3: ||S|| <= Lambda' <= M^3, ||S^-1|| <= M, ||K||, ||L|| <= M^2, and so Lambda' + 1/mu <= 2 M^3 =: M'. (No Hermitian structure was used.) + +(c) *Displacement rank, generators, compiler.* Sec. 4's rank identities (Delta = ZM - MZ, rank Delta(H) <= 2, ranks of A,S <= 2 + depth) and the sec. 5 bounded-generator compiler use only that each node's A, S, K, L are bounded matrices with the displayed ranks; the Stein generators g_j (orthonormal), h_j (norm <= ||E_T|| <= 2 Lambda'') have polynomial norm, and gamma_P = 64 N Lambda'' suffices, where Lambda'' is any common bound for the norms of the node matrices and their inverses, here Lambda'' = M' = 2 (2+Lambda+1/mu)^3 (the inverse bound 1/mu enters through it; it is not a function of Lambda/mu). The recursion D_full(N) <= D_full(N/2) + O(log^3 N) is unchanged. + +(d) *Prefix norms.* With each completed node factor of norm and inverse norm <= poly(N) M'^{O(1)}, B(N) <= max(C B(N/2), N^C M'^C), so log B(N) = O(log N log M') = O(log N log(2+Lambda+1/mu)) as in sec. 9 (M = Lambda, 1/mu there; Lambda'' = Lambda' + 1/mu <= 2 (2+Lambda+1/mu)^3, which changes log only by the constant 3). For H = tI this correctly gives log B >= |log t|. QED. + +**Corollary 4.2 (perturbation, truncation, rounding).** If the symbol is perturbed in l1 by eps' <= mu/2 (band truncation at W, rounding to Cp bits), the Hermitian part stays >= mu/2, the norm stays <= Lambda + mu/2, and Theorem 4.1 applies to the perturbed matrix with (Lambda, mu) replaced by (Lambda + mu/2, mu/2) (guard log(2+Lambda+mu/2+2/mu) <= log(2+Lambda+1/mu)+O(1) as mu <= 3/2). Truncation at any band W >= 0 and rounding of the kernel to Cp bits (fixed or floating point) do not touch the diagonal entry G'_s(0) = 1, which is exactly representable and is the u = 0 term kept by every truncation; so the perturbed kernels remain exactly unit-diagonal and Corollary 4.1b applies verbatim (Lambda >= 1 >= mu for the perturbed matrix). Were the diagonal to drift (e.g. a diagonal conversion with a rounding error), a drift |d-1| <= 2^{-Cp} <= 1/2 is covered by the hypothesis |d-1| <= 1/2 of Corollary 4.1b; the constants change by factors <= 7/2 only (stage F of `check_gaugefree.py`). For the broad kernel the truncation tail is <= 2^{-p}, and mu >= p^{-O(1)}. + +**Corollary 4.3 (boundary pivots for the structured boundary).** The boundary LU pivot P_j of STRUCTURED-BOUNDARY (Schur complement of T[J] onto segment j after eliminating all indices left of it and the interval I_j to its right, J = [0, end of I_j)) satisfies ||P_j^-1|| <= 1/mu, ||P_j|| <= Lambda + Lambda^2/mu, because P_j^-1 is the principal block of (T[J])^-1 and T[J∖seg_j] is a principal submatrix of T[J], both accretive. The couplings are band entries of G_s. The statement holds per constant-offset stretch: each stretch of a cell (maximal run with constant phase offset c, i.e. one Toeplitz matrix T, one shift s) is handled with its own anchor at the segment centre, so that the chirp D' = exp(a theta t^2) (broad: b0 theta t^2) has range a theta B'^2/4 for boundary segments of physical length B' <= B + 2W (and the linear leftover is <= 2 b0 theta B'); a pivot chain crossing a cell cut restarts, since cell cuts are boundary coordinates. Rank-2 Stein/Sylvester displacement for P_j, P_j^-1 holds for every Toeplitz T (Kailath), shifted or not; screened in `check_gaugefree.py` D/E and in the earlier IdeaOuter screen (displacement rank 2 along the whole chain at s = 0.5). Thus the structured-boundary pivot bounds (which used BILATERAL-LDU for the unshifted gauge) transfer without change of statement. + +## 5. Assembly and the guard + +For every capped interior (dyadic length n <= B, with its own anchor and shift s): + broad: Drow' (circuit of G'_s, Theorem 4.1 with mu = mu(b0 lambda, s); unit diagonal, so Corollary 4.1b applies)) Dcol'; + narrow: D' (circuit of G'_s, gamma = a rho) D'^-1 (inverse circuit = reversed gate sequence). +The diagonal factors Dcol', Drow', D' are anchored ratios with logarithmic range + narrow: a theta B^2/4; broad: <= b0 theta B^2/4 (x^2 and y^2 terms) + 4 b0 theta B (linear leftover) . +Per axis the range is O(a B^2/d); direct sums over blocks run in parallel as before. For g simultaneous axes the precision requirement of CAPPED-GAUGE-AUDIT sec. 4 becomes + P >= p + O(a d) + O(g a theta B^2) + O(g log N log p) + ... , i.e. g B^2/d * polylog <= p^{1-r-zeta}: + **beta + 2 xi - eps <= 1 - r - zeta**. +The offline accuracy term a d = o(p) is unchanged. The leftover linear term (exponent beta + xi - eps) is dominated by the chirp (exponent beta + 2 xi - eps) since xi > 0. The circuit norms add g * O(log N log(2+Lambda+1/mu)) bits (Theorem 4.1); here Lambda <= 3 + sqrt(2 pi/gamma) and 1/mu = p^{O(1)} (Section 3), so Lambda + 1/mu = p^{O(1)} directly (and by Corollary 4.1b equivalently Lambda/mu = p^{O(1)}), i.e. g * O(log N log p) bits, polylog per axis, as before. The precision assertion is stated with the absolute-scale guard, not with a ratio alone; the exponent beta + 2 xi - eps and every ledger number are unchanged. +(Why this does not collide with locality: the audited locality/halo constructions use the global rounded matrix only through uniform norms and the row gap, which are properties of the same Toeplitz-similar matrices; the shift is an exact similarity-to-Toeplitz representation of the same blocks, Lemmas 2.1-2.2.) + +## 6. Ledger changes (exact) + +1. `group_gauge_power_slack`: 1 - xi - r - beta -> **1 - (2 xi - eps) - r - beta**; `actual_logarithmic_group_guard`: 1 - xi - beta -> **1 - (2 xi - eps) - beta**. Add `shift_leftover_linear_dominated_by_chirp`: xi > 0, `shift_gap_theta_polynomial`: eps > 0. +2. Parameters: eps = (1 + q beta)/(1 + q) (main = Gaussian), xi = eps(1+q) - c + slack (boundary = main + slack), beta = 1 - 2 xi + eps - r - zeta. Closed form beta = (-1 + 2c - 2 slack + 1/(1+q) - r - zeta)/(1 + 2q - q/(1+q)), c = (3 - tau_eff)/2. +3. `union_tag_work_fits_record` (1 - 2 beta) -> **`union_tag_work_per_visit_fits_record`: 1 - beta** (Section 7). `union_header_fits_record` (1-beta) and `membership_arithmetic_fits_record` (1 - beta - 2(1-eps)) stay: pkg8 values 3.354e-3 and 3.343e-3. +4. Controls added: `old_linear_gauge_guard` (1 - xi - beta < 0), `old_union_tag_2beta` (1 - 2beta < 0) must fire as rejecting controls (they do). +5. Numbers (pkg8-era package, superseded by outer/FINE-PADDING-COMPOSITION.md for the current epsilon = 1 - 1.0009765625e-3080 point): eps = 0.99999437577296, xi = 0.50167412128176, beta = 0.99664613320944; margins above kappa: main = Gaussian = bit router = boundary = union ~ 1e-64; global router 9.4e-9; cap xi - 1/2 = 1.674e-3; group_below_dimension 3.35e-3; epsilon_below_one 5.6e-6; improvement over pkg7 1.4013e-6. +`outer/check_ledger.py` carries exactly these edits relative to the previous (structured-boundary, linear-guard) package; `outer/independent_check.py` re-derives (E,X,B) from the three linear equations by exact elimination. + +## 7. Union tag work: g polylog <= p suffices + +The previous version of UNION-SCHEDULE.md (steps 1 tag and section 3 header paragraph) argued "tag work per record <= O(g^2) in total, o(p) because 2 beta < 1". The cost accounting of Section 2.3-2.4 of that note is per visit: round r touches only records of S_r; each visit costs O(p log g + p log a + 1) head travel (radix passes, transposition) plus the tag extraction O(g) bit operations (scan of the g-bit flag header for the r-th set bit) plus O(g log L) header travel (the header is carried with the record). Sum over rounds = sum_r |S_r| = sum_x |F(x)| <= g mu V (Section 2.4). Total = O(g mu V (p polylog p + g log L)) = O(g mu V p polylog p) as soon as g log L = o(p polylog p), i.e. beta < 1 (and this is already the header check `union_header_fits_record`). The O(g^2) per-record total is a pessimistic restatement: each of the |F| <= g visits of a record is paid for by its own Theta(p) move. UNION-SCHEDULE.md now carries this wording. A reviewer should confirm no other place charges g^2 per record. + +## 8. Obligations / risks (review checklist) + +1. Theorem 4.1 relies on SHALLOW-TOEPLITZ sec. 4-6 (not re-derived); only the descendant bound (Section 4(a)-(b)) is new, and it carries the absolute-scale guard log(2+Lambda+1/mu) (corrected from a ratio-only statement; the application is unaffected because the kernels are unit-diagonal, Corollary 4.1b). The finite replay (`check_gaugefree.py` stage E: the vendored compiler on shifted Gaussian tiles n = 8, 16, 32, gamma in {0.5, 2, 20}, s up to 0.499: max error 1.8e-12, displacement rank <= 3, peak prefix norm <= 91 for n = 8, 16, max generator entry <= 11.4) supports it. The peak prefix norm is tracked only for n <= 16; for n = 32 it is not computed (reported as null). +2. The tile is a contiguous section of the *banded, truncated, rounded* shifted kernel; Corollary 4.2 covers it. The audited global rounded matrix remains the reference for the locality/Neumann/halo arguments; its comparison with the ideal kernel is stated in Section 2 (error poly(p) 2^{-Cp} e^{O(g a theta B^2)} = 2^{-(C-o(1))p}, C > 1). +3. Narrow N formula taken from LOG-A-PHASE-SCHUR-AUDIT (`n_{i,h}`), verified symbolically against the quoted weights of CAPPED-GAUGE-AUDIT sec. 2 (stage A). Broad block formula from CAPPED-GAUGE-AUDIT sec. 3. +4. (pkg8-era, superseded by the fine-padding point beta = 1 - 3.0029296875e-3080) The parameter point sat at beta = 0.9966, 1 - beta = 3.35e-3: header (g log L bits) and membership (g (log s)^2) rows hold only "eventually", with exponent margins 3.3e-3; the claim is an asymptotic statement with the same status as the other margins (1e-64 slacks). +5. Not changed: supplier, bit/complex rows, union schedule beyond Section 7, cap xi > 1/2 (pkg8-era margin 1.67e-3; at the current point xi - 1/2 = 4.079e-1). +6. STRUCTURED-BOUNDARY.md now states the chirp range O(aB^2/d) for consecutive boundary segments (physical length <= B + 2W) and cites Corollary 4.3 for the pivot bounds. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/PROOF-GRID-MAX.md b/research/kappa-observation-integration/release-work/current/package/outer/PROOF-GRID-MAX.md new file mode 100644 index 0000000000..76ab77537a --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/PROOF-GRID-MAX.md @@ -0,0 +1,15 @@ +# Maximum on the declared fixed kappa grid + +Credit: this argument and its parameter point are PR2180 / PR2161 (dataisfire, with OpenAI Codex assistance) on this series' PR2044; the ledger formulas are PR1786 (chafreaky / Chafik Boukhalfa), PR1673 (Vlado Vojdanovski), PR1758 and the Kothe lineage. Re-derived here at this package's own endpoint. + +Let C be the certified complex supplier endpoint, F = 10^3046 the floor denominator and B = 10^-3022 the positive backoff of `outer/check_ledger.py`. Every point of the free-gap family uses 0 < epsilon < 1 and 0 < q < b < C and reports K = floor(F epsilon q)/F - B. Since C F is an integer (asserted exactly) and epsilon q < C, integer ordering gives + + K <= C - 1/F - B. + +This bounds the declared grid/backoff family, not arbitrary suppliers, other backoffs or finer grids. + +Choose r0 = 10^-3080, zeta = r0/100, layout slack = gap = r0, stop = r0/10^6, b = C - r0, q = C - 2 r0, delta = (1025/1024) r0, epsilon = 1 - delta, beta = 1 - 3 delta. Then 0 < C - epsilon q < 1/F, checked exactly, so the reported K equals the bound C - 1/F - B (asserted in `outer/check_ledger.py`, `outer/independent_check.py`). + +All 62 strict inequalities of the earlier ledger and all 17 rejecting controls are retained. The main-binding family condition is now an explicit 63rd strict row, `gaussian_above_main`: Gaussian saving minus main saving = delta (1 - 2q) > 0. It was previously implicit in the checked beta window; it matters for rejection controls because the kappa floor/backoff (1e-3046, 1e-3022) is coarser than the free gaps (1e-3080), so a beta outside the main-binding window could still exceed the rounded kappa. The gamma spare power is r0/1024 > 0. All eventual cutoffs remain asymptotic; no practical cutoff is claimed. + +The supplier, native words, owners, invoice, endpoint and fine engines are unchanged (this package's own verify.py re-admits them in full). The ledger re-encloses the native moment sum at the new b = C - r0 (moment(b) < 1) and at the test endpoint C + 1e-3000 (moment > 1). diff --git a/research/kappa-observation-integration/release-work/current/package/outer/RECT-BOUNDARY.md b/research/kappa-observation-integration/release-work/current/package/outer/RECT-BOUNDARY.md new file mode 100644 index 0000000000..c49777f8be --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/RECT-BOUNDARY.md @@ -0,0 +1,72 @@ +> **Superseded in this package (pkg7) by outer/STRUCTURED-BOUNDARY.md; kept as the documented fallback (pkg6: k = 5, tau_eff = 2.1567081) if the structured-solve argument is not accepted.** + +# Rectangular boundary batching, k = 5 (new in this package) + +**Status.** This construction is a new contribution of this package. The capped phase-cell fusion outer, its boundary algebra and the batching of right-hand sides are Vlado Vojdanovski's +PR1673 (PHASE-LAYOUT-AUDIT.md, "Batched right-hand sides"); the specialisation of the matrix boundary to an established published exponent is Chafik Boukhalfa's PR1699; the +radix-distributed sparse-union schedule is new in pkg4 (outer/UNION-SCHEDULE.md). The change here: the boundary solves are batched with **W^k right-hand-side columns per factor page with k = 5** +(pkg5 used k = 2), so that rectangular matrix multiplication is used. The published input is the bound **omega(1,1,5) = omega(1,5,1) = omega(5) < 6.156708** of Francois Le Gall, +"Faster Rectangular Matrix Multiplication by Combination Loss Analysis", SODA 2024, arXiv 2307.06535 (v2, revised 27 Dec 2023), Section 1.2, Table 3 ("Our new upper bounds on omega(kappa)"), kappa = 5.00 -> 6.156708 +(same table: kappa = 3.00 -> 4.199095, kappa = 2.00 -> 3.250563, kappa = 1.00 -> 2.372537). Convention (Section 1.1): omega(kappa) is the smallest exponent such that the product of an n x ceil(n^kappa) matrix by a +ceil(n^kappa) x n matrix is computable in O(n^(omega(kappa)+eps)), i.e. the exponent of the tensor ; footnote 1 there (citing [8, 9]) states that the arithmetic complexity of n x n times n x m, n x m times m x n and m x n times n x n are the same: +the matrix multiplication tensors , , are cyclic rotations of each other, so and (a W x W factor page against W x W^5 right-hand sides) have the same rank and omega(1,5,1) = omega(1,1,5). +The theorem is used as a cited input and is not re-proved here. All inherited all-size hypotheses of PR1673 remain. (The k = 2 value of ADVWXXZ 2024, omega(1,1,2) < 3.250035, is the pkg5 fallback; for k = 5 only Le Gall's table gives a bound.) + +## 1. Batch width and the per-column cost + +In the audit, all inner coordinates and following tensor axes are independent right-hand sides; their number R is superpolynomial. The audit batches them in a dyadic width between W and 2W. Here a batch has width between W^5 and n^5 W^5, +where n is the fixed base of the rectangular format of section 3 (a constant; the width is a power n^(5j) matching the padded row dimension n^j). +Batching more columns is allowed for the same reason as before: the number of right-hand sides in an axis substream is superpolynomial, so padding the last incomplete batch costs a polynomial amount per axis substream +(at most n^5 W^5 lines of |E_f| records; at most g axes and g rounds per window), negligible against the superpolynomial volume. The batch matrix of |E_f| x n^5 W^5 consecutive lines (each line |E_f| long) is +rearranged so that W-row slices times all batch columns are contiguous by another small-field transpose with O(log(n^5 W^5)) = O(log p) riffle passes, so moving it costs O(log p) times the boundary volume, exactly as in the audit. +**Working set and record count per batch:** one batch is the W x n^5W^5 matrix of a block (and its W x n^5W^5 result) of p-bit records: n^5 W^6 = poly(p) records (about p^3 polylog p), so P-bit entries fit in polynomially many tape cells; +the whole batch matrix |E_f| x n^5 W^5 is |E_f| n^5 W^5 records, polynomial in p. **Superpolynomial supply per factor page (premise used explicitly):** every prefix block of an axis substream contains +L^(g-f) times the spectators lines, which is superpolynomial in p (the spectator payload and the L^(g-f) suffixes), hence >> n^5 W^5 lines: there are enough independent right-hand sides per factor page. For the last axis f = g the suffix is empty and the lines of a block are only the spectator lines: the d - g spectator axes (d = p^epsilon axes in total, g = p^beta in the group) each have length L = p^8, so there are L^(d-g) = p^(8(d-g)) of them, superpolynomial because d - g = p^epsilon - p^beta grows; W^5 = p^2.5 polylog p is far below it. The factor pages depend only on the axis f, the block index along the line and the axis-f coarse label, not on the other coordinates, so lines of every prefix block of the tag-f substream share the same pages. Each W x W factor page is scanned once per batch (page cost W^2 per block-batch versus product W^sigma). + +Cost bookkeeping per axis: a band of boundary rows has V W / B right-hand-side entries per axis. A batch handles W * W^5 = W^6 entries with one product of shape costing W^(omega(1,1,5)+eps). +So per axis the boundary arithmetic is (V W / B) / W^6 * W^(omega(1,1,5)) = (V/B) W^(omega(1,1,5) - 5), i.e. the old form (V/B) W^(tau - 1) with +**tau_eff = omega(1,1,5) - 5 + 1 = omega(1,1,5) - 4** (in general tau_eff = omega(1,1,k) - k + 1; k = 1 is the square case tau = omega). With the published bound, tau_eff < 2.156708. +The boundary row exponent is epsilon - xi + (tau_eff - 1)/2 and its saving is c + xi - epsilon with c = (3 - tau_eff)/2. + +## 2. Blocked triangular solve against W^5 columns, geometric ratio + +The no-pivot band solve of the audit is the block recursion X_j = L_jj^-1 (Y_j - L_{j,j-1} X_{j-1}) on W x W blocks along an axis line, with R = batch width columns: Y_j, X_j are W x R. Each step is two products of a W x W factor +block with a W x R matrix: shape with R = n^(5j), W = n^j padded. The precomputed diagonal inverses L_jj^-1 are part of the factor page. +The product is computed by recursion on the fixed format (n x n by n x n^5 matrices, rank R_f): T(W) <= R_f T(W/n) + A(W), where the additions (linear combinations of blocks and accumulations) at the top level touch +O(W^6) entries (the output W x W^5 and the inputs are of size at most W^6): A(W) = O(W^6). The output size of the format is n * n^5 = n^6 entries, so necessarily R_f >= n^6; the format exponent sigma > 6 satisfies R_f < n^sigma. The level-i additions cost +R_f^i (W/n^i)^6 = W^6 (R_f/n^6)^i, a geometric series with ratio R_f/n^6 > 1 that is dominated by its last term, the leaves (R_f^(log_n W) = W^(log_n R_f) = O(W^sigma)); precisely T(W) = O(W^(log_n R_f)) = O(W^sigma) because the per-level cost +increases geometrically toward the leaves (ledger check `rect_leaf_domination`: sigma - (k+1) = sigma - 6 > 0; if R_f = n^6 exactly there is only a logarithmic factor, absorbed by the strict slack). Equivalently the ratio of the additions' cost +at the top, W^6, to the leaf cost W^sigma is W^(6-sigma) < 1. Per step of the band solve two such products are needed, working set W * n^5 W^5 * P bits (polynomial). The additions are complete block scans: O(W^6) entries per level, +O(W^6) per node up to logarithms, as in the square-case tape accounting of PR1673/PR1699 (depth-first scheduling, fixed block permutations, tape repositioning). + +## 3. Existence of an exact rectangular format and precision + +Let omega(1,1,5) < sigma0 := 6.156708 (cited). By definition of omega(1,1,5) (the exponent of the arithmetic complexity of ; arithmetic complexity and tensor rank have the same exponent up to the constant bilinearisation factor +that is absorbed by the strict slack), for every sigma > sigma0 there are fixed integers n >= 2 and R_f with an exact bilinear formula of rank R_f for and R_f < n^sigma: choose s with sigma0 < s < sigma, a constant C with +rank <= C n^s for all n, and n large (the cited analysis defines omega through the asymptotic rank, and the infimum is over exact formulas for tensor powers). This uses strict slack and never asserts that the infimum is attained by a finite formula. +The rank equations are finitely many polynomial equations over Z. A complex solution can be replaced by one over the algebraic closure of Q: otherwise, by the weak Nullstellensatz over Qbar, 1 would lie in the ideal, giving the same contradiction over C. +Enumerate finite algebraic tuples with isolating descriptions, test the equations exactly, dovetailing over (n, R_f); the search halts by the existence statement and its cost and description are fixed independently of the growing input +(the argument of PR1699/PR1673's matrix lemma with the format in place of ; no finite scheme is claimed as generated here). +Recursion: T(W) <= R_f T(W/n) + O(W^6), with W padded to the next power of n (a constant-factor change of the dimension; the batch width is the padded W^5, so both W and R are powers of n and no axis of the tensor is padded). +Precision: approximate each fixed algebraic coefficient once by precision-doubling Newton iteration, O(M(P')) bit operations for P' bits. The absolute roundoff envelope of the fixed bilinear recursion obeys E_j <= K E_{j-1} + K n^(6(j-1)) +with K fixed by the chosen formula (maximum L1 column sums of the encoders and decoder times the base-case size), hence E_j <= K^j (...) = W^(log_n K) poly(W), polynomial in W. Input magnitudes at most 2^h need fractional precision +P + 2h + O(log W) for absolute error 2^-P; intermediate magnitudes are W^O(1) max(1, 2^(2h)); the total record width is O(P + h + log W) = O(p) at the retained boundary interface (P, h = O(p)), and a larger fixed guard constant absorbs the formula-dependent K. +The bound is absolute, including cancellation. + +## 4. Choice of sigma, controls + +sigma = 6.1567081 is strictly above the published bound 6.156708 (margin 1e-7, costs about 1.4e-13 in kappa), so tau_eff = 6.1567081 - 4 = 2.1567081. Ledger checks: `rect_exponent_above_published_bound` (sigma - 6.156708 > 0), +`rect_leaf_domination` (sigma > k + 1 = 6), `rect_effective_exponent_above_two` (tau_eff > 2, the trivial output-size bound) and `rect_effective_exponent_below_five_halves`. Rejecting controls: sigma = 6.1567079 (below the published bound) +and sigma = 6.156708 (equal to it) must fail `rect_exponent_above_published_bound`; both are asserted to fail. + +## 5. Consequence for the ledger + +c = (3 - tau_eff)/2 = 0.42164595. The boundary row still binds with main = fused Gaussian (xi = epsilon - c + kappa); kappa = 0.0016781238306..., +1.312e-7 over the k = 2 package (pkg5) and +3.04e-7 over the square tau = 2.372864 package (pkg4). +At the new balance epsilon = 0.99903, xi = 0.5791, beta = 0.4209; cap xi > 1/2 holds with margin 0.0791; union saving xi + 1/2 - epsilon = 0.080 (the old g-factor union row would be -0.341); +group scans 0.422; support/key/membership rows about 0.579; union tag work 1 - 2 beta = 0.158. The unproven omega_C <= 9/4 premise is not used. + +## 6. What is not proved + +The rectangular exponent is a published theorem used as a cited input. The existence of the exact format and the stability constants rest on the arguments above and on PR1673's inherited all-size hypotheses +(compiler, precision/tape, exact recovery, analytic transfer); a reviewer should in particular check the claim that the number of right-hand sides in an axis substream is superpolynomial in the regime used +(spectator payload L^g) and the per-column normalisation in section 1. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/RETUNED-SLACKS.md b/research/kappa-observation-integration/release-work/current/package/outer/RETUNED-SLACKS.md new file mode 100644 index 0000000000..75ad18dd4c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/RETUNED-SLACKS.md @@ -0,0 +1,27 @@ +# Strict slacks of the PR1673 capped phase-cell fusion outer in this package + +Construction, proof and every formula are Vlado Vojdanovski's PR1673 (vendored verbatim under `vendor/pr1673`); the classical-matrix boundary +constant c = 3/10 is Chafik Boukhalfa's PR1699 specialisation of Francois Le Gall's published exponent (`CLASSICAL-BOUNDARY.md`). The following +constants are free in PR1673's own text and audits (each only has to be strictly positive) and are set as follows: + +| constant | PR1673 | PR1699 | here | why free | +|---|---|---|---|---| +| b (complex saving) | 17-decimal truncation | endpoint (22 digits) - 1e-30 | certified endpoint - 1e-3080 | must lie strictly inside the certified supplier endpoint | +| q gap (q = b - gap) | 1e-10 | 1e-30 | 1e-3080 | q < a_eff < b strictly | +| backoff after truncation | 1e-11 | 1e-42 | 1e-3022 | strict gap absorbs polylog factors | +| layout slack s0 | 1e-5 | 1e-30 | 1e-3080 | positive layout slack (carried by the boundary row in pkg4) | +| r0 | 1e-6 | 1e-30 | 1e-3080 | only eventual u = alpha^2 <= p^r0 | +| zeta | 1e-8 | 1e-32 | 1e-3082 | r + zeta > 0 gives g a B = o(p) | +| stop exponent | 1e-9 | 1e-36 | 1e-3086 | any t_stop > 0 with (1-t)b > a_eff, q < (1-t)b | +| decimal floor | 1e-18 | 1e-48 | 1e-3046 | rounding grid only | + +The exact enclosure of the native moment sum uses a 2^-10413 grid (directed-rounding integer fixed point, atanh/exponential term counts derived from the grid in check_ledger.py) so that moment(b) < 1 and +moment(endpoint + 1e-3000) > 1 are certified (the supplier endpoint is certified to 1e-3000 by the repaired fine engines). kappa = 0.0016776417313433545642517275958639984231708545722735432722117400060454625429077941422798526440584702 +(exact rational in `LEDGER.json`), above PR1699's exact kappa by 6.69e-25. + + +**Fine-prime package.** epsilon = 1 - DELTA_E with DELTA_E = 1.0009765625e-3080 (any DELTA_E > r0 = 1e-3080 is admissible, outer/FINE-PADDING-COMPOSITION.md), beta = 1 - 3 DELTA_E; all other slacks unchanged. + +**Fixed-grid maximum.** The free gaps above are the point of PR2180 / PR2161 (dataisfire; stacking on this series' PR2044, PR1786 chafreaky / Chafik Boukhalfa and PR1673 Vlado Vojdanovski): r0 = 1e-3080, zeta = r0/100, layout slack = gap = r0, +stop = r0/10^6, b = C - r0, q = C - 2 r0, epsilon = 1 - (1025/1024) r0, beta = 1 - 3 (1025/1024) r0; the certified endpoint C, the floor grid F = 10^3046 and the backoff B = 10^-3022 are unchanged. +The ledger kappa is then exactly C - 1/F - B, the maximum of the declared floor-grid/backoff family (`outer/PROOF-GRID-MAX.md`). This is a statement about this grid and backoff only, not about other suppliers, backoffs or finer grids. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/STRUCTURED-BOUNDARY.md b/research/kappa-observation-integration/release-work/current/package/outer/STRUCTURED-BOUNDARY.md new file mode 100644 index 0000000000..6395bac193 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/STRUCTURED-BOUNDARY.md @@ -0,0 +1,103 @@ +# Structured boundary solves (new in this package) + +> **Package note.** This construction is retained as an alternative but is **not used** by this package's kappa: with fine source-prime padding the boundary row is not binding (xi = 0.908 < epsilon), and the explicit Strassen boundary of PR1786 is used (outer/FINE-PADDING-COMPOSITION.md). The numbers in sec. 7 below belong to the earlier package. + +**Status.** This construction is a new contribution of this package. It builds on Vlado Vojdanovski's PR1673 (capped phase-cell fusion: the interior/boundary factorisation, the no-pivot boundary solver, the audited +conditioning bounds BILATERAL-LDU.md, FORWARD-SMALL-A-AUDIT.md, LOG-A-PHASE-SCHUR-AUDIT.md, PHASE-LAYOUT-AUDIT.md), on the radix-distributed union schedule (pkg4, outer/UNION-SCHEDULE.md) and on the classical structured-matrix +theory of Gohberg and Semencul (inversion of Toeplitz matrices, 1972), Kailath, Kung and Morf (displacement rank, 1979), Bitmead and Anderson (1980) and Pan (structured matrices and polynomial computations). It replaces the dense/fast-rectangular +products of the boundary algebra (pkg1 to pkg6: W x W blocks against W..W^k right-hand sides at cost W^tau) by applications of generator representations of the blocks: O(r W log W) ring operations per right-hand-side column, i.e. polylog(p) per entry. +The boundary row then has exponent epsilon - xi + 1/2 + o(1) (tau_eff = 2 + delta for every fixed delta > 0; tau_eff = 2 itself is not claimed). All inherited all-size hypotheses of PR1673 remain. The package pkg6 (rectangular k = 5, tau_eff = 2.1567081) is the fallback. + +## 1. Where the structure comes from + +Within one phase cell, with the notation of the note and of BILATERAL-LDU.md (rho = 1 + theta, q_i = floor(rho i + 1/2) = i + k on the cell's stretch of constant offset k, beta_j = rho j - q_j = theta j - k), +G_ij = exp(-a[(rho j - q_i)^2 - beta_j^2]). Expanding with rho j - q_i = (j - i) + beta_j gives the exponent (j-i)^2 + 2(j-i) beta_j = rho (j-i)^2 + theta (j^2 - i^2) - 2k(j - i), hence **exactly** + + G = D_r T D_c, T_ij = exp(-a rho (j-i)^2) (Toeplitz), (D_r)_i = exp(a theta i^2 - 2 a k i), (D_c)_j = exp(-a theta j^2 + 2 a k j). + +(The same factorisation, with a, rho and the offsets of the forward map, holds for the broad forward operator H of FORWARD-SMALL-A-AUDIT.md, which is itself a symmetric Toeplitz matrix after the anchored diagonal gauge, and for the augmented Haug = E H C + (I - E C), +which is permutation-similar to H (+) I.) The truncation to bandwidth W (rows farther than W are below 2^-P and are zero in the retained rounded matrix) keeps T banded Toeplitz. Cells are separated by physical cuts, which the note already makes boundary coordinates, so no block of the +boundary system couples two cells; the cell-wise offset k only enters D_r, D_c. + +**Diagonal cancellation.** In the interior/boundary decomposition the Schur complement S = D - C A^-1 B and every step of the block LU are built from products K_{ab} K_{bb}^-1 K_{bc} with K = D_r T D_c; +D_c(b) D_c(b)^-1 and D_r(b)^-1 D_r(b) cancel, so in the coordinates Y = D_r^-1 X D_c^-1 *every block of S, every LU pivot and every inverse pivot is the corresponding block of the Schur complement of the plain Toeplitz matrix T* (checked exactly in struct_check.py). The diagonals are only applied at +the input and output of a boundary solve, as the same anchored ratios (relative to the block anchor) as the capped gauge of CAPPED-GAUGE-AUDIT.md: their range over a block of the line, including consecutive boundary segments, is O(aB) with the linear (old) gauge of CAPPED-GAUGE-AUDIT.md, and O(a theta B'^2) = O(a B^2/d) (B' <= B + 2W) in the fractional-shift representation of GAUGEFREE-INTERIORS.md (chirp only), inside the group guard g a B^2/d = o(p) used by this package (beta + 2 xi - eps < 1). No new gauge is introduced. In the shifted representation the pivot bounds are those of GAUGEFREE-INTERIORS.md Corollary 4.3 (P_j^-1 is a principal block of the inverse of the accretive contiguous section T[J]); the displacement-rank statements are unchanged (they hold for any Toeplitz T). + +## 2. Displacement ranks (the algebraic statement) + +Let W x W be the segment size and Z the W x W down-shift. The displacement of a matrix M with row and column index sets that are intervals (shifts Z of the corresponding sizes) is Delta(M) = Z M - M Z (Sylvester) or M - Z M Z^T (Stein); the two ranks differ by at most 2 (Kailath-Sayed). +Facts for a banded Toeplitz T and interval index sets: +(a) every rectangular block of T has displacement rank <= 2 (the displacement lives in the first row and last column); +(b) for invertible Toeplitz A, Delta(A^-1) = -A^-1 Delta(A) A^-1 has rank <= 2 (Gohberg-Semencul); principal and corner blocks of such a matrix have rank <= 2 + 2; +(c) Delta(XY) = Delta(X) Y + X Delta(Y) with the middle shift matched, so ranks add under products; ranks do not increase under sums beyond the sum; Delta(M^-1) = -M^-1 Delta(M) M^-1, so M and M^-1 have equal rank; +(d) (Kailath) the Schur complement of a leading block of a matrix with displacement rank r has displacement rank <= r (w.r.t. the shift of the remaining indices). +Consequences for the boundary J = G separated W-segments with interiors I_j between them (all interiors eliminated): +* coupling S_{j,j+1} = - T_{j,I_j} A_{I_j}^-1 T_{I_j,j+1}: rank <= 2 + 2 + 2 = 6 (a, b, c); S_{j+1,j} likewise; +* diagonal block S_{jj} = T_{jj} - T_{j,I_{j-1}} A^-1 T_{I_{j-1},j} - T_{j,I_j} A^-1 T_{I_j,j}: rank <= 2 + 6 + 6 = 14; +* the LU pivot P_j = S_{jj} - S_{j,j-1} P_{j-1}^-1 S_{j-1,j} is the Schur complement of T onto segment j after eliminating all indices to its left (a leading block of T: rank <= 2 by (d)) minus the single right-interior correction T_{j,I_j} A^-1 T_{I_j,j} + (rank <= 6; indices farther to the right do not couple to segment j, in T or in its left Schur complement, because the band is W and the separators have length >= W): **rank(P_j) <= 8 uniformly in j and G**, and rank(P_j^-1) = rank(P_j). +A *global* displacement operator on the concatenated J has rank growing like 2G (IdeaLiterature's screen); the claim here is block-wise, with one shift per segment, which is exactly what a block-tridiagonal LU needs. The constants 14 and 8 are proved bounds; the observed ranks are <= 4 (struct_check.py, +and IdeaLiterature's 70-110-digit screens of the real chirp-weighted blocks: coupling 2, diagonal and pivot 2-4, flat in W and in G). + +## 3. Generator representation + +A block M with Stein displacement M - Z M Z^T = sum_{k<=r} g_k h_k^T is represented by (g_k, h_k) and recovered by the Kailath-Kung-Morf formula **M = sum_k L(g_k) L(h_k)^T**, L(g) lower-triangular Toeplitz with first column g +(rectangular off-diagonal blocks use the same formula with the matched shifts). Generators are computed **offline, from generators, never from the dense blocks**: the leading Schur complements of the banded Toeplitz T are propagated by the generalised Schur algorithm (r = 2 generators, +O(r W) work per eliminated index, or the superfast O(r W log^2 W) variant of Bitmead-Anderson and Pan per segment); the interior corner blocks T_{j,I} A_I^-1 T_{I,j'} come from the Gohberg-Semencul generators of A_I^-1 (first and last columns of the inverse of the interior Toeplitz block); +the pivot and inverse-pivot generators are then obtained by the rank-<= 8 recompression of the sum (generators of the Schur piece and of the correction), an exact operation on ranks (QR/SVD at working precision P + poly bits). This is the page regeneration the audit already charges +(polynomial in p per complete microbox and axis, amortised by the superpolynomial payload); the generator pages replace the dense W x W factor pages. Balanced generators from the SVD of the displacement (g_k = sqrt(sigma_k) u_k, h_k = sqrt(sigma_k) v_k) satisfy |g_k||h_k| = sigma_k <= ||Delta(M)|| <= 2||M||, +so each term of the KKM sum has norm <= W ||Delta(M)|| (|L(g)|_2 <= |g|_1 <= sqrt(W)|g|_2) and the sum has norm <= 2 r W ||M|| = poly(W) ||M||: no hidden cancellation beyond a polynomial factor. + +## 4. The solve on the tape + +For one boundary band the block-tridiagonal system S x = y (stage diag(I, S^-1) of the audit; the C and B stages are products with the banded Toeplitz coupling blocks T_{J,E}, themselves rank-2 Toeplitz blocks) is solved by block forward and backward substitution with the audited no-pivot LU: + z_j = P_j^-1 (y_j - S_{j,j-1} z_{j-1}), x_j = z_j - P_j^-1 S_{j,j+1} x_{j+1}. +Each operator application (S_{j,j±1}, P_j^-1 and the Toeplitz couplings) is r pairs of triangular-Toeplitz products L(g) (L(h)^T v), each a convolution of length W computed by FFT of length 2W at working precision: O(r W log W) ring operations of p-bit numbers per right-hand-side column. +Layout: the right-hand sides of an axis substream (the lines of the union schedule, UNION-SCHEDULE.md) are batched in the dyadic width of the audit; the batch matrix |E_f| x batch width is arranged so that the W-axis of every column is contiguous by the small-field transpose (O(log p) riffle passes), the FFT is a sequence of O(log W) butterfly +passes, each a scan with a bit-reversal permutation realised by O(log W) riffle passes: O(log^2 W) scans of the batch data, polylog p overhead. The generator page of a block (r W numbers) is scanned once per batch; with the batch width >= W this is a 1/W fraction of the batch data (as for the dense pages in the audit). +Cost: per band and per column O(r W log^2 W M(p)) bit operations, i.e. per entry O(r log^2 W M(p)) = (M(p)/p) polylog p * p; with the retained short multiplier M(p)/p = polylog p, the boundary arithmetic is **O(V (d/B) W polylog p)**, so the boundary exponent is epsilon - xi + 1/2 + o(1), i.e. tau_eff = 2 + delta (any fixed delta > 0 absorbs the polylog factors since W = p^(1/2) polylog p). +(The ledger uses delta = 1e-3080, the same convention as the other strict slacks.) + +## 5. Precision and stability at Cp bits (the regime: row gap about 1/(16p), poly(p) conditioning) + +The argument reuses the audited bounds and adds the factors introduced by the structured representation. +1. *Pivots and couplings.* By PR1673's no-pivot theorem (BILATERAL-LDU.md Sections 1-2: unit L and U with all entries and all inverse entries bounded by constants times exp(-a rho theta |i-j|), c = prod(1 - r^-m) > 0, ||N^-1||_inf < 16 theta^-2; and the narrow/broad row-gap bounds of LOG-A-PHASE-SCHUR-AUDIT.md and FORWARD-SMALL-A-AUDIT.md: + Schur row gap >= 1/(16 p) survives elimination), every P_j^-1 is a principal block of the inverse of a partially eliminated matrix and satisfies ||P_j^-1||, ||S_{j,j±1}||, ||P_j|| <= poly(p) uniformly in j, with the unit L^-1 / U^-1 bounded independently of G (this is the audited no-factor-equal-to-the-number-of-windows property). In the SPD (broad, Saug/4) case the principal-submatrix inverse has norm <= lambda_min(H)^-1 <= poly(p). +2. *Generators.* From section 3, balanced generators have norm <= sqrt(2 ||M||) = poly(p)^(1/2) and the KKM reconstruction has norm <= 2 r W ||M|| = poly(p). +3. *One application.* An FFT convolution of length 2W in fixed point with Cp-bit coefficients and a precomputed table of 2W roots of unity (computed once, offline, to P + poly bits by Newton iteration in O(M(P'))) has absolute error <= O(log W) 2^-P' times the product of the l1/l2 norms of its operands. So one application of an operator M to a vector v has + absolute error <= poly(p) ||v|| 2^-P' + (the error of the stored generators, <= 2^-P' poly(p)). The diagonal gauges D_r, D_c enter only through the anchored ratios of range O(aB) (section 1), which cost O(aB) = o(p) extra bits: the same guard as the interiors (chirp-only O(a B^2/d) per axis in this package; O(aB) in the earlier linear-gauge version); no exponential growth beyond it. +4. *Accumulation.* The forward and backward substitutions are the unit L^-1 and U^-1 of the audited LU followed by diagonal factors P_j^-1; errors of different steps enter linearly through the unit factors whose inverse is bounded independently of G, so the total absolute error is <= poly(p) 2^-P' (a sum over at most poly(p) = L/B segments of poly(p) terms, with the audited exponential decay of the L^-1, U^-1 entries + making the dependence on G at most polynomial). Hence P' = P + O(log p) + o(p) (the guard), i.e. O(p) bits suffice, the same record width as for the dense algorithm. +5. *Offline generator accuracy.* Generators are generated at working precision P + poly(p) bits in polynomial time; their conditioning is that of the blocks themselves (the generalised Schur algorithm is applied to the banded Toeplitz T with strictly diagonally dominant / SPD structure whose leading Schur complements are bounded below by the row gap), so no loss beyond poly(p) factors occurs. +The one point where structured algorithms are known to be unstable in general (indefinite or nearly singular leading minors) is excluded here by the audited uniform lower bounds on the pivots (item 1); this is where an independent reviewer should look first. + +## 5b. Cleaner conditioning argument, rounding convention, anchors and pages (added after independent review) + +* **Conditioning without the no-pivot theorem.** In the coordinates Y = D_r^-1 X D_c^-1 every block, pivot and inverse pivot is a block or a Schur complement of the plain symmetric positive definite Toeplitz matrix T (a Gaussian symbol is positive for every a; the narrow case has cond(T) = 1 + O(e^-a)). + By interlacing and monotonicity of Schur complements, every principal submatrix and every Schur complement of T has its spectrum inside [lambda_min(T), lambda_max(T)], hence every leading minor and every pivot P_j (and P_j^-1) has condition number at most cond(T), and cond(T) <= (1 + sqrt(pi/gamma))/(sqrt(pi/gamma) e^(-pi^2/(4 gamma))) = O(e^(a/4)) = O(p) + with the eigenvalue bounds of FORWARD-SMALL-A-AUDIT.md (the same constants as the broad forward audit). The Bunch-type danger of structured solvers (indefinite or nearly singular leading minors) cannot occur for SPD T; the Schur/Levinson recursion and the Gohberg-Semencul application on an SPD Toeplitz matrix are weakly stable, losing O(log cond) = O(log p) bits. + (This replaces the appeal to the row gap 1/(16 p) in section 5 item 1 for the SPD blocks; the row-gap bounds remain the justification for the non-symmetric narrow blocks. Independent numerical evidence, LeverRank, 80-160 digits, alpha = 4, a = 50.3, rho = 1 + 1/6000: pivot condition numbers saturate at cond(T) ~ 1.4e5 for W = 16, 32, 64; balanced-generator norm ratio sigma_1(grad L^-1)/||L^-1|| 0.14-0.27 (no growth); generator-to-inverse rounding amplification <= 1; + Levinson/Gohberg-Semencul error about 2^9.5 u flat in n; a perturbation of the first pivot is damped along the chain, so accumulation is independent of G.) +* **Rounding convention (required).** The retained rounded matrix must be rounded *factorwise*: K^ = D^_r T^ D^_c with T^ exactly Toeplitz (symmetric rounding of exp(-a rho x^2) by |i-j|) and D^_r, D^_c the rounded gauge weights. This is the structure-preserving rounding of FORWARD-SMALL-A-AUDIT.md and the Toeplitz-similar cell matrix of LOG-A-PHASE-SCHUR-AUDIT.md. Entrywise rounding of K would destroy the Toeplitz structure (this is the dense non-Toeplitz control of section 6: the displacement rank jumps to W). +* **Anchors.** The diagonal conversions between X and Y coordinates use segment-local anchors. The inter-segment constant ratios D_r(m_j)/D_c(m_{j-1}) (about e^{O(aB)}) are stored inside the coupling pages and cost O(aB) = o(p) guard bits per page; no range over the whole line is ever formed. +* **Pages.** In the interior of a line the pivots are identical (observed), so each cell needs O(1) distinct generator pages (only the cell-end blocks differ): page regeneration is cheap. A batch width W suffices (no W^k batching is needed in this package). + +## 6. Executable check (`outer/struct_check.py`) + +On the exact rational model K = D_r T D_c with a random (non-symmetric) banded Toeplitz T, W = 3..5, G = 3..6 segments, interiors of length 3W: (1) the exact diagonal cancellation S_Y = Schur(T); (2) the Stein displacement ranks of every segment block, coupling block, LU pivot and inverse pivot (all <= 4, flat in the segment index and in G); +(3) exact reconstruction of every block from its rank factorisation by the KKM formula; (4) the block forward/back substitution performed only with generator representations equals the dense solve exactly; (5) controls: a dense non-Toeplitz perturbation of the band raises the pivot and diagonal ranks to the full W (here 6 > 4), and a perturbed pivot is not reproduced by the original generators. +This checks the algebra exactly at small W; it does not check the analytic stability argument of section 5, which is an argument, nor does it replace a check on the real chirp-weighted blocks at 70-110 digits (IdeaLiterature's screens: scratch/IdeaLit/real.py, real2.py). + +Scope of the real-matrix screen (IdeaLiterature, disclosed): block ranks 2-4 were observed for W <= 16 and G <= 8 segments only. In that toy the pivot condition number grows with W (about 3.5 at W = 8 and 1.2e3 at W = 16) because the toy takes a proportional to 1/W^2. +In the regime of the note a = pi alpha^2 >= 1 is fixed independently of W (alpha = ceil(sqrt((4/pi) log 16p))), theta >= 1/(4d), and BILATERAL-LDU.md bounds the periodic inverse by ||N^-1||_inf < 16 theta^-2 and the pivots/factors uniformly (items of section 5), so the conditioning is polynomial in p; the toy's growth with W is therefore not the real regime, but it is exactly the point that is *not* verified numerically here. + +## 7. Ledger consequence + +tau_eff = 2 + delta (delta = 1e-3080): c = (3 - tau_eff)/2 = 1/2 - delta/2. Balance: main = fused Gaussian = boundary - slack with xi = epsilon - c + kappa; the union row xi + 1/2 - epsilon has the same exponent as the boundary row (both epsilon + 1/2 - xi), so both bind at kappa + 1e-3080 together with main and the Gaussian row; +(pkg8-era numbers, superseded by outer/FINE-PADDING-COMPOSITION.md) with the chirp-only guard of GAUGEFREE-INTERIORS.md: xi = 0.50167, xi - 1/2 = 0.00167 > 0 (cap holds), beta = 0.99665, union tag work per visit 1 - beta = 0.00335 > 0. kappa = 0.00167974550880...; +1.40e-6 over the structured-boundary balance with the linear guard (kappa 0.0016783441717...; that was +2.20e-7 over k = 5, tau_eff = 2.1567081). + +## 8. What is not proved + +* The constants c0 = 14, 8 follow from the displacement calculus (a)-(d) above; (d) and the Gohberg-Semencul statement (b) are classical theorems used as inputs; the corner-block increment (+2) and the application to the real chirp-weighted blocks (Haug, Saug, narrow and broad pairs) rest on the exact factorisation of section 1. +* Section 5 is an error argument built from the audited bounds, not a verified stability theorem for structured solvers. The real-regime numerical check (row gap 1/(16 p), p large) has not been run. +* Everything inherited from PR1673 (all-size compiler, precision/tape, exact recovery, analytic transfer) remains an assumption. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/UNION-SCHEDULE.md b/research/kappa-observation-integration/release-work/current/package/outer/UNION-SCHEDULE.md new file mode 100644 index 0000000000..a1ee00a8de --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/UNION-SCHEDULE.md @@ -0,0 +1,132 @@ +# Radix-distributed sparse-union schedule (new in this package) + +**Status.** This schedule is a new contribution of this package. It is not part of PR1673 (Vlado Vojdanovski, whose capped phase-cell fusion +construction and PHASE-LAYOUT-AUDIT.md this document modifies and relies on), nor of PR1699 (Chafik Boukhalfa, Le Gall's exponent). It replaces +one step of PR1673's sparse-union window, the per-axis scan of U, and is conditional on every other hypothesis of PR1673. Its executable finite +check is `outer/sched_test.py`; the arithmetic consequence is `outer/check_ledger.py`. An independent review is required before release. + +## 0. Model and what is replaced + +Machine model: PR1673's fixed-tape model, i.e. a multitape Turing machine with a fixed number of tapes; the cost of an operation is the total head +travel; a record is a p-bit coefficient record with a header of O(g log L) bits (g=p^beta, L=poly(p) the microbox side, beta<1, so the header is o(p) +and moving it multiplies the cost by 1+o(1)). V is the number of records of the microbox (including spectators). All notation (E_i, J_i, T_i, mu, +W, B, d, g, L, the windows) is that of PHASE-LAYOUT-AUDIT.md and the capped interiors of CAPPED-INTERIORS.md; mu = max_i |E_i|/L = O((W/B) polylog p). + +A sparse window applies the Kronecker product `Theta = (x)_{i=1..g} T_i` to the microbox tensor, where `T_i = I_{E_i^c} (+) T_i|E_i` (for the window +containing the C and S^-1 stages, T_i is the product of the two sparse factors of axis i; each is block diagonal in the same way). +The audit's implementation scans the union U = {x : some x_i in E_i} once per axis to copy U_i = E_i x [L]^(g-1): cost O(|U|) per axis, hence +`O(V [g^2 mu + g mu polylog p])` per window and the union row `eps + beta + 1/2 - xi`. Here the g-fold scan of U is removed. + +## 1. Flag sets and class decomposition + +For a record x put `F(x) = {i : x_i in E_i}` (the g membership flags the audit already computes in the single scan that forms U; stored as a g-bit vector +in the header). + +**(F1) invariance.** `T_i` changes a coordinate x_i only inside E_i and fixes E_i^c, and maps E_i to E_i. Hence `F(Theta x) = F(x)`: every T_j preserves every class +`C_F = {x : F(x) = F}`. + +**(F2) commutation.** Each T_i depends only on the axis-i coordinate (and on axis i's own coarse label/page, fixed per microbox), so T_i acts on tensor factor i alone; the T_i act on different tensor factors, hence commute, and on class C_F the operator Theta equals `(x)_{i in F} T_i|E_i` on the coordinates in F +(range E_i) tensored with the identity on the coordinates not in F (the T_j, j not in F, are the identity on E_j^c). + +**(F3) complete lines.** Fix F, f in F and the other coordinates z (all coordinates except x_f). The *E_f-line* of z is {x : x_f in E_f, other coordinates z}. Every point +of it has the flag set `F(z) union {f}` (flags of z-coordinates plus f), so the whole E_f-line lies in one class, and contains exactly |E_f| points. + +## 2. The schedule + +Let `f_r(F)` be the r-th smallest element of F. Define `S_r = {x : |F(x)| >= r}`, kept as a stream in lexicographic order of x (S_1 = U; S_{r+1} is S_r with +the records |F|=r deleted, so lexicographic order is inherited). + +Round r (r = 1,2,...,R, where R = max |F| <= g) applies to every record x of S_r the operator T_{f_r(F(x))} along axis f_r(F(x)), to complete E_f-lines (F3). +After the rounds, every record of class C_F has received T_i exactly once for each i in F (the f_r(F) enumerate F), in increasing axis order; +by (F2) this equals the action of Theta. Since lines of the same round and class are disjoint and independent (F2), applying T_f to each line separately equals +the action of `T_f|E_f (x) identity` on the class. Rounds use the values written by earlier rounds because records carry their current values in the stream. + +### 2.1 One round (cost accounting per step; N = |S_r|) + +1. **Tag.** For each record compute `t = f_r(F(x))` from its header bits: O(g) bit operations (scan for the r-th set bit). Worst case a record has |F| = g and takes g rounds, + so tag work per visit is O(g) bit operations, paid by the Theta(p) cost of that visit (Section 3, header and tag work): g log L = o(p) since beta < 1 (ledger check `union_tag_work_per_visit_fits_record`). +2. **Stable binary LSD radix distribution by tag.** For k = 0..ceil(log2 g)-1: scan the stream; records whose bit k of the tag is 0 go to output tape Z, bits 1 to tape O + (one input tape, two output tapes), then Z is copied followed by O back onto the work tape. Each pass costs <= 2N record moves; the bits of the pass are written (N bits) + to a bit tape together with the count n0 of zeros. *Stability* by induction on k: after pass k the stream is ordered by (bits k..0 of tag) with ties in the original order. + After all passes: ordered by tag, ties in lexicographic order; the stream splits into contiguous axis substreams, one per tag value f. + Inversion (used in step 5): replay the passes in reverse: copy the stream onto Z/O at the stored split point n0 (one pass), then read the bit tape and take the next record from Z or O accordingly. + Cost 3N per pass. Total O(N log g) = O(N log p). +3. **Structure of an axis substream (the key lemma).** Axis f substream = {x in S_r : f_r(F(x)) = f}. Membership depends on F(x) only, and F(x) = F(z) union {f} for x with x_f in E_f (all substream records + have x_f in E_f, because f is in F(x)); so it depends on z only, not on x_f. Therefore for each fixed prefix x_f} is the same for every x_f in E_f. In the stream (lexicographic order of (prefix, x_f, suffix)) the prefix block is the concatenation of + a = |E_f| rows (one per x_f in E_f, in increasing order), each row listing the same b = |Z'(prefix)| suffixes in the same order. No coordinate is invented: every needed point is present. + Block boundaries are detected by comparing the prefix fields of consecutive records (O(g log L) bits per comparison, o(p)); rows by the ordinal of x_f in E_f stored in the header + (computed with the flags: the rank of x_f in E_f costs O(polylog p) with the carving description) -- no counter of unknown length is needed. +4. **Variable-length lockstep transpose of every prefix block** (brings x_f to the line suffix). Let `a = |E_f| <= L = poly(p)`. All rows of a prefix block have the same length b within a tag (point 3), b unknown to the routine. Recursively halve the row index range into the upper part of ceil(a/2) rows and the lower part of floor(a/2) rows (the implementation uses chunk sizes h = floor(a/2) for the first child and a-h = ceil(a/2) for the second; either order works, the tree must be the same for every block and its inverse). + Level structure: a balanced binary tree of row ranges of depth ceil(log2 a) = O(log p), the same tree for every block. *Merge of a node*: its children's outputs `TA` (rows 0..h-1 in suffix-major order, + i.e. for each suffix the h records of A's rows) and `TB` (the a-h rows of B) are read in lockstep from two tapes, taking h records from TA, then a-h records from TB, alternately, until both end markers + coincide; the output is the node's suffix-major order `[row_0[j], ..., row_{a-1}[j]]_j`. The routine never needs the row length b: the chunk sizes h and a-h depend on a alone, and the end of the node's data is + the end marker (prefix-block boundary, or the stream end). A level is realised by two passes over the whole substream. Every record carries in its header the ordinal k of its source row x_f. The data of a node (a range [lo, lo+n) of ordinals) is a contiguous run of records of one prefix block whose ordinals lie in the range; its child A (ordinals [lo, lo+h)) comes first, immediately followed by child B (ordinals [lo+h, lo+n)), so the junction is the first record whose ordinal is >= lo+h, and the end of the node is the first following record whose ordinal is >= lo+n or whose prefix field differs (both read from headers; the tree is the same for every block and nodes appear in row order). Pass 1 copies the A-parts to tape M1 and the B-parts to tape M2, node by node. Pass 2 reads M1 and M2 in lockstep, h records from M1 then n-h from M2, node by node (a node's A- and B-parts end at the same node marker), writing the parent level. So a level costs O(N) record moves and the whole transpose O(N log a). After the transpose, each line (fixed prefix and suffix) occupies a consecutive group of a records with increasing x_f ordinal. +5. **Apply T_f on lines and batch.** Each consecutive group of a records is a complete E_f-line; T_f|E_f is applied exactly as in the audit's step 4: consecutive lines form right-hand sides, batched in dyadic width between W and 2W, + factor pages (regenerated once per microbox and axis from its coarse origins, polynomial in p, amortised by the superpolynomial payload) scanned once per batch, only the polynomial-width row field moved + (O(log p) times the boundary volume). This arithmetic is the *boundary-algebra row*, unchanged and not part of the union movement. +6. **Undo** the transpose (unzip levels), then undo the radix distribution (step 2 inverse): the processed records are back in lexicographic order of S_r. +7. **Split.** One scan: records with |F| = r are final for this window; push them (with the |S_r|-bit indicator keep_r = [|F| > r]) on the LIFO tape; the rest, S_{r+1}, is written to the alternate work tape. + +### 2.2 Wrap-up + +After the last round, pop the LIFO tape: for r = R,...,1 interleave `rest_r` with the already final stream S_{r+1} by keep_r (the audit's merge by indicator), producing the final S_1. +Finally merge S_1 into the microbox using the indicator bits of the single scan that formed U; identity records (F = empty) are never moved. +The two dense-stage scatters/regathers, the one scan forming U and the final merge are the audit's group scans: O(V polylog p) per window, in the group-scan row. + +### 2.3 Tape-head accounting (per round, and why rounds do not multiply the cost) + +Tapes: W1, W2 (alternating work tapes carrying S_r), Z, O (radix outputs), Tb (bit tape), STACK (LIFO), M1, M2 (merge/zip tapes), small header/counter tapes: a fixed number. +* W1 holds S_r written from cell 0; before round r the head is at cell 0 (rewind over the written length |S_r| only; cells beyond are stale and never read). +* S_{r+1} is written to W2 from cell 0, then the roles swap. Rewinding costs the written length, at most |S_r|. +* STACK is only appended to at its end and, in the wrap-up, read backwards from its end; the rest_r are consecutive, so popping costs the sum of |rest_r| <= sum |S_r|. No head ever travels over the whole of + the stack or of U per round. +* Z, O, Tb, M1, M2 are written from cell 0 and rewound over their written length (<= |S_r| records or bits). +Hence the head travel of round r is O(|S_r| p (log g + log a + 1)) and the total is O(p (log g + log L + 1) * sum_r |S_r|). The rounds are sequential, but their *number* does not multiply the cost: |S_r| shrinks and +nothing in round r touches data outside S_r. + +### 2.4 Volume identity + +`sum_r |S_r| = sum_x |F(x)| = sum_i |U_i| = sum_i |E_i| L^(g-1) * (spectators) <= g mu V.` (Each record is counted once per flag; the first equality is exchanging the order of summation.) + +## 3. New cost and row + +Per window: `O(V g mu polylog p)`; no factor g^2 and no per-axis scan of U. Per group, then over d/g groups: + +`O(V (d/g) g mu polylog p) = O(V d W polylog p / B)`, + +i.e. union-movement exponent `eps + 1/2 - xi` (saving `xi + 1/2 - eps`). Old audit row: `eps + beta + 1/2 - xi`. +The identity `sum|F| <= g|U|` shows the new cost is never larger than the old one. + +*Sparsity is not used.* In the regime of this package the expected number of flags g*mu = p^(beta+1/2-xi) is larger than 1 (xi - beta - 1/2 < 0), so U is essentially the whole microbox, and the audit's ledger check +`sparse_union_fraction_decays` is not used. The cost is the total number of record-flag incidences, not |U|. +Header and tag work: the flag header has g bits and g log L bits of coordinates (o(p) since beta < 1); tag work is at most O(g) per visit and a record is visited at most |F| <= g times: each visit costs Theta(p polylog p) anyway, so the total over the sum_r |S_r| <= g mu V visits is O(g mu V (p polylog p + g log L)) = O(g mu V p polylog p) as soon as g log L = o(p), i.e. beta < 1. (An earlier version of this note charged O(g^2) per record against o(p) and required 2 beta < 1; that per-record total is not needed, see outer/GAUGEFREE-INTERIORS.md section 7.) + +## 4. Consequence for the ledger + +Rows: main `eps q`; fused Gaussian `1 - eps + q beta`; boundary `c + xi - eps` with `c = (3 - tau)/2`, `tau` = the explicit Strassen exponent 45/16, c = 3/32, in the ledger (the structured-boundary solves of outer/STRUCTURED-BOUNDARY.md, tau_eff = 2 + 1e-3080, are historical and not used for kappa; this document's schedule is independent of the boundary exponent); +union `xi + 1/2 - eps`; guard `beta = 1 - xi - r - zeta` (linear gauge; in this package the guard is the chirp-only `beta = 1 - (2 xi - eps) - r - zeta`, see Section 5b); `xi > 1/2`. Balance set: main = fused Gaussian = boundary - slack, which gives +`eps = (1 + q + q (c - s - r - zeta))/(1 + q)^2`, `xi = eps (1 + q) - c + s`. The union row is far above kappa at this point (saving about 0.080 here). +Rejecting controls in the ledger: reinstating the old row `1 - eps - beta - 1/2 + xi` at the new point fails (it is about -0.125); the full-key sort row fails; the four inherited controls fail. + +## 5. Executable check (`outer/sched_test.py`, and the independently coded `outer/sched_review.py` by LeverRank) + +The script runs the schedule above on exact integer tensors (g = 3..6 axes, L = 5..8, random E_i and random integer T_i|E_i): tagging, the stable LSD radix distribution with its inversion, per-prefix-block variable-length lockstep zip +transposes with a check that every extracted group of a records is a complete E_f-line with consecutive ordinals and one common z, application of T_f, unzip, undo, the keep/rest split, the LIFO pop and the merge by indicator bits, +and compares with the full tensor product `(x) T_i` computed by tensor contraction: exact equality. It also checks the volume identity `streamed = sum|F| = sum_i |U_i|`, `sum|F| <= g|U|` (), +that the number of record moves is within `3 * sum|F| * (ceil(log2 g) + 2 ceil(log2 a) + 4)`, and a tamper control (skipping the second round) fails. + +## 5b. Rows that depend on beta, re-verified at the balance of this package + +At the fine-prime-padding point (outer/FINE-PADDING-COMPOSITION.md; Strassen boundary c = 3/32, theta ~ p^-2): epsilon = 1 - 1.0009765625e-3080, beta = 1 - 3.0029296875e-3080, xi = 0.90794631...; every row is regenerated exactly by `check_ledger.py` and re-derived by `independent_check.py` (all strictly above kappa or strictly positive): +group scans 1-eps+beta = 1 - 2.001953125e-3080 (saving far above kappa); union xi+1/2-eps = 0.4079 (boundary = main + 1e-3080 binds; the union is no longer binding; the old row 1-eps-beta-1/2+xi = -0.5921 fails: rejecting control, so the radix schedule is still essential); +cap xi-1/2 = 0.4079 > 0 and eps-xi > 0.092; group positive and below dimension (eps-beta = 2.001953125e-3080 > 0); support router span 1-beta = 3.0029296875e-3080 > 0 (the supplier's total moved support O(g polylog p) = o(p) holds eventually, with exponent margin 3.0029296875e-3080); +gauge/guard 1-(2xi-2)-r-beta = 0.184 (fine theta; the linear guard 1-xi-r-beta is negative: rejecting control); group key fits the record (eps-beta = 2.001953125e-3080); membership arithmetic 1-beta-2(1-eps) = 1.0009765625e-3080; union tag work per visit 1-beta = 3.0029296875e-3080 (the old test 1-2beta is negative and not used); union header 1-beta; core-face corrections eps+beta+48-128+2 < 0. +The ledger row `sparse_union_fraction_decays` (xi-beta-1/2 > 0) stays deleted: it is false (g mu = p^(beta+1/2-xi) > 1) and the schedule does not use it. + +## 6. What this does and does not establish + +Established here: the data-movement schedule, its correctness (class invariance, commutation, complete lines, stability of the radix passes, the lockstep transpose, LIFO interleave) and its tape-head cost O(V g mu polylog p) per window, assuming +the audit's carving (|E_i| and mu), the retained polynomial-width row-field movement for batched boundary arithmetic, and the retained dense/gauge/precision lemmas of PR1673. +Not established: nothing in this document changes the arithmetic, the Gaussian conditioning, the supplier, the boundary matrix exponent or the all-size hypotheses of PR1673; those remain inherited assumptions. +The step "variable block length" of the transpose (item 4) and the head accounting (2.3) are the places that most need an independent review. diff --git a/research/kappa-observation-integration/release-work/current/package/outer/check_ledger.py b/research/kappa-observation-integration/release-work/current/package/outer/check_ledger.py new file mode 100644 index 0000000000..d3b6149c5c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/outer/check_ledger.py @@ -0,0 +1,260 @@ +#!/usr/bin/env python3 +"""Exact ledger for fine-prime padding (PR1786, chafreaky / Chafik Boukhalfa, with Tim Trudgian's prime theorem) at the free point epsilon = 1 - (1025/1024)*10^-(DIG+80), explicit Strassen boundary, retuned to the fixed-grid maximum kappa = C - 1/FLOOR - BACKOFF. + +Adapted from PR1786's ledger, itself adapted from PR1731's cost ledger, credited to Jeremy Kothe et al. and +Vlado Vojdanovski (PR1673); uses PR1758's gauge-free chirp and per-visit tag +accounting. New source-prime, coefficient-setup and halo lemmas are in +FINE-PRIME-PADDING.md. Finite arithmetic does not prove the all-size theorem. +""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +from math import factorial +import hashlib, json, sys +sys.set_int_max_str_digits(0) + +if not __debug__: raise RuntimeError('Assertions must be enabled') +HERE=Path(__file__).resolve().parent +WORK=HERE.parent +SPEC=WORK/'new/SPEC.json' +CLAIM=WORK/'CLAIM.json' +BJ=json.loads((HERE/'B.json').read_text()) +spec=json.loads(SPEC.read_text()) +# Free strict slacks of the PR1673 proof (retuned in this package, see RETUNED-SLACKS.md); every one only needs to be > 0. +DIG=spec['outer_refinement']['endpoint_digits'] # the supplier endpoint is certified to 10^-DIG (accept at the endpoint, reject at endpoint + 10^-DIG); every slack sits 10 digits below +# Fixed-grid maximum (PR2180 / PR2161, dataisfire; proof PROOF-GRID-MAX.md): the free gaps shrink together to R0 = 10^-(DIG+80); grid, backoff and certified endpoint are unchanged. +R0=Q(1,10**(DIG+80)); ZETA=R0/100; SLACK=R0; GAP=R0; STOP=R0/10**6 +FLOOR=10**(DIG+46); BACKOFF=Q(1,10**(DIG+22)); B_GAP=R0 +# Fine prime padding: theta is between 1/(4p^2) and 1/(2p^2-1). +# Online chirp guard beta+2*xi-2<1; coefficient setup uses O(p^2 log p) bits. +# epsilon = 1 - DELTA_E is a FREE parameter (outer/FINE-PADDING-COMPOSITION.md sec. 2): every row is strictly satisfied for every DELTA_E > r0 = 10^-(DIG+80); beta = 1 - 3 DELTA_E lies in the admissible window +# [1 - DELTA_E (1 + 1/q), 1 - 2 DELTA_E). DELTA_E = (1025/1024) r0 keeps the gamma spare power 1-eps-r0 = r0/1024 > 0 and puts C - eps*q below 1/FLOOR (PROOF-GRID-MAX.md). +DELTA_E=Q(1025,1024)*R0 +EPS=1-DELTA_E +BETA=1-3*DELTA_E +THETA_POWER=Q(2) +HALO_POWER=48 +CORE_POWER=128 +STRASSEN_TAU=Q(45,16) +KAPPA_1828=Q('0.0016797549561255445213356709939207025769983114385025802954603275201627999999999999999999999999979966404900867489109573286580121585') # exact kappa of rohanarun's PR1828 (slack refinement of PR1800 at 1e-96), from its body +KAPPA_1839=Q('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263049999999997996640490086748910957328658012158594') # exact kappa of PR1839 (previous package: 1e-90 endpoint) +KAPPA_1875=Q('0.0016797549561255445213356709939207025769983114385025802954603275201628645482603256388263056561328446322744513169085862209720704481201654121427713006366929567157349625273906229277148454180995537563371124755588139767841405530386745578819565618798006892541712197674874836946756930569827889227573192555649999999997996640490086748910957328658012158594') # exact kappa of rohanarun's PR1875 (1e-300 endpoint), from its body +KAPPA_1876=Q('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310799999999999999999997996640490086748910957328658012158594846') # exact kappa of rohanarun's PR1876 (1e-400 endpoint), from its body +KAPPA_1897=Q('0.001679754956125544521335670993920702576998311438502580295460327520162864548260325638826305656132844632274451316908586220972070448120165412142771300636692956715734962527390622927714845418099553756337112475558813976784140553038674557881956561879800689254171219767487483694675693056982788922757319255565606685566411725238449650999339827075715283714270157705762309226038611771955655377501414652136546310881185746917580802513640545967852810801680966856133978576162943792322682482152988532923811998938861379689883167389728443314312230815059559906903773756886872476136837071871203030423643505345473735135533') # exact kappa of M6LI's PR1897 (1e-500 grid), from its body +KAPPA_1786=Q('0.0016797547881500489087805478664999999999999999999999999999999998000000199999') # exact kappa of chafreaky's PR1786 ledger (EXPECTED-BALANCE.json), headline 0.001679754788150048 +KAPPA_PKG5=Q('0.001677992607205282520732088774830808372222877897034109604209719298829805473412451530867883424937597') # previous package (rectangular k=2, tau_eff=2.2500351) +KAPPA_PKG4=Q('0.0016778199023144364336310551358776001013829401215770658599799349282716974076737897457922714197448847') # previous package (radix union schedule, square boundary tau=2.372864) +KAPPA_1712=Q('0.0016776417313433545642517275958639984231708545722735432722117400060454625429077941422798526440584702') # pkg of PR1712 (Le Gall tau=12/5, old union row) +KAPPA_1699=Q('0.001677641731343354564251058287063644749065568493') # exact kappa of PR1699's ledger (Chafik Boukhalfa), same supplier profile +certified_complex=Q(spec['expected']['complex_lower']) +bit=Q('0.001733427039589844234749') +# Keep short independently checkable decimals strictly inside both endpoints. +b=certified_complex-B_GAP # strictly inside the certified supplier endpoint; B.json carries the same exact value +assert b==Q(BJ['b']) +q=b-GAP +a=(q+b)/2 +online=Q(spec['expected']['kappa']) +old=Q('0.001673360105405778') +assert b0 +assert m==73 and W==896*T and W==BJ['live'] +EXPECTED_PROFILE={int(k):v for k,v in BJ['profile'].items()} +assert profile==EXPECTED_PROFILE + +GB=int((DIG+130)*3.3220)+16 # enclosure grid 2^-GB, about 50 digits below the slacks +ONE=1<=2:x/=2;k+=1 + lo,hi=unit_log(x) + return down(lo+k*ln2[0]),up(hi+k*ln2[1]) +def exp(x): + assert 0<=x>GB)//j;th=_cd(-((-th*xu)>>GB),j);sl+=tl;sh+=th + tn=_cd(-((-th*xu)>>GB),EN) # x^EN/EN! (rounded up); the tail is at most twice this + return Q(int(sl),ONE),Q(int(sh+2*tn+2),ONE) +def moment(endpoint): + lo=hi=Q(0) + for r,n in profile.items(): + l,h=log(Q(m,r)); weight=Q(n*r,m*W) + lo+=weight*exp(endpoint*l)[0] + hi+=weight*exp(endpoint*h)[1] + return down(lo),up(hi) + +native=moment(b) +assert native[1]<1 +b_up=Q(BJ['b_above_endpoint']) +assert b_up>certified_complex and moment(b_up)[0]>1 # a larger test endpoint is rejected + +def ledger(kind, *, q_=q, zeta=ZETA, slack=SLACK, + old_scan=False, full_gauge=False, full_key=False, old_union=False, old_theta=False, old_linear=False, old_tag=False, + eps_=EPS, beta_=BETA, tau_eff=STRASSEN_TAU, halo_power=HALO_POWER, core_power=CORE_POWER): + r=R0; stop=STOP + # Existing explicit Strassen format, 7^16<2^45; c=3/32. + # Union: radix-distributed schedule (outer/UNION-SCHEDULE.md, new in this package): row exponent eps+1/2-xi. The boundary row binds with main; the Gaussian row has slack at this explicit point. + c=(3-tau_eff)/2 + eps=eps_; beta=beta_ + xi=eps*(1+q_)-c+slack + saving=eps*q_ + # Deliberate strict gap (BACKOFF) after decimal truncation to the FLOOR grid. + kappa=Q((saving*FLOOR).__floor__(),FLOOR)-BACKOFF + # kappa is derived, not silently asserted to equal an expected decimal. + tau=1-a; sigma=1-b + internal=tau+(1-stop)*max(sigma-tau,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q_+a)/2 + key=beta+1-eps + margins={ + 'main_transform':saving, + 'global_coordinate_router':a, + 'fused_gaussian':1-eps+q_*beta, + 'group_bit_router':1-eps+a*beta, + 'group_scans':1-eps+beta, + 'boundary_algebraic_matrix':c+xi-eps, + 'sparse_union_movement':xi+Q(1,2)-eps, + 'polynomial_products':eps, + 'microbox_face_corrections':1-(eps+beta+halo_power-core_power+2), + } + if old_union:margins['old_g_factor_union_row']=1-eps-beta-Q(1,2)+xi + if old_scan:margins['old_per_axis_scan']=1-eps + if full_key:margins['unnecessary_full_key_sort']=margins['sparse_union_movement']-key + checks={ + 'actual_bit_above_effective':bit-a, + 'complex_above_effective':b-a, + 'q_positive':q_, 'complex_endpoint':b-q_, + 'q_below_effective':a-q_, 'stop_positive':stop, + 'stop_below_one':1-stop, + 'leaf_saving_above_effective':(1-stop)*b-a, + 'q_below_internal':1-internal-q_, 'q_below_leaf':1-leaf-q_, + 'lambda_above_tau':lam-tau, 'lambda_above_sigma':lam-sigma, + 'lambda_above_internal':lam-internal, + 'lambda_prime_above_lambda':1-q_-lam, + 'lambda_prime_above_leaf':1-q_-leaf, + 'epsilon_positive':eps,'epsilon_below_one':1-eps, + 'cap_above_band':xi-Q(1,2),'cap_below_cell':eps-xi, + 'group_positive':beta,'group_below_dimension':eps-beta, + 'support_router_span':1-beta, + 'group_gauge_power_slack':1-(2*xi-THETA_POWER)-r-beta, + 'actual_logarithmic_group_guard':1-(2*xi-THETA_POWER)-beta, + 'global_gamma_power_slack':1-eps-r, + 'actual_logarithmic_gamma_guard':1-eps, + 'alpha_logarithm_absorption':r, + 'group_key_fits_record':1-key, + 'membership_arithmetic_fits_record':1-beta-2*(1-eps), + 'microbox_halo_volume':core_power-halo_power-beta, + 'positive_layout_slack':slack, + 'short_record_fallback':eps-a, + 'improvement_over_online_final':kappa-online, + 'improvement_over_previous_local_final':kappa-old, + 'complex_below_one_over_32':Q(1,32)-b, + } + checks['strassen_integer_power_gap']=Q(2**tau_eff.numerator-7**tau_eff.denominator) + checks['theta_positive_polynomial_power']=THETA_POWER + checks['cap_below_phase_cell_power']=THETA_POWER-xi + checks['halo_covers_generic_inverse_radius']=Q(halo_power-43) + checks['union_tag_work_per_visit_fits_record']=1-beta + checks['union_header_fits_record']=1-beta + checks['source_axis_superpolynomial']=1-eps + checks['improvement_over_pr1786']=kappa-KAPPA_1786 + checks['improvement_over_pr1828']=kappa-KAPPA_1828 + checks['improvement_over_pr1839']=kappa-KAPPA_1839 + checks['improvement_over_pr1875']=kappa-KAPPA_1875 + checks['improvement_over_pr1876']=kappa-KAPPA_1876 + checks['improvement_over_pr1897']=kappa-KAPPA_1897 + checks['improvement_over_pr1758']=kappa-Q('0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983') + checks['improvement_over_pr1699']=kappa-KAPPA_1699 + checks['improvement_over_pr1712']=kappa-KAPPA_1712 + checks['improvement_over_previous_package']=kappa-KAPPA_PKG4 + checks['improvement_over_pkg_k2']=kappa-KAPPA_PKG5 + if full_gauge:checks['unrecentered_group_gauge']=1-THETA_POWER-r-beta + if old_theta:checks['old_theta_chirp_guard']=1-(2*xi-eps)-r-beta + if old_linear:checks['old_linear_guard']=1-xi-r-beta + if old_tag:checks['old_union_tag_2beta']=1-2*beta + # Main-binding family invariant (PR2180): the Gaussian saving stays above the main saving. The fixed kappa grid (FLOOR) is coarser than the free gaps, + # so the *_above_kappa rows alone would no longer exclude a beta outside the main-binding window; this row is delta*(1-2q) > 0. + checks['gaussian_above_main']=margins['fused_gaussian']-saving + checks.update({n+'_above_kappa':v-kappa for n,v in margins.items()}) + assert margins['boundary_algebraic_matrix']==saving+slack + assert margins['sparse_union_movement']==saving+slack+(tau_eff-2)/2 + return {'kind':kind,'kappa':kappa,'epsilon':eps,'xi':xi,'beta':beta, + 'q':q_,'a':a,'r0':r,'zeta':zeta,'slack':slack, + 'margins':margins,'strict_checks':checks} + +def encode(x): + if isinstance(x,Q): + with localcontext() as c: + c.prec=DIG+200 + return {'rational':str(x),'decimal':str(Decimal(x.numerator)/Decimal(x.denominator))} + if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()} + if isinstance(x,(list,tuple)):return [encode(v) for v in x] + return x + +rows={};controls={} +for kind in ['struct_boundary_radix_union']: + row=ledger(kind) + assert all(v>0 for v in row['strict_checks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + # fixed-grid maximum: C*FLOOR is an integer and 0 < C - eps*q < 1/FLOOR, so floor(FLOOR*eps*q)/FLOOR = C - 1/FLOOR exactly and kappa attains the grid/backoff ceiling + assert (certified_complex*FLOOR).denominator==1 and 0= 1/(4 p^THETA_POWER)), an exponential factor <= p^2 eventually, +# alpha^5 polylogarithmic; a further half power is allowed for the polylogarithmic and bounded-constant factors of the augmented matrices. The umbrella must exceed that budget. +UMBRELLA_POWER=10 +def umbrella_margin(theta_power):return UMBRELLA_POWER-(2*theta_power+2+Q(1,2)) +assert umbrella_margin(THETA_POWER)>0 and umbrella_margin(Q(15,4))<=0 # control: the umbrella would fail silently for THETA_POWER >= 3.75 +HALO_UMBRELLA={'umbrella_power':UMBRELLA_POWER,'theta_power':THETA_POWER,'exponent_budget':2*THETA_POWER+2+Q(1,2),'margin':umbrella_margin(THETA_POWER),'fails_from_theta_power':Q(15,4)} +out={'status':'PASS_FINE_PRIME_PADDING_EXACT_ARITHMETIC', + 'scope':'Conditional fine-prime padding composition. Fresh native admission is required by verify.py. The cited prime theorem, written gauge-free theorem and inherited all-size interfaces are not proved by this arithmetic. No independent human review is claimed.', + 'geometry_changes':{'theta_power':THETA_POWER,'halo_power':HALO_POWER,'core_power':CORE_POWER,'matrix_exponent':STRASSEN_TAU}, + 'halo_umbrella':HALO_UMBRELLA,'complex_lower_short':b,'complex_lower_pinned':certified_complex, + 'native_moment':native,'profile':profile,'m':m,'W':W, + 'online_final_kappa':online,'rows':rows,'rejecting_controls':controls, + 'pr1673_kappa':Q('0.001677160151714381'),'improvement_over_pr1673':rows['struct_boundary_radix_union']['kappa']-Q('0.001677160151714381'), + 'source_sha256':{p.relative_to(WORK).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in [SPEC,CLAIM]}} +(HERE/'LEDGER.json').write_text(json.dumps(encode(out),indent=2)+'\n') +print(out['status']) +for name,row in rows.items(): + print(name,'FINAL KAPPA',encode(row['kappa'])['decimal'], + 'positive checks',len(row['strict_checks']), + 'gain over online',encode(row['kappa']-online)['decimal']) diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md new file mode 100644 index 0000000000..abdd30cc1d --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md @@ -0,0 +1,190 @@ +# Independent algebraic-coefficient and stability transfer lemma + +This note uses no claimed finite certificate from any particular matrix multiplication paper. It isolates what follows from an exponent bound alone, including effectiveness and ordinary bit complexity. + +## Statement + +Let `omega_C` be the matrix multiplication exponent over the complex numbers. Suppose + +\[ +\omega_{\mathbb C}<\tau, +\] + +where `tau > 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md new file mode 100644 index 0000000000..5cad6ee19f --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/BILATERAL-LDU.md @@ -0,0 +1,131 @@ +# Bilateral Gaussian LDU and a fixed-width periodic inverse bound + +Independent audit of the parent's infinite-LDU observation. This is an analytic norm theorem. It does **not** assert that the bilateral factors already have a fully priced fast implementation. + +## Statement + +Let `1=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-SIGNOFF.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-SIGNOFF.md new file mode 100644 index 0000000000..fc3456387b --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/CAPPED-SIGNOFF.md @@ -0,0 +1,89 @@ +# Capped-interior layout extension: positive signoff + +The earlier sparse-union layout proof extends to dyadic interior cap +`B=p^b infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md new file mode 100644 index 0000000000..60d59bb378 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/HALO-THEOREM.md @@ -0,0 +1,271 @@ +# Fast tensor convolution by core blocks and sparse face corrections + +This gives a constructive replacement for g dense halo scans. It applies to +a **translation-invariant separable compact stencil**, with optional +diagonal conjugations whose coefficients and tile gauges satisfy the +precision and sequential-access conditions below. It does not claim that +the separate Gaussian product-tree factorization has this form. + +Let g consecutive active axes have dyadic lengths S_i, with +`h=max_i log2 S_i`. Earlier coordinates are outer prefix fibers. Later +coordinates are a complete payload R, so they need no keys or transposition. +Let V denote the total bit volume, including B-bit coefficient records and +all spectators. Assume B=Theta(p), `g h=O(p)`, and the active group grows +superpolynomially. A fixed number of tapes may hold O(V) bits. + +The operation is + +``` +(Tf)(x) = sum_{z in [-L,L]^g} [product_i k_i(z_i)] f(x-z mod S), +``` + +where `L=poly(p)`. A fixed or polylogarithmic number of these operations and +diagonal scans is allowed. Kernel coefficients, coefficient products and +the required intermediates have O(p)-bit representations under the supplied +normalization contract. The retained grouped convolution engine on a +binary field of width k has cost `O(V k^(1-q) polylog p)`. Polynomial-length +one-axis convolution also has its unconditional quasilinear bit algorithm; +it does not invoke the improved bound being proved. + +Choose a dyadic tile side N dividing every S_i, with `N>4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. This resolves the length caveat in section 5 for the intended phase-cell application; it does not prove the separate simultaneous tensor-layer routing claim. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. The first theorem's records are not overwritten. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires gkappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. Exact positive inequalities alone do not discharge those interfaces. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md new file mode 100644 index 0000000000..bbeeee7431 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md @@ -0,0 +1,242 @@ +# Grouped phase-cell layout audit + +The sparse-union layout in `REFINED-PHASE-FUSION.md` has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. The outer conclusion remains conditional on exactly the previously retained supplier/computational assumptions. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/README.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/README.md new file mode 100644 index 0000000000..f7f3348ee3 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/README.md @@ -0,0 +1,162 @@ +# Capped phase-cell fusion: conditional final kappa = 0.001677160151714381 + +This package gives the conditional final integer-multiplication bound + +\[ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{1677160151714381}{10^{18}}=0.001677160151714381. +\] + +The contribution changes the **outer Gaussian resampling reduction**. It uses +shallow in-place Toeplitz circuits, a common Hadamard schedule for multiple +axes, recentered and capped Gaussian cells, and explicitly charged sparse +boundary operations. The finite supplier is the unchanged, pinned +[PR1661](https://github.com/CrocSwap/integer-mult-bounds/pull/1661) T74 construction. +Its complete native verification was run afresh, including both new provider +words and the retained parent. The final exponent is a conditional theorem +from the written transfer argument; finite checks alone do not prove it. + +Read [the proof](phase_fusion_note.tex), especially its T74 substitution section, +and the [independent substitution review](frontier_fusion/INDEPENDENT-SUBSTITUTION-REVIEW.md). +The proof's earlier numerical specializations remain valid historical cases. +“Independent” in this package means separate agent implementations/reviews, +not an independent human referee or a proof-assistant verification. + +## Exact paid inputs and result + +| Quantity | Value | +|---|---:| +| Retained bit saving c | 0.001733427039589844234749 | +| Chosen complex saving b | 0.00167971337478 | +| Effective tensor saving q | b - 10^-10 | +| Effective bit saving a | (q+b)/2 | +| Native dimension m | 73 | +| Native live width W | 66304 | +| Final kappa | 0.001677160151714381 | + +The complete native profile is +`{1:1559766, 2:337668, 3:278058, 4:84912, 5:54612, 8:5328, 48:22748}`. +All decimals above are exact rationals. Exact epsilon, xi, beta, every outer +cost power and all 45 strict inequalities are in +[FRONTIER-FUSION-ARITHMETIC.json](frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json). +Five deliberate bad variants reject. The native moment has an outward rational +upper bound below one; a larger test endpoint is rejected. + +For the same pinned supplier, PR1661's original outer reduction gives +0.0016768966690212993077507..., so the new outer transfer improves its final +exponent by about 2.6348269308e-7. This is an exponent comparison, not a measured +runtime improvement or a claim of global priority. The selected-result record +is not changed by this contribution. + +The supplementary branch gives 0.001677600129305161 **only with the additional +assumption omega_C <= 9/4**, used through a fixed algebraic algorithm of exponent +19/8. That extra premise is not needed for the headline Strassen-based result. +The supplied 9/4 paper, its audit and historical local locator are recorded in +PROOF-AND-COVERAGE.md and MATRIX-SOURCE-COVERAGE.json; it is treated as an explicit +additional premise, not as a theorem certified by this package. + +## Scope and remaining assumptions + +Retained assumptions include the all-size compiler, exact common-label and +mutable-source realization, whole-call hosting, the semantic precision guard +with C1=1, fixed-tape/prime-packing interfaces, paid arbitrary-width equal-chunk +interchange, exact recovery, and the original analytic transfer. The construction +does not prove an unconditional multiplication theorem, a complete Lean theorem, +or a practical finite cutoff. The written proofs address the new outer steps; +finite algebra and layout checks support, but do not replace, those arguments. + +Original assignment: improve the bound using broad mathematical resources. +For repository indexing this is a changed scope of `uniform-transfer-audit`: +it constructs and audits a replacement outer transfer rather than only checking +one inherited implication. [TASK.json](TASK.json) records the scope and next steps. + +## Reproduce + +Python 3.12, with SymPy 1.14.0, is the registered environment. The exact checks +also ran locally with the Python version recorded in PUBLICATION-CHECKS.json. +The package verifier checks its manifest and source pins, works in a temporary +copy, and compares the regenerated mathematical receipts. It does not modify +this directory. From the repository root: + +```sh +python3 -m pip install sympy==1.14.0 +python3 -B research/capped-phase-fusion/verify.py +python3 scripts/research_package.py --package research/capped-phase-fusion --check-only +``` + +The first command after installation is the bounded outer-proof evidence replay, +not a cold native-supplier replay. The native run already completed with all 12 +top-level stages passing, 319680 owners, prefix 1441 charged at 2048, physical row +coefficient 19545 below 20161, and rejected controls: 114 admission, 12 bit, +15 controller, 8 independent owner, 6 geometry. The retained compact RESULT +and [native binding](frontier_fusion/NATIVE-ADMISSION-BINDING.json) record it. + +To reproduce the full native dependency, obtain the exact source commit +`f852af5e178ce035cd46181a29d9330bcbcaa206` from CrocSwap/integer-mult-bounds, +install its documented dependencies (NumPy, SciPy, SymPy, a C++17 compiler and +Boost headers), and run its unmodified command without cache/reuse flags: + +```sh +python3 -B /absolute/pr1661/research/gmx-t74-p566-ep400/verify.py \ + --work /absolute/new-native-work --audit-shards 4 +python3 -B research/capped-phase-fusion/verify.py \ + --native-source /absolute/pr1661/research/gmx-t74-p566-ep400 \ + --native-work /absolute/new-native-work +``` + +The second command binds a completed external native run to this package's +source and mathematical result. It compares regenerated proof-relevant content; +fresh timings and log bytes need not match the historical run. The full 121 MB +assembly and vendor tree are not duplicated here. The receipt capsule adds canonical hashes for 46 JSON receipts. The 58 hashes +in the historical native binding identify external run artifacts; they are not all bundled files. +The original bind_native_admission.py is retained as historical provenance; +use the new top-level verifier for portable binding. + +PACKAGE-MANIFEST.json is the authoritative publication manifest. MANIFEST.json, +MANIFEST-UPDATE.json and earlier review receipts are explicitly historical +snapshots. The copied upstream CLAIM.json is immutable and contains older pending +language; the later completed native RESULT records this contribution's replay. +Hosted CI and independent human review are separate from the local checks. + +## Sources and overlap + +Prepared by **Vlado Vojdanovski with substantial AI assistance**. New material is +Apache-2.0. The original manuscript and source-specific notices retain their +licenses. This is not an official OpenAI release or endorsement. + +The main research snapshot was `e6f9716621f4e01fc190617934cdfaf072bd976f`; +the publication base is recorded in SOURCES.json. Exact T74 source pins are in +PUBLICATION-SOURCES.json and the source manifest. +The [unaltered T74 NOTICE](target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE) +is included. The original +OpenAI #109 framework and the Harvey–van der Hoeven reduction remain inputs. + +The in-place algebra builds on Giorgi–Grenet–Roche, +[*Fast In-place Algorithms for Polynomial Operations*](https://arxiv.org/abs/2002.10304), +and Dumas–Grenet, [*Fast in-place accumulation*](https://arxiv.org/abs/2302.13600). +SHALLOW-TOEPLITZ.md distinguishes their results from the new parity-split, +low-displacement and prefix-stability arguments. The Gaussian factorization +also relates to +[PR489](https://github.com/CrocSwap/integer-mult-bounds/pull/489), which already +gives a branch-free local Gaussian inverse but leaves joint evaluation, +routing and boundary composition open. The present package supplies a different +in-place tensor compiler and capped boundary transfer for those outer costs. +[PR1141](https://github.com/CrocSwap/integer-mult-bounds/pull/1141) concerns a +reversible quadratic phase stream and is related background, not a dependency. +The overlap check does not establish novelty beyond these identified sources. + +## Open next questions + +1. Can a new finite supplier exceed the present complex rate by a substantial + amount? Even a lossless outer transfer cannot reach 0.002 from b<0.00168. + First step: emit a complete side circuit for a lower-cost decoder and measure + its actual rank profile, helper storage and restoration. The missing piece is + the physical circuit, not an idealized central-rank estimate. +2. Can the capped Gaussian transfer be independently checked or formalized with + its complete fixed-tape schedule? First step: independently reconstruct the + sparse-union and support-local routing lemmas. Existing finite tests cover + algebraic identities, not an end-to-end all-size tape implementation. +3. Can the same outer transfer consume a future reviewed supplier without + changing its proof? First step: rebind the endpoint, prefix and completed-call + restoration contract, then replay the exact ledger. The supplier must satisfy + the full functional interface; a numerical endpoint alone is insufficient. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md new file mode 100644 index 0000000000..bc4349b2da --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SHALLOW-TOEPLITZ.md @@ -0,0 +1,197 @@ +# Shallow, stable, dimension-preserving Toeplitz circuits + +Status: a positive algebraic circuit construction for **dyadic local tiles**. It gives the missing full-Toeplitz primitive in the normalized Sylvester merge. It does not reconcile the anchor coordinates with the original Gaussian gauges; that remains a separate obligation. The construction uses no data ancilla, no zero-initialized coordinate and no fan-out. Fixed coefficient generation is allowed polynomial time in the local tile size and precision. + +## 1. Result + +Let N be a power of two, and let H be an N by N real or complex Toeplitz matrix with + + max(||H-I||_1, ||H-I||_infinity) <= delta < 1/4. + +There is an N-wire linear circuit applying H, with O(log^4 N) rounds of disjoint invertible 2 by 2 gates and diagonal gates, O(N log^4 N) gates, and polynomial-in-N bounds for all coefficients and all partial circuit operator norms and inverse operator norms. Constants depend only on delta. + +For any requested output accuracy 2^-p, rounding the coefficients to p+O(log N) bits gives operator error at most 2^-p; arithmetic at that precision also gives uniform forward error 2^-p on bounded inputs, after allowing the same O(log N) guard bits. A rational approximation to the input Toeplitz coefficients and all required gate coefficients can be generated using poly(N,p) bit operations. This coefficient preprocessing may use polynomial extra memory; the **data circuit** still has exactly N live coordinates. + +The same result applies to upper or lower triangular Toeplitz matrices with max(||T-I||_1,||T-I||_infinity)<=eta for **any fixed eta<1**, with only O(log^2 N) rounds and constants depending on eta. In particular eta=1/2 is permitted for the generator-normalization step below, even though the full-Toeplitz statement uses delta<1/4. Descendants need bounded norm and inverse, not continued distance below 1/4. This independent triangular construction is important: a contiguous bottom-recursion / shear / top-recursion has linear depth even though its work is quasi-linear. + +## 2. Dirty Toeplitz shear is a shallow circuit + +For disjoint m-coordinate vectors x and y, implement + + (x,y) -> (x,y+Tx), + +where T is Toeplitz, in O(log m) rounds when m is dyadic. It is an invertible square map on the existing 2m wires. + +Write the first column and row of T as c_0,...,c_(m-1) and c_0,r_1,...,r_(m-1). Set + + g_0=h_0=c_0/2, + g_j=(c_j+r_(m-j))/2, + h_j=(c_j-r_(m-j))/2, 1 <= j < m. + +Then T=C(g)+C_-(h), where C is a circulant and C_- is a negacyclic circulant. For a unitary m-point FFT F, C(g)=F^* diag(DFT(g)) F. A circulant shear therefore consists of F on x and y in parallel, the disjoint pairwise shears y_j+=DFT(g)_j x_j, and F^* on x and y. Every FFT stage consists of disjoint unitary 2 by 2 butterflies; bit reversal is a disjoint swap stage. + +For gamma=exp(pi i/m), D=diag(1,gamma,...,gamma^(m-1)), + + C_-(h)=D^-1 C(Dh) D. + +Use diagonal twists before and after the same circulant-shear circuit. The two completed shears commute. Source x is restored after each. No circulant inverse is needed, so zero Fourier multipliers cause no problem. Every gate is invertible even when T itself is singular. + +All intermediate norms are polynomial in m and the l1 norm of the Toeplitz symbol. In the 2-norm, FFT and twist stages are unitary, and the spectral diagonal has norm at most the symbol l1 norm. + +## 3. Independent triangular primitive: parity splitting + +Let T_N(f) be lower triangular Toeplitz multiplication by f modulo z^N. Write + + f(z)=e(z^2)+z o(z^2). + +Reorder the existing coordinates into even and odd powers, N=2m. Then + + T_N(f) = [ E ZO ; O E ], + +where E=T_m(e), O=T_m(o), and Z=T_m(z). All these m by m triangular Toeplitz matrices commute. Set + + k=o/e mod z^m, + s=e-z o^2/e mod z^m. + +The exact factorization is + + [ E ZO ; O E ] + = [ I 0 ; T_m(k) I ] + diag(T_m(e),T_m(s)) + [ I T_m(zk) ; 0 I ]. + +Thus apply the right Toeplitz shear, the two child triangular circuits **in parallel**, and the left Toeplitz shear. The recurrence for rounds is + + D_tri(N) <= D_tri(N/2)+O(log N), + +giving D_tri(N)=O(log^2 N). The two recursions act on disjoint coordinates. Inverses are obtained by reversing the gate sequence and inverting every gate, at the same depth. + +Stability is not proved by multiplying coefficient l1 bounds afresh at each depth. Each child is a principal submatrix or principal Schur complement of the original triangular matrix, after a coordinate permutation. If the initial T_N(f) differs from I in both 1- and infinity-norm by at most eta<1, every descendant M satisfies + + ||M-I|| <= eta+eta^2/(1-eta)=eta/(1-eta) + +in both norms, and ||M^-1||_2 <= 1/(1-eta). This follows directly by expressing the descendant as a Schur complement of a principal block of the original matrix; nested elimination and restriction preserve that description. Hence all local diagonal blocks and local Schur multipliers have constant 2-norms. Their triangular Toeplitz coefficient l1 norms are at most sqrt(N) times these constant bounds. The dirty shear circuits from section 2 therefore remain polynomially bounded. + +## 4. Full Toeplitz recursion: displacement rank grows additively + +Use the Sylvester shift displacement + + Delta_m(M)=Z_m M-M Z_m. + +A Toeplitz H has rank Delta_N(H)<=2. At a node of size 2m, partition + + M=[ A B ; C D ], + Z_(2m)=[ Z 0 ; E Z ], E=e_0 e_(m-1)^T, + K=A^-1 B, L=C A^-1, S=D-C A^-1 B. + +The usual block factorization is + + M=[ I 0 ; L I ] diag(A,S) [ I K ; 0 I ]. + +Again A and S run in parallel. To bound ranks, let U=[-C A^-1,I] and V=[-A^-1 B;I]. Then M V=[0;S], U M=[0,S], and direct block multiplication gives + + Delta_m(S)=U Delta_(2m)(M) V-S E A^-1 B. + +Therefore rank Delta(S)<=rank Delta(M)+1. Also, the upper-left block of Delta_(2m)(M) is Delta_m(A)-B E, so rank Delta(A)<=rank Delta(M)+1. At depth d, both children thus have displacement rank at most 2+d. + +The top-right block of Delta(M) equals Delta(B), while its bottom-left equals E A+Delta(C)-D E. Consequently + + rank Delta(B)<=r, + rank Delta(C)<=r+2, + rank Delta(K)<=rank Delta(A)+rank Delta(B)<=2r+1, + rank Delta(L)<=rank Delta(C)+rank Delta(A)<=2r+3. + +All shear matrices have displacement rank O(log N). + +Every recursive M is a principal Schur complement of the original H. Since H=I+R and both ||R||_1 and ||R||_infinity are at most delta, + + max(||M-I||_1,||M-I||_infinity) <= delta/(1-delta), + ||M^-1||_2 <= 1/(1-delta). + +The inverse assertion also follows because M^-1 is a principal submatrix of the inverse of a principal submatrix of H. Thus A,S,K,L are uniformly bounded in 2-norm independently of recursion depth. + +## 5. Bounded-generator shear compiler + +For an m by m matrix T define its Stein displacement + + E_T=T-Z T Z^T=-Delta_m(T) Z^T+T e_0 e_0^T. + +Its rank is at most rank Delta(T)+1. Choose an orthonormal basis g_1,...,g_r for the column space of E_T, and put h_j=E_T^* g_j. With conjugates interpreted appropriately, + + E_T=sum_j g_j h_j^*, + T=sum_j L(g_j) U(conj(h_j)). + +Here L(g) is lower triangular Toeplitz with first column g, and U(h) is upper triangular Toeplitz with first row h. The second identity is the finite telescoping sum T=sum_(k=0)^(m-1) Z^k E_T (Z^T)^k. All g_j have 2-norm 1, and all h_j have 2-norm at most ||E_T||_2=O(1). Their coefficient l1 norms are O(sqrt(m)). + +A zero or tiny first coefficient of h is harmless. Choose an integer gamma >= 2||h||_1 (a fixed conservative gamma=64N suffices under the stated bounds), and set + + P=I+U(h)/gamma. + +P and P^-1 have constant norm, and P is upper triangular Toeplitz. On the same two existing vectors, one generator term is implemented by + + x <- P x, + y <- y+gamma L(g) x, + x <- P^-1 x, + y <- y-gamma L(g) x. + +The result is y<-y+L(g)U(h)x and x is restored. Source transforms use section 3; the other two operations use section 2. Each term costs O(log^2 m) rounds. There are O(log N) terms, hence O(log^3 N) rounds for a full low-displacement-rank shear. + +Although the partial generator sum can be larger than its final T, its norm is at most the sum of O(log N) products of O(sqrt N)-norm factors, hence O(N log N). Within a term the source norm remains constant times its entry norm and the target norm is polynomially bounded. Terms are completed as shears with one common source, so their partial sums add; one must not charge a product of polynomial factors for every generator. + +The full recurrence is therefore + + D_full(N) <= D_full(N/2)+O(log^3 N)=O(log^4 N). + +## 6. Global prefixes, rounding and preprocessing + +At each block-LU node the completed right shear, completed middle block, and completed left shear have constant norm and constant inverse norm. At an intermediate time there is at most one unfinished generator compiler or one unfinished triangular compiler along each recursive path; disjoint paths form a direct sum. If B(N) bounds a full circuit prefix and inverse prefix, this gives B(N)<=max(C B(N/2),N^C), hence B(N)=N^O(1). The same argument applies to suffixes and to the triangular compiler. + +There are N polylog(N) arithmetic gates. Perturbation of one bounded gate by epsilon contributes at most N^C epsilon to the whole operator, by the prefix/suffix bounds. The sum over all gates remains N^O(1) epsilon. Therefore p+O(log N) coefficient/arithmetic bits suffice; there is no N or N^2 precision term on the data. + +The coefficients are fixed before seeing the data. One concrete polynomial-time generation route is: + +1. Approximate the original Toeplitz symbol by dyadic rationals at p+O(log N) bits, preserving Toeplitz structure. The operator perturbation is at most N times the entry perturbation. +2. Compute every recursive Schur matrix as a principal Schur complement of this original rational matrix. Its entries are ratios of minors of that original matrix, so numerator and denominator bitlengths are polynomial in N and p. Avoid an unnecessarily expanded recursive rational expression. +3. For each Stein displacement, use exact rational elimination to select independent columns and rational, unnormalized Gram-Schmidt to obtain orthogonal vectors v_j. Then g_j=v_j/sqrt(v_j^*v_j) and h_j=E_T^*v_j/sqrt(v_j^*v_j). The resulting generators have the bounded norms proved above. Nonzero rational residual norms are at least 2^-poly(N,p), so poly(N,p) offline precision suffices for their normalization. Equivalently a high-accuracy SVD may be used. +4. Compute the finite series divisions and FFT roots to p+O(log N) final bits. All final data-gate coefficients are polynomially bounded. Offline intermediate rational numerators may be longer; they never become data wires. + +This separates polynomial-time coefficient setup from bounded-precision data application. Applying an algebraic-RAM theorem directly to arbitrarily large integers would not provide this separation. + +## 7. Normalized Sylvester merge + +For two b-point children, N=2b, the normalized map supplied by the radical-reduction branch is + + S(B,A)=f(z)B(z)+z^b g(z^-1)A(z), + f(z)=product_lower(1+(x/zeta)z), + g(z^-1)=product_upper(1+(zeta/x)z^-1). + +Let h=g(z^-1)/f(z), K_ij=h_(i-b-j), H_ij=h_(i-j), 0<=i,j0. Every principal submatrix has spectrum in [mu,M]. Every principal Schur complement S also does: S^-1 is a principal block of the inverse of a principal submatrix, whose spectrum is in [1/M,1/mu]. Thus every recursive A,S has norm at most M and inverse norm at most 1/mu. The shears K,L have norm at most kappa=M/mu. The displacement-rank formulas are unchanged. + +Use orthonormal g generators and h of norm O(kappa), and choose gamma>=2||h||_1. The triangular source transform P is still near I, so the depth O(log^4 N) is unchanged. Generator-stage amplitudes and coefficients are poly(N,kappa). Across the O(log N) completed ancestor factors, the bound becomes + + log(max prefix/inverse-prefix norm) + = O(log N * log(2+M+1/mu) + log N). + +If M and 1/mu are exp(O(u)), this is O((u+1)log N). The corresponding data precision is p+O((u+1)log N), and g-axis tensorization adds O(g(u+1)log N) guard bits. This is a usable broad-Gaussian extension once a separate spectral argument supplies the stated eigenvalue interval. Reversing the circuit applies H^-1; the inverse itself need not be Toeplitz. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md new file mode 100644 index 0000000000..72404c42d7 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md @@ -0,0 +1,139 @@ +# Independent audit: support-local simultaneous-H supplier + +Conclusion: the retained compact phase supplier can be used at width g with cost + + O(V g^(1-q) polylog(p)) + +without the previously charged additive V p^(1-a), **provided the call is made globally on the complete dyadic tensor, not separately on one fixed coarse-label microbox**. This is conditional on the same accepted finite complex circuit and arbitrary-width equal-chunk interchange as the retained supplier, and on the support-router transfer proved in `work/halo_layout_ultra/SUPPORT-ROUTER.md`. The new support-router's unequal-spacing algebra and its use here have been checked below. + +The full address still has O(p) bits. That quantity remains in descriptor arithmetic, keys and error precision; it no longer determines the width of any newly required fast coordinate interchange. + +## 1. What the actual phase frames touch + +The primary source is `work/integer-mult-current/upstream/build/sections/05-layers.tex`. + +At lines 86–139, a residual basis is a matrix M in GL(m,F_2), extended **inside F_2^m**. At width e=mf it is applied separately to the column + + (x_(1,j),...,x_(m,j)), 0<=j T as prescribed, + T\S -> S\T bijectively, + identity outside S union T + +is a coordinate permutation supported on at most 2gK positions. Its inverse restores the original coarse-coordinate interpretation. The target can be chosen before the payload and a reserved untouched suffix. No stable reordering of every coarse field is required. + +Apply the H supplier to this full global stream, whose complete coarse-coordinate ranges remain present. A fixed coarse-label microbox alone does **not** contain enough complete K-bit fields: its high coarse bits are fixed. Borrowing those fixed labels inside such a microbox would violate the supplier's complete-field hypothesis. This audit approves the global call only. + +The tensor compiler's H schedule is independent of all coarse labels. Those labels affect only diagonal coefficients between complete H calls. The diagonal scan can still be organized by coarse-label microboxes after undoing the global exposure permutation. Coarse-dependent coefficients therefore do not require separate supplier invocations. + +## 3. Deferred reservoirs and the row prefix + +At a phase with h unprocessed selected bits, reserve whole K-bit chunks for two front hG fields, one back hG field, and a leading row prefix. With K=128G, the field reservation is at most 3h/128+O(1) chunks. Row preparation needs O(log h) further chunks, including the product stock for nested bit and complex calls. For h above a fixed polylog(p) threshold the total R(h) is at most h/16. The remaining bounded/polylogarithmic h can be processed individually. + +Every such chunk is one of the current selected chunks. The phase applies C only to the other h-R(h) selected bits and identity to the reserved chunks; the latter are deferred to the next phase. + +This requires one explicit modification to the original exposition: **do not first apply C to the reserved row/front/back selected bits**. That preliminary work in the old all-axis construction only ensured those selected bits were eventually covered. It is not needed to create rows or complete temporary fields. Source `compact-control-layout.tex` lines 30–62 proves that the active invocation is the required C tensor separately on every row and identity on all reserved/spectator coordinates, for arbitrary incoming payloads. Hence dirty, unprocessed reserved selected bits are valid. + +The leading q0 complete chunks have cardinality 2^(q0 K). Choose q0 from the actual product row stock Q_rows=W_complex^D_complex W_bit^D_bit. This preserves the accepted nesting contract; replacing the product by a maximum would be incorrect. Pad the leading row count to a multiple of Q_rows once for this phase, by adding whole rows. The factor is below two. All descendants inherit the same complete within-row fields and the same row stock; no new independent bit coordinates are allocated. + +A completed active invocation is identity on the row label and every temporary coordinate. Each padded zero row is therefore zero at completion and can be deleted. Delete padding and restore the exposure layout **before the next deferred phase**. Consequently row padding does not multiply by 2 at every phase or tensor axis. + +The next deferred phase gathers only its h'=R(h) remaining neighborhoods. A gather/restore can again be extended with support at most 2h'K; it may move previously processed coordinates temporarily but fixes them after the inverse placement. Their values need not be recomputed. There is no need to move all p coordinates back into a stable global axis order. + +## 4. The logarithmic K choice is legal after tightening the statement + +The original compact-control proposition states the sufficient asymptotic condition K/log p -> infinity. K=128G does not satisfy that literal condition. The source proof, however, uses it only at lines 149–154 to obtain + + K >= G+4 ceil(log2 p)+10, + delta_bad <= 5/(128p^3). + +K=128G satisfies the displayed finite inequality directly. The earlier/later-source guard proof, complete-field layout, row stock, scalar gates and semantic guard have no other use of K/log p -> infinity. Thus the transfer must cite the tightened finite guard hypothesis; it must not claim that the new K satisfies the old stronger asymptotic wording. + +With this change, all compact interchanges retain width O(h log p), and the per-node movement overhead is O(V_node h^(1-a) polylog(p)). Here h is the current selected width, not the ambient number d of original Gaussian axes. The supplier's stopping threshold and branching analysis likewise use its own top width g; global p remains only in logarithmic guards and O(p)-bit descriptors. + +## 5. Check of the support-router extension + +The original arbitrary-source router colors active targets by their current location modulo K. Targets in one color have disjoint K-bit guards. The new router stores only the r active control digits, H=rG, even when the target spacings are unequal multiples of K. + +The four-update proof depends solely on disjointness of the guard intervals: on a good address the first y update adds 2zt within a guard, the t update adds its old low bit e without carry, and the second y update adds z(1-2(t+e)). Their net is z(1-2e), changing exactly the selected low bit. The final t update restores t. Nothing uses equality between successive target gaps. A complete F therefore fixes every nontarget source bit, even inside another guard. F; load u from the sources; F; unload implements the desired simultaneous XOR for sources outside the target set. + +All rotations remain globally bijective, because their controls precede the target after the prescribed back-field shuttle. Their current-control inverse is used on bad addresses. The ideal selected-bit XOR preserves the guard/control bad predicate, so the old sparse repair applies unchanged. Its O(p)-bit keys and polynomial per-record arithmetic remain explicit; they are not replaced by a unit-cost sort. + +For a permutation moving k positions, factor it into two involutions and three XOR stages per matching. Three disjoint 3kG-bit reservoirs can be chosen when n>=9kG; the three matching cohorts avoid the respective reservoirs. Each target residue class then uses width at most kG in each control field, rather than ceil(n/K)G. Original equal-width interchanges place and restore the fields. This gives O(V k^(1-a) polylog p), plus descriptor setup. + +When records are short, choose only ell_s=ceil(log2 p)^2 untouched binary bits, not a whole long original axis. Here is explicit noncircular suffix bookkeeping. For a requested permutation pi with moved support at most k, if n>2(k+1)ell_s, a gap in its support contains a consecutive ell_s-bit field S disjoint from the actual last ell_s bits. Use one original equal-ell_s interchange A to move S to the actual suffix. Route the conjugate A pi A^-1 on the n-ell_s prefix only, then undo A. The conjugate fixes that suffix pointwise because pi fixed S. Choose all control reservoirs inside the prefix, after this interchange, and recompute positions and residue classes after their placements. Thus no control operation touches the grouped suffix. Grouping its 2^ell_s complete values into a record preserves the data volume and gives a superpolynomial record length. If n<=2(k+1)ell_s, the retained full-span bound is already O(V(k+1)^(1-a) polylog p), so no suffix grouping is needed for this support bound. The placement costs O(V polylog p), not V ell^(1-a) for the original long axis length ell. + +For the calls here, k=O(hK)=O(h log p), and n/(k log p) tends to infinity because g=p^beta with beta<1 while n is of order p in the retained full tensor. If this inequality fails in a small regime, n=O(k polylog p), so the retained full-span bound is already O(V k^(1-a) polylog p). The support theorem's own temporary reservoirs may have O(kG) bits; this is a polylogarithmic enlargement of support, already priced in the same bound. + +## 6. Complete charge table + +| Operation | Active coordinate/data support | Charge | +| --- | --- | --- | +| Initial and inverse neighborhood exposure | At most 2gK moved bit positions | O(V g^(1-a) polylog p) | +| Deferred phase exposure at width h | At most 2hK positions | O(V h^(1-a) polylog p) | +| Fixed binary residual adapter | h selected bits; compact controls O(hG) | O(V_node h^(1-a) polylog p) | +| Control-field shuttle | At most O(hG)-bit chunks | Original O(V_node(hG)^(1-a)) | +| Modular rotations and exceptional repair | Full scan; O(p)-bit descriptors/keys | O(V_node) after record grouping | +| Residual child concatenation | Existing contiguous first-r slots | Descriptor change only | +| Source/sink/inverse phases | Selected-bit masks and coefficients | O(V_node) | +| Scalar role gates, parking and restoration | Fixed role streams at aligned addresses | O(V_node) | +| Whole-row padding and deletion | One current-phase row prefix | O(V), factor below two, removed before next phase | + +No row in this table requires an interchange of width p. + +Let tau=1-a. The retained stopped branching argument gives O(V h^(1-q) polylog p) for each phase, with q 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md new file mode 100644 index 0000000000..4f39fa9eaf --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/SUPPORT-ROUTER.md @@ -0,0 +1,157 @@ +# Coordinate routing measured by moved support + +This is a conditional extension of the retained compact arbitrary-source +router. It uses the same equal-width chunk interchange with cost +`O(V H^(1-a))`, original fixed-tape modular rotations, counters and sparse +radix repair. It changes the packed-control length from the full address span +to the number of active target positions. No computed-key sort of the full +input is assumed. + +Let a supplied permutation of `n=O(p)` binary address coordinates move at +most `k` positions. The stream is a complete binary rectangle, with arbitrary +preceding fibers and following payload. In the regime used below, +`n/(k log p) -> infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 0=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. This resolves the length caveat in section 5 for the intended phase-cell application; it does not prove the separate simultaneous tensor-layer routing claim. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md new file mode 100644 index 0000000000..35af6c7b99 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md @@ -0,0 +1,180 @@ +# A fast, stable application of the general Newton lower factor + +Status: a proved one-dimensional **nonperiodic lower-factor** algorithm and exact controls. It does not by itself implement the entire periodic inverse. The upper factor has a polynomial norm bound but its fast normalization/fusion remains to be supplied. No final multiplication exponent is claimed. + +## 1. Exact Gaussian LDU + +Let `rho=1+theta`, `q_i=floor(rho*i+1/2)`, `beta_i=rho*i-q_i`, `x_i=r^q_i`, `r=exp(2a rho)>1`, and `d_i=exp(-a q_i^2)`, for `0<=i=j`, zero otherwise. + +The ordinary monic Newton expansion of `X^j` gives exactly + +`G=L diag(C_i) U`, + +where `L=diag(d_i) T diag(d_i)^-1`, `L_ii=U_ii=1`, and + +`(U^-1)_ij=(-1)^(j-i) (e_j/e_i) e_(j-i)(x_0,...,x_(j-1))`, + +`e_j=exp(-a rho^2 j^2+a beta_j^2)`. + +The pivot calculation is exact: + +`d_i P_i(x_i) e_i = C_i`. + +Define + +`A_i=d_i x_i^i C_i/prod_(k<=i)x_k` + +` =C_i exp(-a beta_i^2-a rho^2 i+2a rho sum_(k<=i) beta_k)`. + +Divided differences give the exact inverse lower entries + +`(L^-1)_ij=(-1)^(i-j) (A_i/A_j)/D_(i,j)`, + +`D_(i,j)=prod_(j=1)(1-r^-m)`. Because `q_(i+1)-q_i>=1`, + +`c<=C_i,D_(i,j)<=1`. + +For `i>=j`, direct summation gives + +`log |A_i/A_j| <= -log c+a/4-a rho theta(i-j)`. + +Consequently + +`||L^-1||_infinity <= c^-2 exp(a/4)/(1-exp(-a rho theta))`. + +For the upper factor, compare every elementary-symmetric term with the product of the largest `j-i` variables. The sorted deficits form a partition, hence + +`1<= e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k=-1/2` gives + +`||U^-1||_infinity <= c^-1 exp(a/4)/(1-exp(-a rho theta))`. + +Also `||diag(C_i)^-1||<=c^-1`. For `a>=1`, `r>=e^2`, so `c` has an absolute positive lower bound. For `a=O(log(1/theta))`, both factor norms are polynomial in `theta^-1`. These bounds apply to **arbitrary signed/complex input vectors**, not only correlated outputs of another factor. + +## 3. One stable scan plus a local correction + +Let + +`D_j^+=prod_(j=0`, + +`0<=1-H_i(x_j)<=sum_(k>i)x_j/x_k <= r^(-h-1)/(1-r^-1)`. + +The sum of discarded correction entries for `h>=W` is therefore at most + +`c^-2 exp(a/4) r^(-W-1)/[(1-r^-1)(1-exp(-a rho theta)/r)]`. + +Thus `W=O((p+a+log(1/theta))/a+1)` makes the omitted correction less than `2^-p` in operator norm. At `a=O(log(1/theta))`, `theta>=p^-O(1)`, this is `O(p/a+1)`. + +Partition the line into pages of W indices. Compute the exact correction on each diagonal page and each immediately lower adjacent page, and omit it elsewhere. Every omitted entry has `h>=W`; every retained tile has O(W) rows/columns. A diagonal page is recursively split into ordered cross tiles plus its diagonal. Only O(log W) tree levels are required. The adjacent-page rectangle can use the tile algorithm below directly. Subtracting the scan's contribution on those same tiles gives exactly the desired retained correction. + +## 4. The bounded-polynomial cross-tile algorithm + +Let every column in J precede every row in `I=[l,b]`, and suppose both sets lie in an index interval of length O(W). Put + +`z_j=x_j/x_l` (j in J), `v_k=x_l/x_k` (l=0) M_m Z^m=sum_j w_j/(1-z_j Z)=Q(Z)/P(Z)`. + +Build a balanced tree with leaf pairs `(P,Q)=(1-z_j Z,w_j)` and merge by + +`P=P_left P_right`, + +`Q=Q_left P_right+Q_right P_left`. + +Truncate to the needed degree at each node. Since `sum z_j<=1/(r-1)`, all product coefficient l1 norms are at most `exp(1/(r-1))<2`; numerator norms are at most twice the absolute input weight sum. The power-series reciprocal of P has l1 norm at most `c^-1`. Standard fast power-series inversion, or doubling the geometric series of `1-P` (whose l1 norm is <1/2 for r>=e^2), is therefore stable with polynomially bounded intermediates. Product-tree complexity is soft-O(|J|+|I|) scalar operations. + +### Fast suffix-polynomial evaluation + +The second operation is the transpose of multiplication by the suffix products. Split `[l+1,b]` at m. Right-hand output rows use the moment vector restricted to their degree range. For left rows, replace the moment vector by + +`M'_h=sum_k [Z^k]prod_(m=1; at a=1 this is `<16 theta^-2`. The independent exact negative-origin controls are BILATERAL-CONTROLS.json. +2. **Complete one-dimensional data application:** LOWER-FACTOR.md supplies the lower factor with logarithmic guard overhead. UPPER-DATA-ALGORITHM.md supplies the upper factor and a direct periodic implementation with `O(s p polylog p)` data cost, shared per-axis coefficient preprocessing, and **Cp internal precision**. This is stated in the target `s>>poly(p)` regime. Exact controls are LOWER-CONTROLS.json, UPPER-DATA-CONTROLS.json, and CYCLIC-SCAN-CONTROLS.json. +3. **Length-preserving upper merge:** UPPER-LIFTING.md proves the Newton merge and bounded exact shear algebra. UPPER-MERGE-CONTROLS.json checks its degree indexing and factorization. +4. **Still open:** a realization of the normalized Hankel/Toeplitz steps with only logarithmic internal guard bits, and a bounded-depth width-preserving tensor circuit. Neither the analytic norm theorem nor the one-dimensional time bound establishes those, and no improved final multiplication exponent is claimed. + +Replay all finite controls with Python 3: + +``` +python3 work/gaussian_fast_ultra/check_lower.py +python3 work/gaussian_fast_ultra/check_upper_merge.py +python3 work/gaussian_fast_ultra/check_bilateral.py +python3 work/gaussian_fast_ultra/check_upper_data.py +python3 work/gaussian_fast_ultra/check_cyclic_scan.py +``` + +The package uses no third-party source result beyond the elementary Newton polynomial identities stated and proved by their constructions. It retains the parent Gaussian notation and does not modify the frozen upstream manuscript. + +Focused phase-fusion audit added later: + +- `LOG-A-PHASE-SCHUR-AUDIT.md`: verifies strict row dominance at a=Theta(log p), polynomial Schur bounds, actual rounded Toeplitz interiors, and dyadic interior subdivision with only a logarithmic increase in boundary count. `LOG-A-SCHUR-CONTROLS.json` and `DYADIC-INTERIOR-CONTROLS.json` contain independent exact checks. +- `FORWARD-SMALL-A-AUDIT.md`: handles the distinct broad forward Gaussian S via b=pi/(alpha^2 rho^2), gives polynomial finite-block/completed-matrix inverse bounds, the exact target-grid chirp factorization, and the SPD Toeplitz circuit guard estimate. `FORWARD-CELL-CONTROLS.json` checks the coordinate and matrix identities. + +These are interface components for the parent's focused partial-fusion proposal. They do not independently certify its full tensor schedule or final exponent. + +- `CAPPED-GAUGE-AUDIT.md` independently validates capped dyadic interiors and exact reanchoring of the already-rounded narrow matrix, with local log range O(aB) rather than O(ad). It also validates the broad O(B/alpha^2) gauge and whole-boundary-group solve order. `CAPPED-GAUGE-CONTROLS.json` contains 27,216 partition checks and exact narrow/broad gauge controls. The capped final exponent ledger remains a separate assembly artifact. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md new file mode 100644 index 0000000000..f07f2c68c7 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md @@ -0,0 +1,166 @@ +# A complete upper-inverse data algorithm with linear precision overhead + +Status: the construction below closes the one-dimensional **data application** of the nonperiodic upper inverse in `O(s p polylog p)` bit operations, after fixed per-axis coefficient preprocessing, using `C p` internal precision for an absolute constant C. It does not obtain logarithmic internal guard bits and does not prove tensor fusion. The lower-factor algorithm in LOWER-FACTOR.md does have logarithmic guard overhead. + +Let `p` be the required output precision, `a>=1`, `theta>=p^-O(1)`, and `a=O(p)`. In the target family a is fixed or logarithmic and every line length s grows faster than any fixed power of p. Preprocessing depending on the axis is shared by all its lines, as allowed by the parent reduction; it is not charged anew for every line. The coefficient words actually read by one data application have total `O(s p polylog p)` bits. + +## 1. Upper inverse is a stable suffix scan plus bandwidth O(p/a) + +Use the column diagonal `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Define + +`B_j=e_j prod_(0<=k=i`, write + +`C_i,j=e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k0. Put `d=m-i`, + +`gamma_i=(U^-1)_i,m`, + +`tau_j=(-1)^(j-m)(B_j/B_m)`. + +An exact Laurent-polynomial coefficient identity gives + +`(U^-1)_ij=gamma_i tau_j sum_(h>=0) (alpha_(d+h)/alpha_d) beta_j,h`. + +The signs are already in gamma and tau; the remaining coefficient sum is nonnegative. Both gamma and tau are polynomially bounded, with the same exponential-distance primary bounds as before. + +To verify the identity, write X=x_m Z and factor + +`P_j(X)=X^m prod_(k=1)(1+r^-k) <= 2/c`. + +A truncated product/sum tree computes M_0,...,M_R in `O((|J|+R) polylog p)` scalar operations. At a split, its exact rule is `S=S_left+P_left S_right`, with P a fixed product. Intermediate norms are bounded by a constant times the absolute weighted input sum, independently of tile length. No large Gaussian diagonal occurs in this step. + +## 3. Exponentially small high degrees + +The partition bound on alpha and beta gives + +`alpha_(d+h)/alpha_d <= c^-1 r^(-d*h-h(h+1)/2)`, + +`beta_j,h <= c^-1 r^(-h(h-1)/2)`. + +Thus the h-term, after summing input weights, is bounded by + +`c^-2 r^(-d*h-h^2) sum_j |tau_j u_j|`. + +Group row distances dyadically: D<=d<2D. For that group retain only + +`R=min(|J|, O(sqrt(P)), O(P/D))`, `P=(p+a+log(1/theta))/a`, + +with sufficiently large absolute constants. If |J| is smaller, the polynomial is already exact. Otherwise all omitted terms sum to less than `2^-p` after allowing the polynomial gamma/tau factors. There are O(log p) row groups. + +## 4. Fast normalized Hankel application, with Cp internal bits + +The remaining map for a row group is + +`y_d=sum_(0<=h<=R) (alpha_(d+h)/alpha_d) M_h`. + +Split the d-range into consecutive blocks of length at most R (or the whole group if shorter). For a block beginning at d0, divide all alpha coefficients used by alpha_d0. Then the numerator is one reversed ordinary convolution of that coefficient segment with M. Divide output d by alpha_d/alpha_d0. + +The decisive height bound is local, not global. If k>=0, comparing dominant products and using q-gaps in {1,2} gives + +`c r^(-2*d0*k-k(k+1)) <= alpha_(d0+k)/alpha_d0 <= c^-1`. + +Here the lower bound is needed only when the coefficient exists; every queried alpha index is <=m. If alpha vanishes because its degree exceeds m, store exact zero and never divide by it. Since d0=O(D), k=O(R), and `D R+R^2=O(P)`, every nonzero coefficient ratio and every divisor in this block lies between `exp(-O(aP))` and an absolute constant. Thus **Cp internal precision** suffices for these convolutions and the final divisions. + +The number of scalar operations for one row group is `O(D polylog p)`; its separate bounded-prefix computation costs `O((|J|+R) polylog p)`. Summing the O(log p) row groups gives `O((|I|+|J|) polylog p)` scalar operations per ordered tile. Recursing through one W-page or processing adjacent pages therefore costs `O(W p polylog p)` bits. Over all pages the data cost is + +`O(s p polylog p)`. + +The Gaussian factors gamma and tau can be evaluated from bounded dominant-product ratios rather than raw e_i or B_i. Precomputed alpha tables can similarly be stored with dominant-product gauges. Every alpha segment actually read by the data algorithm has length O(|I|+|J|) per row-group and tile level; this accounts for the coefficient tape scans in the same bound. + +A conservative preprocessing implementation may compute every required alpha table from O(poly(p)) nearby nodes separately, and can therefore cost `s poly(p)` per axis. This is a one-time axis cost, not an unpriced per-line cost. In the parent regime it is negligible compared with the full tensor data volume. A uniformly quasi-linear preprocessing theorem is not asserted here. + +## 5. Complete nonperiodic and periodic application + +Combining this upper map with the lower map and the bounded diagonal gives the complete nonperiodic inverse `G^-1=U^-1 C^-1 L^-1` in `O(s p polylog p)` data time, at Cp internal precision, with polynomial operator norms. Requested final error is achieved by a fixed additional factor in precision plus O(log p+log(1/theta)) endpoint guard bits. + +For the periodic matrix one may either implement the bilateral factors in BILATERAL-LDU.md or use a finite boundary correction. The latter needs only two nonperiodic inverse data applications: + +- Truncate nonzero Gaussian aliases to the two corner blocks of width `b=O(sqrt((p+log(1/theta))/a)+1)`, obtaining `K=G+UV` and a row-norm tail smaller than `2^-Cp`. +- U inserts O(b) boundary coordinates and V reads a bounded O(b)-by-O(b) corner matrix. +- Precompute `S^-1`, `S=I+V G^-1 U`. Its norm is polynomial because `S^-1=I-V K^-1 U` and the periodic inverse norm plus the tiny discarded tail bounds K inverse. +- Apply `x=G^-1 z`, `w=S^-1 Vx`, and return `x-G^-1 Uw`. + +The boundary data work is polynomial in p. For the target line lengths s growing faster than every fixed polynomial in p, it is negligible relative to sp. The updated matrix in the Woodbury inverse identity is K, not the base G. The periodic Gaussian tail and final correction error are bounded using the fixed-a theorem in BILATERAL-LDU.md. + +## What is still not established + +The normalized Hankel step uses Cp working precision because its local alpha divisors may be exp(-Cp). It has not been converted to a circuit whose every intermediate operator has polynomial norm. Therefore this result does not meet the stronger **logarithmic guard at every internal stage** requirement needed by the proposed full-dimensional fusion. It supplies a complete one-dimensional data algorithm and a precise remaining normalization target; it supplies no improved final kappa. + +## 6. Direct periodic implementation from the bilateral factors + +The bilateral theorem also permits the Woodbury step above to be omitted. This is cleaner in the target regime `s>>W`. + +For the bilateral lower inverse, use + +`C_i=prod_(ki)(1-x_i/x_k)`. + +The same lower formula and scan/correction split hold, with the defining products now bilateral. The adjacent A ratio is still given by the beta formula in LOWER-FACTOR.md. For the upper inverse, use elementary polynomials in the infinite prefix `x_k, k=1-exp(-a)`. Intermediate partial products and scan norms have the polynomial bounds already proved. This supplies a stable two-pass scan on the original s records. + +Retain the correction only for one-sided distances below W on the integer cover of the circle. With `s>>W`, each retained coefficient belongs to its diagonal page or the single adjacent cyclic page; virtual negative or beyond-s indices are generated by periodic q/beta formulas. The same local tile algorithms apply. Periodization cannot increase the row norm of any discarded tail, so all the previous error estimates remain valid. + +In the upper tile algorithm, alpha_d is now the coefficient of the infinite small-root prefix + +`prod_(k=d+O(q/a+1)`: a contribution involving an omitted node has partition deficit at least K-d, and summing the corresponding partition tail gives `O(r^(-(K-d)/2))` relative error (bound the tail by r to half the negative deficit times the convergent partition generating product). For the O(W)-degree tables at q=Cp, this is O(W+p/a) nodes, independent of s. Normalized table storage and data access are unchanged. + +Thus, after fixed per-axis coefficient preprocessing, the bilateral factorization itself gives the **periodic** inverse data application in `O(s p polylog p)` time with Cp working precision, in the target regime `s>>poly(p)`. It needs neither a large raw Vandermonde nor a boundary Schur system. The unresolved logarithmic-internal-guard and tensor-fusion qualifications remain exactly as stated above. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md new file mode 100644 index 0000000000..0985ad16cd --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md @@ -0,0 +1,57 @@ +# The upper factor has a length-preserving Newton merge + +This is a structural advance, not yet the missing fast normalized FFT implementation. It concerns the nonperiodic factor from LOWER-FACTOR.md. + +Write `d_i=exp(-a rho^2 i^2+a beta_i^2)` in this note (this is the column diagonal called `e_i` in LOWER-FACTOR.md). The upper inverse is + +`(U^-1)_ij=(d_j/d_i) [X^i] prod_(k=m and h=j-i, the logarithm of the last Gaussian factor is at most + +`a/4-a rho^2 h^2-a rho(2rho-1)h-2a rho^2(i-m)h`. + +Thus D inverse also has row norm `O(exp(a/4))`, and the shear B D inverse has polynomial norm. These statements hold on arbitrary signed/complex records. + +## Remaining realization problem + +The algebra is length preserving and the exact blocks have bounded norm. Directly executing their diagonal-Toeplitz-diagonal presentation is not yet safe for full-dimensional tensor fusion: the raw d_i ratio can have logarithmic range `Theta(a(r-l)^2)`. A local FFT using that presentation therefore reintroduces an O(p) guard for blocks of length sqrt(p/a). + +A plain unit triangular Toeplitz map can be implemented by recursive Toeplitz shears and in-place cyclic/negacyclic Fourier transforms, as suggested by the parent investigation. To close this upper-factor route, that primitive needs a version for the **Gaussian-conjugated** Toeplitz blocks above whose intermediate norms stay polynomial, or a different bounded lifting factorization of them. The final exact block norms alone do not prove the intermediate FFT normalization claim. + +The numerator Sylvester merge in the lower-factor moment tree remains a separate lifting problem; the one-product Newton merge here avoids that two-product merge for the upper factor only. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/HALO-THEOREM.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/HALO-THEOREM.md new file mode 100644 index 0000000000..60d59bb378 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/HALO-THEOREM.md @@ -0,0 +1,271 @@ +# Fast tensor convolution by core blocks and sparse face corrections + +This gives a constructive replacement for g dense halo scans. It applies to +a **translation-invariant separable compact stencil**, with optional +diagonal conjugations whose coefficients and tile gauges satisfy the +precision and sequential-access conditions below. It does not claim that +the separate Gaussian product-tree factorization has this form. + +Let g consecutive active axes have dyadic lengths S_i, with +`h=max_i log2 S_i`. Earlier coordinates are outer prefix fibers. Later +coordinates are a complete payload R, so they need no keys or transposition. +Let V denote the total bit volume, including B-bit coefficient records and +all spectators. Assume B=Theta(p), `g h=O(p)`, and the active group grows +superpolynomially. A fixed number of tapes may hold O(V) bits. + +The operation is + +``` +(Tf)(x) = sum_{z in [-L,L]^g} [product_i k_i(z_i)] f(x-z mod S), +``` + +where `L=poly(p)`. A fixed or polylogarithmic number of these operations and +diagonal scans is allowed. Kernel coefficients, coefficient products and +the required intermediates have O(p)-bit representations under the supplied +normalization contract. The retained grouped convolution engine on a +binary field of width k has cost `O(V k^(1-q) polylog p)`. Polynomial-length +one-axis convolution also has its unconditional quasilinear bit algorithm; +it does not invoke the improved bound being proved. + +Choose a dyadic tile side N dividing every S_i, with `N>4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md new file mode 100644 index 0000000000..fdfe26d197 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md @@ -0,0 +1,89 @@ +# True periods inside a near-one dyadic container + +The primary halo theorem extends to arbitrary true periods s_i when an +already supplied dyadic rectangle t_i>=s_i has bounded total inflation +`product t_i / product s_i=O(1)`. Simultaneous zero embedding and its inverse +cost one ordered scan each by `RECTANGULAR-EMBEDDING.md`. This prerequisite +is satisfied only where the inherited near-one capacity bound applies; it +must not be replaced by independently rounding every axis to its next power. + +Take N dividing every t_i. Write `s_i=q_i N+r_i`, `0<=r_i0, one ragged core in axis i. The initial +support-limited gather uses the dyadic storage coordinates, not mixed-radix +division. It may produce invalid all-zero storage tiles; discard those. + +Apply ordinary flat cyclic convolution only to tiles whose every active +side has valid length N. Their interior outputs remain exact. Regular tile +face corrections use the earlier first-boundary-axis rectangles, but every +halo coordinate is now reduced modulo **s_i**, never t_i. + +If any side of a tile is ragged, discard its entire flat-convolution output. +For its actual side lengths r_i(tile), compute every valid output in +`product [0,r_i(tile))` using the input rectangle +`product [-L,r_i(tile)+L)`. All sides are at most N+2L. The same g successive +polynomial-length convolution/crop passes therefore produce exact outputs. +Full ragged tiles and regular face outputs form a disjoint correction set. + +Let Q_i=q_i+1_(r_i>0). The total full-halo volume of *all* valid tiles is + +``` +H_all = product_i [s_i + 2L Q_i]. +``` + +Their all-full-tile contribution is `product_i q_i(N+2L)`. The sum of the +ragged-tile halo volumes is their difference. Expanding by a first ragged +axis, or simply telescoping the product, gives + +``` +H_ragged <= H_all * sum_i (N+2L)/s_i + = O(S gN/s_min), +``` + +since `H_all/S <= exp(2L sum_i Q_i/s_i)=O(1)` when gL/N=o(1). +Regular face input volume is O(S gL/N). Thus total sparse copy volume is + +``` +M=O(V [gL/N + gN/s_min]). +``` + +The new term is smaller than V divided by every fixed polynomial in p when +`s_min=2^Theta(p/d)` and `N,g=poly(p)`. It pays all the same key sorting, +metadata and local computation factors. No new per-axis full scan appears. + +## One source-driven enumerator + +For each valid source coordinate x_i, list the valid destination tiles t +for which an unwrapped representative + +``` +y_i = x_i - tN + k s_i +``` + +lies in `[-L,r_i(t)+L)`. Only the home and neighboring core intervals can +qualify; there are at most three choices because L5/6, +all sparse halo work is O(V polylog p). The initial gather costs +`O(V g^(1-a_bit) polylog p)` and the grouped convolution costs +`O(V g^(1-q) polylog p)`. The cited bounded-gauge variation is still o(p). +Actual parameter exponents are recorded in `PARAMETER-CHECK.json`. + +Read: + +1. `SUPPORT-ROUTER.md`: arbitrary-spacing guarded shears, support-sized + reservoirs, fixed-tape repair, and the support-at-most-2k gather. +2. `HALO-THEOREM.md`: tensor identity, exact sparse halo volume, all record + movement, key lengths, local transposes, final output order and costs. +3. `RECTANGULAR-EMBEDDING.md`: simultaneous near-one zero embedding and crop + in one scan, using B=Omega(p) record width and an O(p)-bit descriptor. +4. `RAGGED-PERIODS.md`: arbitrary true periods inside that dyadic container, + using full ragged tiles as exponentially sparse additional exceptions. + +No rank/select oracle or sort of the complete tensor by a computed key is +used. Full scans preserve stream order. Only sparse correction records +are sorted, with every key bit pass charged. + +Reproduce, from the chat workspace: + +``` +python3 -B work/halo_layout_ultra/check_sparse_shear.py +python3 -B work/halo_layout_ultra/check_halo.py +python3 -B work/halo_layout_ultra/check_embedding.py +python3 -B work/halo_layout_ultra/check_ragged.py +``` + +Controls include an exhaustive 32,768-address shear permutation, 4,000 +unequal-spacing probes, 5,632 regular-grid convolution outputs and 6,538 +true-period inputs for ragged gather/correction checks. Wrong naive cyclic +flattening is explicitly detected before correction. Source/result hashes +are in `MANIFEST.json`. + +This is not a complete Gaussian resampler or a larger multiplication +exponent certificate. The kernel must be translation invariant and +separable; a fixed/polylogarithmic composition with bounded and sequentially +generated diagonal conjugations is admitted. A product tree with many +different short kernels or per-axis factor-two padding is not admitted by +renaming it a convolution. The missing Gaussian representation/normalization +theorem remains separate. The original bit/complex supplier, fixed-tape, +precision and all-size interfaces remain conditional inputs. + +Prepared with substantial OpenAI Codex assistance, continuing the retained +arbitrary-source router and bulk-resampling proofs in +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. +No frozen source or selected certificate was changed. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md new file mode 100644 index 0000000000..191867c72e --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md @@ -0,0 +1,51 @@ +# One-pass near-one rectangular embedding and crop + +This supplies the dyadic-container qualification for the halo theorem. It +does not require one full payload scan per axis. + +Let input axes have arbitrary lengths `s_1,...,s_D` and let dyadic target +lengths `t_i>=s_i` be given, with total address width +`A=sum_i log2 t_i=O(p)`. The input and output use lexicographic axis-major +order, the last coordinate changing fastest. Every payload record has at +least `c p` bits, as in the retained coefficient representation. Assume +`T=product t_i <= C product s_i = C S`; C is the separately proved +near-one capacity bound, not a product of per-axis dyadic roundings. + +Enumerate the complete target rectangle in lexicographic order with its +ordinary A-bit binary counter. For each target address x, output the next +source record if `x_i infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md new file mode 100644 index 0000000000..6a5ecf4229 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md @@ -0,0 +1,197 @@ +# Independent adversarial review of capped phase fusion + +This is an independent review of the algorithms and their growing-parameter +costs, not merely of the parameter arithmetic. It does not certify the finite +suppliers or turn a conditional multiplication statement into an unconditional +one. The exact reviewed core-source hashes are in `CORE-SOURCES.json`. + +## Current disposition + +No blocking gap found in the shallow Toeplitz compiler, the growing-g tensor +compiler, the support-local deferred supplier transfer, Gaussian precision, +or sparse layout, conditional on the interfaces stated in +`phase_fusion_note.tex:60-65`. The completed Gaussian and sparse-layout +reviews are `gaussian_layout_review.md`, `layout_subreview.md`, and +`union_subreview.md`, and must be read with this report. + +The additional matrix-exponent corollary is conditional on its additional +matrix exponent premise. The effective algebraic-coefficient transfer in +`phase_fusion_note.tex:327-345` does not itself introduce a further unproved +uniformity assumption: strict exponent slack permits one fixed finite formula, +its coefficients can be algebraic and computable, and recursion has only +polynomial-in-W error and magnitude amplification. + +## 1. Shallow Toeplitz construction + +I checked the identities and the norm mechanism in the complete +`work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +* Lines 25-39 implement a dirty Toeplitz shear by circulant and negacyclic + shears. They invert FFTs, never Fourier multipliers. A singular Toeplitz + multiplier therefore causes no illegal inverse, and no clean coordinate is + used. +* Lines 43-73 give a genuinely parallel triangular recursion. The Schur child + `e-z o^2/e` and the two shears give the stated parity block exactly. Children + are restrictions/Schur complements of the original near-identity matrix, + so their norms do not acquire a new multiplicative condition factor at every + depth. +* Lines 77-111 use `ZM-MZ` displacement. The displayed Schur displacement + differs by one rank-one boundary term, giving additive rather than + multiplicative rank growth. +* Lines 115-139 turn each bounded Stein generator into four operations using + `P=I+U(h)/gamma`. The subtraction restores x and cancels the unwanted + `gamma L(g)x`. Intermediate generator sums add; treating their norms as a + product would be an unjustified pessimistic bound, not an actual failure. +* Lines 145-158 separate long offline exact rational calculations from online + bounded coefficients and establish polynomial prefix and suffix norms. + This is essential: merely citing an arithmetic operation count would not + have supplied an O(p)-bit data algorithm. +* For the SPD extension, lines 188-197 correctly bound principal Schur + complements using the principal-block inverse identity. The log-prefix + budget is `O(log N log(2+M+1/mu))`, which is sufficient for the actual + Gaussian spectral bounds. + +One nonblocking wording error was found and corrected by the parent author +during this review. The original `phase_fusion_note.tex:103` wrote a general +SPD prefix bound using only the condition number M/m. This omits absolute +scale. For example, +`G=exp(-p) I` has condition number one but inverse-prefix norm at least exp(p). +The supporting proof at `SHALLOW-TOEPLITZ.md:194-195` has the correct +`log(2+M+1/mu)` expression. The current phase note now uses that expression and +requires both M and 1/m polynomial in p. The actual Gaussian has those bounds, +so the correction does not change either claimed exponent. Both pre-correction +and final source hashes are retained in the source manifest. + +## 2. Growing-g tensorization + +I read all of `work/inplace_algebra_ultra/TENSOR-COMPILER.md` and checked the +following potential failure points. + +* Its exact six-H words cover zero shears: `t+c` and `-c` are bounded away + from zero, so the diagonal inverses do not divide by the original t + (lines 31-60). +* The Beneš route uses identical bit schedules across axes and coarse labels. + Switch controls, including inactive pairs, are diagonals. Thus a common + global H call really exists even when every local circuit differs + (lines 62-97). +* The Kronecker equality at lines 89-95 is literal multiplication of tensor + factors. It introduces neither a sum of 2^g subproblems nor padded tensor + dimensions. +* Completed route/identity words have exact norm-one or identity endpoints; + only one local word is unfinished at a time. Local prefix bounds therefore + do not multiply once per compiled stage. Tensor norms do multiply across g, + but their logarithms add to the explicitly charged `O(g polylog p)` + (lines 99-110). +* The suffix-product odometer (lines 114-132) is a valid fixed-tape amortization, + not a RAM lookup assumption. A carry visiting k tables costs O(kN), and its + frequency is at most V/N^k. Table initialization and polynomial generation + must be charged per visited coarse block; lines 130 and the phase note + 283 explicitly do so. Since g grows, N^g dominates every fixed polynomial + setup cost. No extra gV diagonal row is forced by this implementation. + +The supplier is called globally, across complete coarse-coordinate ranges, +and exposure is undone before a label-dependent diagonal is evaluated. The +distinction made at `phase_fusion_note.tex:120-134` and 260-262 is essential. +A separate supplier call inside one fixed coarse-label microbox would not +satisfy the complete-neighborhood hypothesis. The actual schedule avoids that +invalid call. + +## 3. Deferred reservations and support-local routing + +I read the whole support audit and router, and checked their main claims +against the primary compact-control and whole-child sources. + +* `compact-control-movement.tex:149-154` is the only use there of the stronger + asymptotic K/log p condition: it obtains a finite guard inequality and an + O(p^-3) bad density. K=128G directly satisfies that finite inequality. + The tightened hypothesis is therefore legitimate for this use. +* `compact-control-layout.tex:30-62` supplies whole-row splitting and exact + identity on arbitrary dirty reserved coordinates. It does not require the + earlier individual C operations in order to make the active invocation + legal. Deferring those reserved selected bits preserves the tensor target. +* `whole-complex-concatenation.md:102-139` and the leading-row qualification + use the PRODUCT of nested bit and complex row stocks. The new audit retains + this product. It does not replace it by a maximum. The required number of + complete reserved chunks is O(log h), and the three compact fields use at + most 3h/128+O(1) chunks, so the eventual h/16 contraction is valid. +* A requested mapping of k bits can be extended on S union T, moving at most + 2k positions (`SUPPORT-ROUTER.md:128-140`). It is unnecessary to stably + compact all coarse fields. +* The unequal-gap XOR algebra at `SUPPORT-ROUTER.md:23-81` uses only disjoint + guard intervals, not equally spaced targets. Distinct sources outside the + target set remain unchanged between loading and unloading, even if they + lie in guards. Every elementary operation is bijective, and the ideal map + preserves the bad predicate, validating exact bad-address repair. +* The fixed-size reservoirs are reserved before target coloring, and target + positions are recomputed after placement (`SUPPORT-ROUTER.md:92-109`). This + avoids a circular bound based on a residue class before its layout changes. +* For short records, the suffix conjugation at lines 112-125 is valid. + Conjugation preserves moved-support cardinality. Choosing an untouched + log^2 p-bit field and routing on the complementary prefix leaves a complete + superpolynomial record; control reservoirs are chosen inside that prefix. + This pays polynomial descriptor arithmetic without spending an entire + long original axis or introducing new data. +* Individual K-neighborhoods fit one per original axis because + `p^(1-epsilon)/log p -> infinity`. Their aggregate and all helper reservoirs + fit the full address because beta<1. There is no necessary constraint + beta<1-epsilon; a group is allowed to be wider than one original axis. +* The width-dependent phase sums are geometric. With q=2`, the exponent is at least +`h^2-|h|`; the two-sided tail is at most `4 exp(-2a)`. Thus the claimed +`exp(-2a theta)+5 exp(-2a)` bound follows. In particular, counting both nearest +neighbors separately at the larger bound would be invalid, but the supplied +argument does not do so. + +The actual H is specified by globally fixed rounded weights and a globally +fixed truncated Toeplitz kernel inside each physical phase cell. Its diagonal +is one, with the omitted aliases included in the perturbation. The candidate's +perturbation budget is strictly below the displayed row-dominance gap. Principal +submatrices preserve this row gap. Adding identity hole rows/columns preserves +it as well, and increases the physical bandwidth by at most the q-image spacing +factor, which is bounded by two. + +For `F=H_aug-I`, `||F||_infinity<=1-delta` with +`delta>=1/(16p)`. Powers through order J agree on a core row whose distance from +the principal-window edge exceeds J times the bandwidth. The two inverse tails +are bounded by `2(1-delta)^J/delta`. With `J=C p^2`, this is exponentially small +in p. The distance requirement is `O(p^(5/2) polylog p)`, below the candidate's +`A=C_A p^3` eventually. The preserved physical period cut and the fact that +true axes exceed every polynomial in p exclude spurious short wrap edges. + +## 2. Broad Gaussian conditioning is a different argument, and it works + +Sources: candidate lines 151–201; +`work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md` lines 13–73 and 75–105; +`work/gaussian_fast_ultra/BILATERAL-LDU.md` lines 25–119. + +For the finite LDU formulas, I checked the origin-shift correction and the +triangular inverse estimates. If `h=j-i`, the elementary-polynomial estimate +for the upper inverse reduces its exponential part to + +`b[beta_j^2-beta_i^2-rho^2 h-2 rho sum_{k=i}^{j-1} beta_k]`. + +Using `|beta_k|<=1/2` gives at most `b/4-b rho theta h`. The lower inverse uses +the corresponding expression with the other sign on the beta sum and has the +same decay. The factors `c_b^-1` and `c_b^-2` come from the explicitly bounded +products. This proves the stated uniform finite-interval bound; integrality of +the q-values is not needed, only gaps at least one and the beta bound. + +The broad target-cell identity also checks directly. With +`q'_k=k+rho c`, `beta'_k=theta k-rho c`, and +`b=b0/rho^2`, one obtains + +`K=alpha^-1 N_b diag(exp(-b beta'^2))`. + +At a selected target index, `beta'_k` equals the original source beta. Therefore +the finite LDU bound really applies to the block K used in the candidate; it is +not a bound for a different narrow matrix. The product estimate +`log(c_b^-1)<=pi^2/(12 b rho)` gives +`c_b^-4<=exp(pi alpha^2 rho/3)=p^O(1)`. The reciprocal gap is also polynomial +because `theta^-1=poly(p)` and `alpha^2=O(log p)`. + +The square identity-hole completion has block form `[A 0; R I]`. Its inverse is +`[A^-1 0; -R A^-1 I]`. Hence bounded Gaussian row sums and the selected-row +inverse bound give a polynomial full inverse bound. Dividing the completion +by four is contractive even on numerical hole errors. Its extra factor of +one half per axis is exactly compensated by the stated `2^D` shift. + +For the Toeplitz core, the Poisson symbol bound supplies a polynomial lower +eigenvalue, since its exponent is `O(alpha^2)=O(log p)`. An exponentially small +truncation/rounding perturbation preserves that bound. The extension from +these SPD cores to shallow circuits is outside this subreview, but the +Gaussian side supplies the conditioning claimed for that extension. + +## 3. Capping changes the runtime gauge, without changing H + +Sources: candidate lines 203–221 and 273–283; +`work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md` lines 31–107; +`work/phase_assembly_ultra/CAPPED-INTERIORS.md` lines 13–29 and 68–70. + +Within a fixed narrow phase cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Its absolute value is at most `a|i-j|`. Taking exact ratios of the already +rounded positive weights at one local anchor cancels the same common scalar +from both sides of the similarity, so the matrix is unchanged. Relative +rounding error of the original weights remains small because their minimum +is `exp(-O(ad))` and the absolute setup precision is `Cp`, with `ad=o(p)`. +Thus the local circuit sees range `O(aB)`, not the original `O(ad)`. + +For the broad split, expansion of the displayed exponents gives exactly +`-b0(delta+x-lambda y)^2`. The two logarithmic derivatives are +`-2 b0 lambda beta'_i` and `2 b0 lambda^2 beta'_k`; each is bounded in absolute +value by b0. Changing anchor changes the two ideal diagonals by reciprocal +constants. Applying the same exact reciprocal change to previously rounded +diagonals preserves their product with the Toeplitz core. Thus the broad +rounding scheme can consistently use the same global matrix and locally +recenter its capped restrictions. + +The number of pieces does not multiply the norm of their direct sum. The +group logarithmic range is consequently `O(gaB+g polylog p)`. The inherited +absolute weight accuracy adds `O(ad)` once to the precision requirement, rather +than once for every tensor axis. The candidate's condition +`beta+xi+r0=1-zeta`, with `a=O(log p)<=p^r0` eventually, makes this range o(p). +The total precision is therefore `p+O(ad+gaB+g polylog p)=O(p)` in both parameter +families. This conclusion concerns asymptotics; no reasonable finite cutoff +is supplied or required by the statement. + +## 4. Boundary factorization and rounding do not introduce a window-count factor + +Sources: candidate lines 225–240; +`work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md` lines 78–98; +`work/integer-mult-current/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md` +lines 203–301. + +The five-factor identities check by ordinary block multiplication. The shears +use B and C from the original banded matrix. They do not use the generally +dense products `A^-1 B` or `C A^-1`. For forward expansion, F is only applied; +its inverse is not required by that formula. + +For a strictly row-diagonally-dominant matrix, one elimination step gives + +`gap_i(new)>=gap_i(old)+|a_ik/a_kk| gap_k(old)`. + +Its absolute row norm does not increase. Thus the narrow Schur matrix retains +the polynomial gap and bounded row norm. A noncyclic cut of its banded matrix +also retains row dominance. In its LU factors, U has bounded row norm and the +unit-lower factor has only O(W) entries per row, each bounded by the reciprocal +pivot gap. Therefore `||L||=poly(p)`. The identities +`L^-1=U A^-1` and `U^-1=A^-1 L` then give polynomial inverse-factor norms. +The periodic correction has rank O(W) and is controlled by the full completed +inverse, as in the retained identity `K^-1=I-V A^-1 U`. + +The exact partial right-hand sides are consequently polynomially bounded. +Rounding a completed block update creates a row residual bounded by +`poly(p) 2^-P`. Equivalently, rounding and coefficient error can be treated as +a perturbation whose norm is small enough to close the same bound on computed +values. Applying the polynomial full inverse bounds the output error by that +residual. Infinity norm takes the maximum over windows rather than summing +their count. Strassen's fixed recursion contributes polynomial W growth and +O(log W) guard bits, already covered by the precision allowance. + +## 5. Density, stream layout, and global calls + +Sources: candidate lines 207–211, 242–271, and 368–385; +`work/phase_layout_audit_ultra/AUDIT.md` lines 15–188; +`work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md` lines 7–65; +`work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md` lines 16–58; +`work/halo_layout_ultra/HALO-THEOREM.md` lines 88–196. + +The capped carve contributes O(d/B+log B) separators per physical phase cell. +Retaining W-wide separators ensures distinct interiors are uncoupled. Each +elimination only joins adjacent O(W)-sized boundary groups, so the Schur +bandwidth remains O(W). The extra clipped-endpoint term in an L-core is +O(W polylog p), negligible relative to `L W/B polylog p`. + +For the union U, every sparse factor changes its own axis only inside E_i; +therefore it preserves U. The dense A stage does not preserve U, and the +candidate explicitly scatters and regathers around it. For one sparse axis, +`U_i=E_i x [L]^(g-1)` is a complete rectangle inside U. Extracting it in order, +transposing only its polynomial-width axis field, processing, undoing the +transpose, and merging back needs no full g-axis key sort. The full union +scan is charged once per sparse axis, producing the stated g-squared factor +per group. The matrix corollary cannot afford an extra full-key union sort, +but this algorithm does not require one. + +The q-image predicate checks the exact half-open interval +`s(2k-1)<=2tj 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 0 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md new file mode 100644 index 0000000000..1a98011b1f --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md @@ -0,0 +1,91 @@ +# Matrix boundary transfer: full-source audit and interface signoff + +Status: the supplied 9/4 source supports the matrix boundary substitution, with strict exponent slack. This is a theorem-level, computable asymptotic transfer. It is not a numerical finite-size rank certificate and it does not remove any supplier or computational-interface assumptions in the outer integer-multiplication argument. + +The independent reader for this note read all 1,022 lines of +`/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex` +in order, including the full character-existence appendix and all bibliography entries, and also read the README. The companion 2.258 paper is unnecessary for this transfer and is not claimed as read in this new audit. + +## 1. Source coverage + +| Source lines | Material read | Audit result | +|---|---|---| +| 1–36 | Preamble, title, abstract | Establishes complex arithmetic setting. | +| 37–141 | Introduction, theorem and proof route | Explicitly promises an asymptotic exponent, not a competitive finite size. | +| 142–203 | Tensor semiring, rank, characters, detecting lemma | Uses exact rank and mutual restriction. Rank need only be subadditive; no false additivity assumption. | +| 204–235 | Oriented dot-product exponents | Monotonic completely multiplicative functions give power laws; tensor product of the three orientations is square matrix multiplication. | +| 236–269 | Degenerations and interpolation | Fixed degeneration degree is handled before the tensor-power limit. No uniform-in-power degree assumption is made. | +| 270–349 | Finite shared-leg separation | Fourier relation is exact since the phase has absolute size below 5M. Surviving weights equal `(g-h)^2`; the retained auxiliary dot index is distinct from the direct-sum block index. | +| 350–391 | Shared-leg entropy | The factor M cancels correctly before the N-th-root limit. Fixed-frequency counts and continuity cover every probability vector. | +| 392–432 | Polynomial multiplication and symmetrized profile | Rank is exactly a+b−1 by interpolation and output flattening. All six leg permutations are needed for the later middle branch. | +| 433–523 | Determinant filtration and discrete concavity | Fixed input/output bases work for every first-input slice. Multiplication preserves the determinant submodule; the forbidden upper-right block vanishes. The residual lower-left block receives positive weight. | +| 524–604 | Three sectors and shifted tripling | Outer sectors equal C(a,h); the middle is obtained by reversing the first basis and exchanging its second/third legs. This is an identification of an individual block and does not apply incompatible maps to the shared first leg. | +| 605–651 | Diagonal growth | Concavity gives the stated two-increment recurrence; H_a grows at least as a^(1/3). | +| 652–689 | Spectral bound, rank bound, recursion | Detecting characters may depend on d. The conclusion is `nu<=9/4`; selecting a fixed rank scheme requires a strict larger exponent. Rectangular block substitution preserves product order. | +| 690–699 | Algebraic-coefficient remark | The source explicitly allows replacing a complex solution of rational rank equations by a point over Qbar. | +| 700–769 | Appendix: invariant state family and multiplicativity | The fixed-point map is continuous, not asserted affine. Freezing its scalar eigenvalue produces a compact convex invariant subset, permitting finite intersections. | +| 770–905 | Appendix: detecting state existence | Finite rational separation gives an additive catalyst. Reuse keeps one s factor; no tensor cancellation is assumed. Amplification with a second multiplication tensor gives the contradiction to k9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. The outer conclusion remains conditional on exactly the previously retained supplier/computational assumptions. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. The first theorem's records are not overwritten. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md new file mode 100644 index 0000000000..974b8db67f --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md @@ -0,0 +1,93 @@ +# Phase-cell fusion: outer assembly audit + +Status: assembly audit in progress. The exact arithmetic in this directory is a **conditional target**, not a replacement of the selected certificate. The remaining proof interfaces are recorded explicitly below. + +## Conditional transfer statement + +Assume the same finite bit/complex suppliers, common-label arithmetic compiler, semantic guard and all-size fixed-tape interfaces as the selected conditional PR1094 result. Also assume the new deferred-reservation, support-limited H supplier, shallow-circuit tensorization, and grouped sparse/core-halo lemmas below have their stated uniform implementations. Then the retained integer-multiplication reduction runs in + +`O(n (log n)^(1-kappa))`, `kappa=0.001672664764353193`. + +Here p=Theta(log n) and the live scalar bit volume V=Theta(n). Global one-axis preprocessing is polynomial in the original axis sizes and p, hence `n^o(1)`. Table construction or page loading repeated per coarse block is charged to its superpolynomial `L^g` payload by the tensor/scan lemmas; it is included in the displayed V costs, rather than called one global preprocessing operation. Every cost row below is `O(V p^e polylog p)` with `1-e>kappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. Exact positive inequalities alone do not discharge those interfaces. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md new file mode 100644 index 0000000000..c9bd9bfd6f --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_assembly_ultra/README.md @@ -0,0 +1,15 @@ +# Reviewed conditional phase-cell fusion + +With the explicit additional supplied matrix-exponent premise `omega_C<=9/4`, the reviewed local corollary gives **`kappa=0.001673360105405778`**, a gain of `6.9990832886e-7` over the selected conditional exponent. Its status is `PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW`. + +The separate Strassen-based transfer gives `kappa=0.001672922349588871` with status `PASS_CONDITIONAL_PHASE_FUSION_REVIEW`, without that additional matrix premise. Both retain the original supplier and all-size computational assumptions. The upstream selected certificate is unchanged. + +- `phase_fusion_note.tex` is the standalone proof note. +- `CAPPED-INTERIORS.md` records the local gauge and boundary-count change. +- `check_capped_outer.py` recomputes 44 positive rational slacks and five rejecting controls; `CAPPED-OUTER-ARITHMETIC.json` records the reviewed local status. +- `MATRIX-COROLLARY.md` and `check_matrix_corollary.py` give the additional corollary; `MATRIX-COROLLARY-ARITHMETIC.json` contains 47 positive slacks and six rejecting controls. `MATRIX-COROLLARY-REVIEW.json` records its separate review and source hashes. +- `OUTER-AUDIT.md`, `check_outer.py`, and `OUTER-ARITHMETIC.json` preserve the weaker uncapped arithmetic as history. + +Run `python3 -B work/phase_assembly_ultra/check_capped_outer.py` and `python3 -B work/phase_assembly_ultra/check_matrix_corollary.py` from the workspace root. + +The old `Vd` row and the unrecentered full-cell gauge both reject this result. The improvement uses the reviewed Gaussian circuits, support-limited compact supplier, anchored capped gauges, and grouped sparse/halo scans. Tightening the old arithmetic alone cannot produce it. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md new file mode 100644 index 0000000000..bbeeee7431 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md @@ -0,0 +1,242 @@ +# Grouped phase-cell layout audit + +The sparse-union layout in `REFINED-PHASE-FUSION.md` has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires g1, theta=rho-1, q_i=floor(rho i+1/2), beta_i=rho i-q_i for all integer i. Define + +``` +x_i=exp(2a rho q_i), e_i=exp(-a rho^2 i^2+a beta_i^2), +G_ij=exp(-a q_i^2) x_i^j e_j, +r=exp(2a rho), c=product_(m>=1)(1-r^-m)>0. +``` + +The sequence x_i increases at least by a factor r at every step and tends geometrically to zero toward minus infinity. + +## Finite-origin identities + +For an integer origin l, Newton interpolation of columns j>=l gives, on every finite principal interval starting at l, + +``` +C_i(l)=product_(l<=k=j, +U_ij(l)=(e_j/e_i) h_(j-i)(x_l,...,x_i), j>=i, +G=L(l) diag(C_i(l)) U(l). +``` + +Here h denotes complete homogeneous symmetric polynomials, and both triangular factors have diagonal one. The powers arising from moving the monomial origin from zero to l cancel exactly: the pivot is C_i(l), not a residual power of x_i. + +Since C_i(l)>=c and the normalized complete-homogeneous coefficient is at most c^-1, + +``` +|L_ij(l)| <= c^-1 G_ij, +|U_ij(l)| <= c^-1 G_ij. +``` + +All limits as l tends to minus infinity exist, including fixed-degree h over the infinite summable past. The displayed Gaussian domination permits passing the factorization to the bilateral limit. + +## Inverse factors + +For i>=j and h=i-j, + +``` +(L(l)^-1)_ij=(-1)^h C_i(l)/C_j(l) + * exp(a[-beta_i^2+beta_j^2-rho^2 h+2rho sum_(j=i and h=j-i, + +``` +(U(l)^-1)_ij=(-1)^h (e_j/e_i) e_h(x_l,...,x_(j-1)). +``` + +The latter e_h is elementary symmetric, distinct from the scalar normalization e_j. Its dominant product consists of x_i,...,x_(j-1), so its normalized value is at most c^-1. + +Using |beta_i|<=1/2 in the two exact exponents gives uniform estimates + +``` +|(L(l)^-1)_ij| <= c^-2 exp(a/4-a rho theta h), +|(U(l)^-1)_ij| <= c^-1 exp(a/4-a rho theta h). +``` + +These pass to the bilateral limits. Products of triangular factors and their inverse factors have only finitely many terms for each supported matrix entry, so both inverse identities survive directly. All four factors define bounded absolutely row-summable operators. + +## Periodization + +Every limiting factor is simultaneously s-periodic in its row and column indices. For a bounded absolutely row-summable periodic array A, define + +``` +Pi(A)_ij=sum_(n in Z) A_(i,j+ns), 0<=i,j=1**, this is O(theta^-2), with a constant depending only on a. Thus a fixed admissible integer Gaussian width alpha suffices for the norm and perturbation estimates. No logarithmic growth of alpha is needed for conditioning. + +The theorem changes an analytic parameter constraint. Applying these factors quickly on the tape remains a separate constructive obligation. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md new file mode 100644 index 0000000000..59d9128d75 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md @@ -0,0 +1,211 @@ +# Independent checks of Gaussian interpolation and holes completion + +Status: exact algebraic identities and analytic bounds below; no complete new inverse implementation, and no final kappa claim. This continues the parent investigation after the first outer-reduction package. + +## 1. Local scaled Vandermonde identity + +Write `a=pi*alpha^2`, `rho=t/s`, `q_i=floor(rho*i+1/2)`, and `beta_j=rho*j-q_j`. On any nonwrapping index interval, + +``` +N_ij = exp[-a*(rho*j-q_i)^2 + a*beta_j^2] + = exp[-a*q_i^2] * (exp[2a*rho*q_i])^j + * exp[-a*rho^2*j^2+a*beta_j^2]. +``` + +Translations of both index origins preserve a diagonally scaled Vandermonde representation. If the local span is L, centered nodes have logarithms O(aL), and the product-tree polynomials, Lagrange denominators, and outer diagonals have logarithmic height O(aL^2+L). Thus L=O(sqrt(p/a)) is compatible with Cp-bit working precision. Distinct nodes are explicitly separated, so the inverse of a *single supplied principal block* can be computed by multipoint interpolation with that precision and O(L p polylog p) bit cost. + +This does not justify independent short blocks. The current inverse has strong directed coupling near the end of a phase cell. Its exact q-LDU below shows why a halo of width merely sqrt(p/a) can lose much more than 2^-p. + +## 2. Exact Gaussian Toeplitz LDU in a no-gap phase cell + +Within a cell `q_i=i+c`, put `theta=rho-1`, + +``` +q=exp(-a*rho), Q=q^2, +P_j=product_(k=1..j) (1-Q^k), P_0=1, +D_i=exp(a*theta*i^2-2a*c*i), +G_ij=q^((i-j)^2). +``` + +Then `N=diag(D_i)*G*diag(D_i)^-1`. With the Gaussian binomial + +``` +{i choose j}_Q = P_i/(P_j*P_(i-j)), +``` + +the exact lower factor and its inverse are + +``` +L_ij = q^((i-j)^2) * {i choose j}_Q, i>=j, +Linv_ij = (-q)^(i-j) * {i choose j}_Q, i>=j, +G = L * diag(P_j) * L^T. +``` + +These follow from the ordinary Newton factorization of the Vandermonde matrix at nodes Q^-i; eliminating its successive monic Newton polynomials gives the displayed Gaussian binomial factors and pivot P_j. The inverse follows from the finite q-binomial theorem, or multiplying the two triangular factors and grouping equal powers of Q. `check_q_ldu.py` verifies both identities exactly at 30 rational q/size choices (1,155 matrix entries), including diagonal conjugations. + +Each triangular application is a diagonally scaled Toeplitz convolution: + +``` +L = diag(P_i) T_(q^(h^2)/P_h) diag(P_i)^-1, +Linv = diag(P_i) T_((-q)^h/P_h) diag(P_i)^-1. +``` + +The N factors add only the D_i conjugations. Their inverse lower entries have exponent + +``` +a*((2*beta_i-rho)*h - theta*h^2), h=i-j. +``` + +Since beta_i<=1/2, + +``` +abs((D Linv D^-1)_ij) + <= C exp[-a*theta*h*(h+1)], +C = product_(k>=1)(1-Q^k)^-1. +``` + +The upper inverse obeys the same bound using beta_i>=-1/2. For a>=pi and rho>1, C<2 by a geometric bound on the logarithm of the infinite product. Hence both inverse triangular factors have row norm O(1+(a theta)^-1/2). The P_j diagonal is bounded between 1/C and 1. This avoids using cancellation of exponentially large factors to bound the *completed* inverse. Over one cell, the explicit D_i range costs O(a/theta) guard bits, so the retained condition alpha^2/theta=O(p) still suffices for a straightforward implementation. + +This supplies a clean alternative to the retained Gohberg–Semencul interior solve. The latter already has linear convolution cost; a new exponent requires treating the gaps between cells or their boundary system. + +## 3. Exact complementary hole positions + +Let 11, x_k=r^k, and V_t=(x_k^j)_(0<=k,j=k, and 0 otherwise, +(V_holes^T)_(j,h) = x_h^j. +``` + +The coefficient of X^(t-1-j) in P(X)/(X-x_h) equals +`sum_(k=0..j) p_(t-k) x_h^(j-k)`, by synthetic division from the leading end. Lagrange interpolation gives the identity. + +Given values only on selected nodes, set the H missing values initially to zero and do one full interpolation. Solve A for the missing values that cancel the H degree-tail coefficients, then interpolate with the completed values. The resulting polynomial has degree=1, + +``` +abs(B_ij) <= exp[-a*rho*(rho'*k^2-k)]. +``` + +Indeed the relevant distance is at least rho'k-1/2, and the subtractive delta_j^2 is at most 1/4. Thus off-diagonal row sums are bounded by a convergent geometric Gaussian tail, with first exponent a rho(rho'-1). For small theta, rho' is Theta(1/theta), so this exponent is Theta(a/theta). To obtain error 2^-p a direct truncated solve only needs complementary width + +``` +O(1+sqrt(p/(a*rho*rho'))) = O(1+sqrt(p theta/a)). +``` + +This rigorously confirms the conditioning and narrow-band part of the parent's holes-completion proposal. It is not a cost proof for the complete original inverse: the Toeplitz T and full geometric interpolation must first be implemented with their enormous diagonal scalings canceled symbolically. + +## 6. A useful normalized erasure-coefficient bound + +Let hole exponents h_0<...=g, let R=r^g>1, and e_k be the kth elementary symmetric polynomial in r^h. Let M_k be the product of the k largest nodes. Then + +``` +1 <= e_k/M_k <= product_(j>=1)(1-R^-j)^-1. +``` + +For any selected k-index subset, compare its indices with the k largest indices. Its nonnegative deficits form a partition, and its product divided by M_k is at most R to the negative partition size. Summing over partitions proves the bound. The contribution from deficit >=m is at most + +``` +R^(-m/2) * product_(j>=1)(1-R^(-j/2))^-1. +``` + +For R>=4 the latter product is below 4, so retaining deficit less than `m >= (2p+4)/log2 R` suffices for relative error below 2^-p. Thus normalized coefficients have bounded height even when unnormalized coefficients contain Theta(a theta t^2) exponent bits. Only O(p theta/a) partition energy near the subset boundary is relevant. + +Likewise, for a nonhole i, pull `product_h max(r^i,r^h)` out of |E(r^i)|. The residual is `product_h(1-r^-|i-h|)`, which lies between `product_(k>=1)(1-r^-k)^2` and 1. It is uniformly bounded away from zero for fixed r>1. These are positive products and sums; the bounds do not depend on fortuitous numerical cancellation. + +The unresolved algorithmic question is performing the needed polynomial multiplication/division while storing these *degree-dependent* scales, rather than one common scale. Feeding the raw coefficients to an ordinary FFT requires Theta(a theta t^2) bits; declaring that cost absent is invalid. The positive partition estimates give a concrete starting point for a banded or locally normalized implementation. + +## 7. Correct boundary of the result + +- Exact q-LDU, exact hole indexing, exact completion, complementary conditioning, and the normalized erasure bounds above are established algebraically. +- `Q-LDU-CONTROLS.json` and `HOLES-CONTROLS.json` contain finite independent checks. +- No stable all-size global holes solver is implemented here. Its fully paid precision/tape cost remains open. +- No final multiplication exponent follows from these identities alone. + +## 8. Weighted full-geometric interpolation: a stable local implementation + +This is a further positive lemma for one component of holes completion. It does **not** complete the Toeplitz/complement adapters. + +Let beta=H/t=theta/rho, let h_l be the actual holes above, and let h*_l=(l+1/2)/beta be the ideal equally spaced holes. For x in [0,t], define + +``` +R(x) = 2a*rho * sum_l (x-h_l)_+ - a*theta*x^2. +``` + +This is the logarithm of the dominant erasure evaluation after canceling the full-geometric Gaussian input diagonal, up to a constant independent of x. That constant can be kept symbolically and canceled at the final adapter. + +For ideal holes, put m=floor(beta*x+1/2). Then + +``` +sum_l (x-h*_l)_+ = m*x - m^2/(2beta), +R*(x) = -(a*rho/beta)*(m-beta*x)^2, +-a*rho/(4beta) <= R*(x) <= 0. +``` + +Since |h_l-h*_l|<=1/2, compare increments on [x,x+v]. Holes below both endpoints give the same increment v under both locations, and holes above both give zero. Only O(beta*v+1) holes whose ideal or actual locations meet the interval can contribute; each contributes an error at most 1/2. Consequently, for 0<=x<=x+v<=t, + +``` +|[R-R*](x+v)-[R-R*](x)| <= a*theta*v + O(a*rho), +|R(x+v)-R(x)| <= a*theta*v + K, +K <= 4a/theta (01. It is a norm lemma, not a complete fast inverse algorithm. + +Let s>=5, s=1. The exact periodic normalized Gaussian matrix is + +``` +N_(i,j mod s) = sum_(k congruent j-i mod s) + exp[-a*((beta_i+rho*k)^2-beta_(i+k)^2)]. +``` + +Its principal k=0 term is one. Let L be its k=-1 weighted cyclic shift and U its k=+1 shift. Thus + +``` +L_i = exp[-a*((beta_i-rho)^2-beta_(i-1)^2)], +U_i = exp[-a*((beta_i+rho)^2-beta_(i+1)^2)]. +``` + +All other k terms form R. The aliases k=±s and beyond remain in R; they are not discarded. + +For h consecutive lower factors the exponent telescopes to + +``` +beta_i^2-beta_(i-h)^2 - 2rho sum_(j=0..h-1) beta_(i-j) + h rho^2 + >= h rho theta - 1/4. +``` + +The upper factors obey the same bound. Therefore both alternating Neumann inverses exist, including over arbitrarily many cyclic wraps, and + +``` +||(I+L)^-1||, ||(I+U)^-1|| + <= B = exp(a/4)/(1-exp(-a rho theta)) + <= 2 exp(a/4)/theta. +``` + +The final inequality uses a>=1 and 0=2, + +``` +(beta_i+rho*k)^2-beta_(i+k)^2 + >= (rho*|k|-1/2)^2-1/4 + >= k^2-|k|. +``` + +Consequently + +``` +||R|| <= 2 exp(-2a)/(1-exp(-4a)) < 4 exp(-2a). +``` + +Write `A=(I+L)(I+U)`. Then N=A+(R-LU), and + +``` +||A^-1(R-LU)|| <= 5 B^2 exp(-2a) + <= 20 theta^-2 exp(-3a/2). +``` + +If `a>=4 ln(16/theta)`, this is bounded above by +`20 theta^4/16^6 < 1/2`. Hence + +``` +||N^-1|| <= 2 B^2 <= 8 exp(a/2)/theta^2. +``` + +Choose the integer `alpha=ceil(sqrt(4 ln(16/theta)/pi))`. With +`a0=4 ln(16/theta)`, the elementary ceiling inequality gives +`a<=2a0+2pi`; therefore + +``` +||N^-1|| <= 8 exp(pi) 16^4 theta^-6. +``` + +This is an explicit polynomial bound. For theta=Theta(1/d), alpha^2=O(log d) and the total original tensor scale has logarithm O(d log d)=o(p) for every fixed d=p^epsilon, epsilon<1. The period and aliases introduce no exception. + +## Exact controls + +`check_nearest_neighbor.py` checks 69 admissible finite (s,t) pairs and 93,976 product identities/bounds, including all skip gaps and two complete cyclic wraps. It uses exact rational exponents and no floating point. The exponential conclusions follow from monotonicity and the displayed convergent geometric sums. The output is `NEAREST-NEIGHBOR-CONTROLS.json`. + +## Remaining algorithmic obligation + +This lemma establishes that logarithmic alpha squared is analytically permitted. The old inverse implementation still incurs too much work at that alpha. A fast stable inverse, such as a fully normalized holes-completion algorithm, must still supply its actual movement, rounding and operation count. Erasing the inverse cost from the outer ledger would not follow from this norm proof alone. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md new file mode 100644 index 0000000000..876f44a23e --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md @@ -0,0 +1,60 @@ +# Partial tensor fusion: a strictly better prospective outer balance + +This is a prospective cost theorem, not an implemented resampler. It changes the full-fusion target to a smaller interface whose precision is payable. + +Suppose the normalized holes construction is completed so that a g-axis Gaussian group is a fixed number of tensor geometric/Toeplitz convolutions, diagonal factors, and corrections supported on the hole rows, and suppose its halo gather/scatter has the same asymptotic cost. Let d=p^epsilon and a=Theta(log d), theta=Theta(1/d). + +The exact normalizations developed in HOLES-AND-Q-LDU.md have local logarithmic range O(a/theta)=O(a d) per axis. Therefore g axes can be fused with O(p) guard bits if + +``` +g*a*d = O(p). +``` + +Taking `g=p^(1-epsilon-zeta)` for fixed zeta>0 pays the logarithm and leaves strict slack. There are d/g groups. A group convolution using the retained supplier with saving q costs, after division by full volume, + +``` +g^(1-q) polylog p. +``` + +Thus all Gaussian groups cost + +``` +(d/g)*g^(1-q) = d/g^q + = p^[epsilon-q*(1-epsilon-zeta)]. +``` + +Its saving is `(1+q)*(1-epsilon)-q*zeta`. The main synthetic transform, using deferred reservations, retains saving epsilon*q. Ignoring the arbitrarily small fixed zeta and strict absorption gaps, balancing yields + +``` +epsilon=(1+q)/(1+2q), +kappa=q*(1+q)/(1+2q). +``` + +This exceeds the old `q/(1+q)` by exactly + +``` +q^3 / ((1+q)*(1+2q)). +``` + +That is a new outer balance, rather than tighter rational margins in the old family. At the selected complex endpoint the difference is about 4.7e-9. For example take q equal to the selected complex saving minus 10^-10, zeta=10^-8, the displayed epsilon, and kappa ten^-11 below the smaller main/Gaussian margin. The resulting exact arithmetic target is approximately + +``` +0.00167266475042434534254271472, +``` + +about 4.55335e-9 above the selected 0.001672660197076918. This is only a prospective target until both missing interfaces below are established. + +## Other paid terms + +The bit routing saving is larger than q at the selected pair. Moving a group of O(g log p) small coordinate bits across spectator fields costs O(V g^(1-a_bit) polylog p) per group, hence O(V d/g^a_bit polylog p), below the convolution cost since a_bit>q. A superpolynomial untouched spectator suffix remains available while processing a group, so the compact supplier's descriptor work can still be paid; pointwise small-polynomial products require a separately paid small-field transpose but only logarithmic-width fields. + +The g-axis scalar normalization costs O(g a), and the stronger local weights cost O(g a d); both fit O(p) by construction. For local convolution window L=O(p/a+d), take a dyadic microtile N≫gL. The total halo volume is bounded by `(1+O(L/N))^g=O(1)`. The gauge increment bound implies total logarithmic variation O(g[a theta N+a/theta]); with N=Theta(gL), the extra condition is g^2<=O(d). It holds at the proposed epsilon near one, since 2(1-epsilon)q`, and requires `dK=o(p)`. Thus even if all Gaussian work disappeared, retaining that reservation argument would still force `epsilon<1/(1+q)`. It is insufficient to erase just the Gaussian row from a ledger. + +The selected outer logic was read in `research/complex-pr615-ceiling/vendor/multicold/engines/outer.py`; its seven margins and 47 slacks match the current 1094 assembly. The underlying detailed interfaces are in upstream sections 02, 04, 05, 06, 07, 08, 09, 10 and the cited `references/semantic-bulk/rad20/reports/` transfer notes. In particular, the current bulk Gaussian proof already uses local principal windows and short fields; proposing that same improvement again would not change the selected result. + +## 2. Deferred reservations: removing the power-width reservation condition + +This is an all-size **conditional transfer argument**, assuming the retained complete rectangular compact gadget, exact common-frame circuit, arbitrary coordinate router, and linear semantic guard. It does not require a new finite supplier identity. + +Let `G=4 ceil(log2 p)+6`, and choose `K=128G`. At a layer on `h` still-unprocessed selected bits, reserve enough complete K-bit chunks to provide the two front H-bit fields, one back H-bit field, and row bits, where `H=hG`. The number of reserved chunks is + +``` +R(h) = ceil(3hG/K) + O(log h). +``` + +For `h` above a fixed polynomial in `log p`, `R(h) <= h/16`. Select those chunks from the still-unprocessed axes. Apply the retained compact circuit to the other `h-R(h)` bits, borrowing all three dirty fields from the reserved subset. At the completed-call boundary it is exactly `C` on the active bits and identity on every reserved or spectator bit, for arbitrary initial coefficients and arbitrary dirty addresses. Therefore all reserved bits retain their original complete ranges and can be processed next. + +Repeat on the reserved subset. The processed subsets are disjoint. Stop once `h=polylog(p)` and apply those remaining kernels individually. This is **not** a recursive call on a truncated record or a clean-scratch claim: the entire original stream is still present, with the same full polynomial suffix, and every previously processed coordinate is an untouched spectator. Completed kernels on disjoint selected bits commute. All row padding is removed before moving to the next phase, so temporary factor-two padding never multiplies across phases. + +To establish the time bound, retain the original `m,W,s` recurrence and stop its internal recursion at `d^beta`. With `sigma=1-b`, any `lambda>max(1-a,sigma)` gives, with logarithmic factors suppressed, + +``` +F(h) <= O(h^lambda + h^sigma d^(beta*(1-sigma)) + 1). +``` + +This follows because a packed local basis change has width `O(hG)` and hence cost `O(V h^(1-a) polylog p)`. The term formerly written `(h d^c)^(1-a)` is now `h^(1-a) polylog p`. All compact-gadget exceptional fractions remain polynomially small with K=128G; the retained repair proof uses the current call's O(p)-bit address and is not replaced by a global exception count. + +Since `h_j <= d/16^j` above the polylogarithmic tail, + +``` +sum_j F(h_j) + = O((d^lambda + d^(sigma+beta*(1-sigma))) polylog p). +``` + +The extra physical rearrangements of complete existing fields cost +`O(V p^(1-a) polylog p)` by the retained arbitrary coordinate router, over O(log d) phases. Its polynomial descriptor work is paid by the unchanged superpolynomial polynomial-record suffix, at every internal call. This is additive, not free routing. + +Consequently, for every fixed `q < min(a,(1-beta)*b)`, the whole layer has + +``` +O(V [d^(1-q) + p^(1-a)] polylog p). +``` + +The common guard remains O(d): each completed phase changes semantic numerator/denominator bounds by O(h_j), and `sum h_j=O(d)`. Only the currently active phase contributes its internal excess. There is still exactly one truncation after the entire normalized layer. No power K=d^c and no condition c>q remains. K=o(ell) now follows from `ell=p^(1-epsilon)` for every fixed epsilon<1. + +The extra top round on the long axes can use the same layer after a paid coordinate rearrangement collecting its selected bits and K-bit complete neighborhoods. The common rounds use a single paid global layout router, rather than `p/K` separate K-swaps. Thus the transformed version of the synthetic cost is + +``` +O(V [p^(1-a) + ell*d^(1-q)] polylog p). +``` + +This is the strongest positive transfer established by the present note. It removes the geometric source of `q/(1+q)` in the compact compiler. It does not remove the independent O(Vd) Gaussian and scalar costs. + +### Checks + +`check_constructions.py` checks 1,000 exact reservation schedules and 5,460 entries of actual Gaussian-rational `C` tensor matrices. Every axis is processed once; the phasewise output equals the independently calculated closed tensor matrix. Omitting the deferred subset is a retained failing control. These checks support the written argument; they are not an all-size compiler implementation. + +## 3. Batched right-hand sides for the Gaussian boundary inverse + +This refinement applies to the precomputed noncyclic banded L/U factors in the retained principal-window solver. It uses the many spectator lines carrying the same axis matrix. + +For a triangular matrix with scalar bandwidth w, group consecutive rows into blocks of a power-of-two width `B in [w,2w)`. There is only one preceding block in a lower solve (one following block for an upper solve). Precompute each diagonal-block inverse. For B simultaneous right-hand sides, the update is + +``` +X_j = L_jj^-1 * (Y_j - L_j,j-1 * X_j-1). +``` + +Two B-by-B products implement each update. The seven-product identity gives `O(B^omega)` scalar operations, `omega=log_2(7)`. For z boundary rows and R>=B right-hand sides, splitting the columns into B-column batches gives + +``` +O(z R B^(omega-2)) +``` + +scalar work. Incomplete edge blocks and the last RHS batch are zero-padded only for the current product; this is a constant temporary volume increase, not a tensor-axis padding factor. The matrices may vary from row block to row block. No Toeplitz assumption is made. + +The original boundary has `z=O(M w/d + w)` rows in a length-M local window. Summed over axes and lines, the old regular contribution `V w^2` becomes + +``` +O(V w^(omega-1) polylog p). +``` + +The artificial-end contribution is handled by a polynomial-size microbox L; it is +`O(V d w^(omega-1)/L polylog p)`. The O(Vd polylog p) local field motion and interior-convolution cost is still paid and is still present. + +### Tape layout + +Expose the same axis and factor page used by the retained solver. Gather B adjacent RHS indices with a B-valued local field, moving that O(log p)-bit field across the current row field by the retained small-field transposition. It costs O(V log p) per axis. This charge belongs to the existing d-pass term. Read factor pages and RHS row blocks sequentially. Keep the immediately preceding solved B-by-B block on a fixed work tape. Products use a depth-first seven-product recursion, parking/restoring the other finitely many matrices at each node; scanning and quadrant rearrangement is O(B^2) per level and is dominated by B^omega. There is no random factor lookup and no separate tape for every RHS. + +### Precision + +Keep the retained O(p)-bit rounded factor entries and compute each B-by-B product exactly as a dyadic matrix product before one component rounding. An individual seven-product recursion has depth log B. Additions grow integers by only O(log B) bits at every product boundary, so exact products and their sums have O(p+log B)=O(p) bits. A block inverse is precomputed at the retained high precision, with its row-norm perturbation charged explicitly. + +Round after a completed block update, not between scalar products. The residual of the completed triangular system is bounded rowwise by `poly(p)*2^-P*(1+||X||)`, with P the retained large constant multiple of p. The retained inverse-norm bounds then give a uniform global solution error after absorption, independent of the number of row blocks. This is the same residual closure used by the original banded solver; it must be instantiated with new block-factor rounding constants in any adoption. An unproved claim of numerical stability of fast floating-point multiplication is unnecessary. + +For `u=p^r`, `w^2=O(p^(1-r))`. The revised regular boundary exponent is + +``` +((omega-1)/2)*(1-r), +``` + +whose saving at r=0 is `(3-log_2 7)/2 > 0.096`. Thus an implemented version could remove the old requirement r>kappa while keeping `epsilon+r<1`. It cannot remove the separate O(Vd) scan term by itself. + +### Exact checks and remaining check + +The script independently compares seven-product multiplication against classical multiplication, then compares 60 varied-size blocked triangular solves against exact Gaussian-elimination inverses (1,976 output entries). A dropped-predecessor-block control fails. This establishes finite algebraic orientation, padding and dependencies. A complete adaptation of the original numerical residual constants is still required before adopting the changed analytic interface. + +## 4. A concrete route toward tensor Gaussian fusion + +This section is **conjectural construction work**, not an established algorithm. The missing step is stated so it can be attacked. + +A possible way to avoid d separate Gaussian passes is to compute the tensor Gaussian on a microbox through one tensor convolution. For `rho=s/t`, a real Gaussian row has the exact scalar identity + +``` +exp[-a(j-rho*k)^2] + = exp[-a(1-rho)j^2] + exp[+a*rho(1-rho)k^2] + exp[-a*rho(j-k)^2]. +``` + +Integer translations of j and k add only linear factors, so each local rectangle is a diagonal, Toeplitz convolution, diagonal. Across axes, the diagonal factors fuse into one per-record multiplication. The central convolution is a tensor cyclic convolution after sufficient halos. + +With the current theta≈1/d, a local Gaussian halo has radius +`A=O(alpha*sqrt(p))`. To keep product halo volume constant, the core needs length `L=Omega(dA)`. But the fused diagonal logarithmic range is then +`O(d*theta*L^2/alpha^2)=O(d^2 p)`, not O(p). Thus the most obvious tensor-chirp fusion fails its precision ledger. This explicit failure should prevent a false positive. + +One changed hypothesis makes that favorable: choose source primes much closer to t, with `theta=Theta(d^-3)`. Then at `L=Theta(dA)`, that logarithmic range is O(p), and the product halo overhead is still constant. Such prime lengths are available eventually for every fixed epsilon<1: the explicit prime-counting error from Fiori–Kadiri–Swidinsky, Corollary 22, is `O(x sqrt(log x) exp(-c sqrt(log x)))`; on an interval of width `x/d^3`, with `log x=Theta(p^(1-epsilon))`, its error is o(x/(d^3 log x)). The interval contains far more than d primes, so distinct choices and `S/T=1-o(1)` follow. Trial division setup remains `n^o(1)`. This is an asymptotic existence argument with an enormous cutoff, not a practical prime-search claim. + +However two serious costs remain unproved: + +1. A power-of-two convolution window N has a useful core N-2A. Reinterpreting an axis by these variable non-dyadic cores is not the known binary-coordinate router. Padding every N-2A field to the next power of two costs 2^d. Performing the known small-field reshape one axis at a time costs Vd. A new simultaneous variable-core gather/scatter is needed, with full payload movement charged. +2. The inverse Gaussian correction is not itself one convolution. With theta=d^-3, the current phase-cell similarity weights range like `exp(O(u/theta))`, too large for its old O(p)-bit proof. The generic banded solve can avoid those enormous weights, but the present batched bound still has a d factor outside its favorable structured regime. The tensor inverse needs a different stable factorization or a sparse-boundary construction with a fully paid ledger. + +Therefore “choose closer primes” plus “use a multidimensional FFT” is not a proved bypass. It is an explicit construction program with two specified lemmas. + +## 5. Exact optimistic target and fully paid comparison + +Assume a new tensor-resampling interface gives + +``` +O(V [p^(1-a) + d^(1-q) + w^(omega-1)] polylog p) +``` + +for the complete expansion and inverse, including variable-core layout, coefficient factors, stable inverse, all restoration and only O(p)-bit records. Combine it with Section 2. The complete normalized saving is at least + +``` +min(a, epsilon*q, 1-epsilon*(1-q), + 1-((omega-1)/2)*(1-r), epsilon), +``` + +with strict `epsilon+r<1`, `q 0`, the complex endpoint exceeds q by `0.000000462676808103522925`, and the gamma gap is `0.000005`. The target exceeds the selected kappa by `0.000002319802923082` (about 0.1386894%). Letting epsilon approach one and q approach b would recover the complex saving itself, up to arbitrary strict slack. + +**Fully paid ledger today:** without the unproved tensor-resampling interface, the unchanged Vd term remains. It imposes `kappa<1-epsilon`, so maximizing the proven/reused terms still gives at most `b/(1+b)`. Sections 2 and 3 alone do not improve final kappa. Under the displayed sample, the paid Gaussian margin is only `0.00001`, so that sample FAILS the current interface. It is deliberately not emitted as a passing outer certificate. + +## 6. Reproduction and evidence + +Run from the parent workspace: + +``` +python3 work/radical_reduction_ultra/check_constructions.py +python3 work/radical_reduction_ultra/check_ledger.py +``` + +`CONTROLS.json` contains exact finite controls. `LEDGER.json` contains the optimistic row and the correctly rejected fully paid row. Neither imports a producer or mutates the source checkout. + +## Open next questions + +- Instantiate Section 2 against the exact current compact field-layout code and reject malformed reservoir choices. This is a bounded integration task; it could remove a real interface restriction even before a new final exponent is obtained. +- Carry the block residual argument in Section 3 through the retained P=32768p constants and factor-page format. The algebraic and layout reductions are explicit; numerical certification is the remaining bounded task. +- For the fusion program, first prove or disprove a simultaneous variable-core gather/scatter with O(V p^(1-a) polylog p) cost and no 2^d padding. This isolates the data-layout bottleneck before more analytic effort. +- Seek a stable tensor inverse for near-identity Gaussian resampling with theta≈d^-3 that avoids both enormous diagonal similarities and d sequential full scans. A finite small-d result is not enough; require the exact operator, record width and mass recurrence. + +## Primary sources and attribution + +- CrocSwap/integer-mult-bounds pinned upstream and transfer reports, in particular 02-streams, 04-swap, 05-layers, 06-transforms, 07-resampling, 08-assembly, 09-exact-arithmetic, 10-transposition; reviewed contributions by Douglas Colkitt and the cited independent contributors are retained unchanged. +- Harvey and van der Hoeven's original Gaussian resampling and multiplication interfaces are inherited through those source proofs. +- V. Strassen, *Gaussian Elimination is Not Optimal*, Numerische Mathematik 13 (1969), 354–356, DOI `10.1007/BF02165411`. The seven-product identity is reproduced and independently checked here; no modern matrix multiplication exponent is needed. +- A. Fiori, H. Kadiri, J. Swidinsky, *Sharper bounds for the error term in the prime number theorem*, arXiv:2206.12557v2, Corollary 22: https://arxiv.org/pdf/2206.12557 . Used only for the clearly marked close-prime construction program. + +## Later constructive developments + +The positive interpolation-tree and global two-tail results are in `STABLE-POSITIVE-TREE.md`, with exact controls in `POSITIVE-TREE-CONTROLS.json`. They provide bounded gauges for every signed-data intermediate and show that all high-level rank-one contributions collapse to two global running sums. Its Section 6 gives a normalized Sylvester factorization into a dirty Toeplitz shear, one well-conditioned full Toeplitz map, and one triangular Toeplitz map; `SYLVESTER-CONTROLS.json` checks that identity. + +The stronger fixed-width Gaussian norm theorem is independently audited in `FIXED-A-PERIODIC-NORM-AUDIT.md`, with `FIXED-A-LDU-CONTROLS.json`. It replaces the earlier sufficient logarithmic-width condition: for any fixed admissible a>=1, the periodic inverse has norm O(theta^-2). This is an analytic parameter improvement, not a completed fast inverse. + +The outer partial-fusion candidate in `PARTIAL-FUSION.md` remains prospective. Neither the stable, shallow in-place algebra for plain Toeplitz matrices nor bounded final Gaussian conditioning alone proves that the curved Gaussian coefficient gauges can be processed with only O(p) total guard bits when many axes are fused. That interface, and the actual halo movement cost, must be resolved before claiming a new kappa. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md new file mode 100644 index 0000000000..51920b3b76 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md @@ -0,0 +1,125 @@ +# A narrower positive route: fuse retained phase-cell interiors + +Status: a complete parameter ledger for a concrete conditional transfer. It avoids the unresolved normalized-erasure adapter. It still requires the new shallow in-place structured-matrix circuit to be integrated with the retained fast tensor supplier and a grouped sparse boundary gather. No new selected certificate is asserted. + +## Construction + +Keep the already reviewed phase-cell Schur decomposition, with + +``` +d=p^epsilon, u=alpha^2=Theta(p^r), w=Theta(p^((1-r)/2)), +0(1-r)/2. +``` + +The positive r may be arbitrarily small; it eventually dominates the old logarithmic lower bound on u. Thus this route can retain the established diagonal-dominance, rounded H, local inverse, and Schur residual proofs. The stronger fixed-a theorem is not needed here. + +Within each no-gap phase cell, the interior matrix is exactly + +``` +A=diag(a_j) G diag(a_j)^-1, +G_ij=G_(i-j), log(max a_j/min a_j)=O(u d). +``` + +The retained G is a symmetric truncated Gaussian Toeplitz matrix with `||G-I||<1/8`; hence G and G^-1 are uniformly conditioned. Apply its inverse by the new shallow in-place low-displacement circuit, rather than summing the two Gohberg-Semencul branches. The circuit's Toeplitz coefficients and Schur generators are precomputed. This is the algebraic connection to the new in-place work; the old source proof did not provide such a tensor-fusible circuit. + +Fuse + +``` +g=p^beta, beta=1-epsilon-r-zeta, +``` + +axes at a time, with fixed zeta>0. The total diagonal similarity guard is +`O(g u d)=O(p^(1-zeta))=o(p)`. Polylogarithmic circuit depth and bounded polynomial factor growth add only `O(g polylog p)=o(p)` bits. Every completed group has polynomial norm; error is allocated at the completed-group boundaries, preserving the retained O(p)-bit record model. + +### Dyadic interiors without exponential padding + +Inside a phase cell, repeatedly choose the largest power of two K such that `K+2w` fits the remaining interval, while its remaining length exceeds 4w. Make a piece of length K+2w, designate w endpoints on each side as boundary, and use its K remaining indices as the interior. The leftover interval of length at most 4w is all boundary. + +The remaining length at least halves on each step. Thus there are O(log d) pieces per original cell, every interior has dyadic length, and the total boundary size grows by only O(log d). No padding by a factor two per tensor axis is used. Schur fill still joins only the two w-sized endpoint groups of a piece, so its scalar bandwidth remains O(w). Distinct dyadic size classes require only O(log p) circuit schedules; inactive axes use identity stages. + +### Boundary arithmetic + +Let R denote the number of spectator right-hand sides, which eventually exceeds every polynomial in p. Batch them in groups of w. Every dense w-by-w block multiplication in the banded forward/backward solve, the border correction, and the boundary/interior couplings then costs O(w^omega) scalar products for w right-hand sides, omega=log_2(7)<2.8081. The retained no-pivot factors are precomputed; Strassen creates only polynomial intermediate growth in w, hence O(log p) extra guard bits. Use the existing global residual argument to bound the completed solve error. + +There are O(s*w*log(d)/d) boundary rows per axis. Consequently the total boundary arithmetic over all d axes is + +``` +O(V * w^(omega-1) * polylog(p)), +``` + +where V counts p-bit data volume. This includes H_BI and H_IB couplings; doing those couplings scalarly would incorrectly reintroduce V*w^2. + +### Five stages separate fused interiors from sparse corrections + +For the interior/boundary order write H=[A B; C D] and S=D-C A^-1 B. The exact identity + +``` +H^-1 = diag(A^-1,I) [I -B;0 I] diag(A,S^-1) + [I 0;-C I] diag(A^-1,I) +``` + +is verified by multiplication. Thus the only full-interior stages are A and A^-1. The two shears use original banded B and C, whose support is within O(w) of each artificial cell boundary. The middle factor splits into the commuting interior A and boundary S^-1. + +For a g-axis group, collect the union U of the O(w log d)-wide endpoint slabs. Its volume is O(V*g*w*log(d)/d). All sparse shear and boundary stages preserve this union; outside it they are identity. Process the g axis operators sequentially on U, without g scans of the full V data. Even a deliberately loose movement cost is + +``` +O(V*g^2*w*polylog(p)/d) per group, +O(V*g*w*polylog(p)) over all d/g groups. +``` + +Add one full scan per group to gather/scatter U, for O(V*d/g). This is below the proposed fused interior cost at the displayed parameters. Dense boundary arithmetic is still batched as above and costs O(V*w^(omega-1)*polylog p); scanning inactive entries in U is included in the movement bound. + +The union-slab gather/scatter and the evaluation of its piece catalogues must be implemented in the retained macro-routing model. The address fields have polynomial length and g*polylog(p)=o(p), but that observation alone is not a tape routing proof. Importantly, the cost budget now tolerates a sequential factor g on the **sparse union**, so the required route is weaker than a fully fused boundary solver. + +## Paid exponents + +Let q be strictly below the supplied complex saving and below the bit saving. The proposed dominant costs, normalized by V, are + +| Component | Power of p | Saving | +|---|---:|---:| +| Main transform with deferred reservations | 1-epsilon*q | epsilon*q | +| Fused Gaussian interior / convolution groups | epsilon-q*beta | 1-epsilon+q*beta | +| Group bit routing | epsilon-a_bit*beta | 1-epsilon+a_bit*beta | +| Sparse group scans | epsilon-beta | 1-epsilon+beta | +| Boundary algebra | (omega-1)*(1-r)/2 | 1-(omega-1)*(1-r)/2 | +| Sequential operations on sparse union | beta+(1-r)/2 | (1+r)/2-beta | + +Every entry has an explicit cost source above. The similarity/normalization condition is `epsilon+r+beta<1`. The old full-axis scan Vd is not part of the claimed implementation; until its replacement is proved, this remains conditional. + +Ignoring arbitrarily small zeta and strict absorption margins, balancing the first two rows gives + +``` +epsilon=(1+q-q*r)/(1+2q), +beta=(q-(1+q)*r)/(1+2q), +kappa=q*(1+q-q*r)/(1+2q). +``` + +This is feasible for `0 0. +``` + +Thus positive r is compatible with a strictly larger exponent, while retaining the old phase-cell analytic hypotheses. + +One explicit arithmetic target uses + +``` +q=0.001675462576808103522925, +r=0.000001, +zeta=0.00000001, +epsilon=(1+q-q*(r+zeta))/(1+2q), +beta=1-epsilon-r-zeta, +kappa=epsilon*q-0.00000000001. +``` + +It yields + +``` +epsilon = 0.9983301313358845139763404147924773... +beta = 0.0016688586641154860236595852075227... +kappa = 0.0016726647643531934852000135266363... +``` + +The proposed kappa exceeds the selected 0.001672660197076918 by about 4.5672763e-9. Its boundary saving is greater than 0.09595, its sparse-scan saving is greater than 0.0033387, its sparse-union saving exceeds 0.49833, and the bit saving is strictly larger than q. This target is deliberately **not** promoted: the new circuit/supplier integration and grouped sparse tape route must first be proved. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md new file mode 100644 index 0000000000..687c668a9d --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md @@ -0,0 +1,187 @@ +# Positive interpolation trees and two global inverse tails + +Status: exact algebra and an error bound for a compressed inverse representation. This note does not yet claim that the local tree operators admit the pure convolution factorization required for tensor fusion. + +## 1. Setup and signs + +Let `x_i=r^{q_i}`, `r=exp(2 a rho)>1`, and `q_i=floor(rho i+1/2)` for `0<=iu) x_j, +W_I(k)=d_(u-k)/M_I, 0<=k<|I|. +``` + +The equality `u-k=s-1-(k+c_I)` is the connection to the final reversed degree. + +Suppose `||y||_infinity<=1` and `||N0^-1||_infinity<=B`. Replace all leaves by their absolute values and call the resulting polynomial `R_I^+`. Completing the interval by the single outside monomial that chooses every node above I and no node below I gives + +``` +M_I * [z^k] R_I^+ <= [z^(k+c_I)] R_all^+ <= B d_(u-k). +``` + +Consequently + +``` +|[z^k] R_I| <= [z^k] R_I^+ <= B W_I(k). +``` + +This holds for every node of any contiguous binary interpolation tree, for arbitrary signed inputs. It does not require roots to remain correlated after rounding. In particular, it avoids the independently-rounded-erasure-quotient failure in ERASURE-ROUNDING-CONTROL.json. + +The merge identity is + +``` +R_(A union B)=F_B R_A + F_A R_B, +``` + +where A is the lower child and B the upper child. All fixed polynomial coefficients in this identity are positive. Each intermediate contribution, after multiplication by the chosen outside monomial, is a sub-sum of the final absolute Lagrange sum. Therefore bounded intermediate amplitude follows from the completed inverse bound, including in the presence of cancellation between input leaves. + +This is a height bound, not by itself a fast multiplication theorem. The gauges are degree dependent. Ordinary convolution of their unscaled coefficients can still require excessive precision. + +## 3. Partition bound for arbitrary selected nodes + +For any increasing nodes satisfying `x_(i+1)/x_i>=r`, let `e_k` be the kth elementary symmetric polynomial, and `M_k` the product of the k largest nodes. Comparing a selected k-subset to the k largest indices gives a partition of nonnegative index deficits. Hence + +``` +1 <= e_k/M_k <= C_r, +C_r=product_(h>=1)(1-r^-h)^-1. +``` + +For `r>=4`, `C_r<2`. This applies to every interval and to intervals with a node deleted. It controls the normalized fixed polynomial factors used by the tree. + +## 4. Two globally compatible far fields + +Write `E_k=e_k(x_0,...,x_(s-1))`, with E_0=1. Let `k=s-1-j`. + +If `i<=j-m`, then + +``` +0 <= 1-e_k(X without x_i)/E_k + = x_i e_(k-1)(X without x_i)/E_k + <= C_r x_i/x_(j+1) + <= C_r r^(-m-1). +``` + +The endpoint k=0 is exact and is read separately. If `i>=j+m`, then + +``` +e_k(X without x_i) + = E_(k+1)/x_i - e_(k+1)(X without x_i)/x_i, +``` + +and the second term relative to `E_(k+1)/x_i` is at most + +``` +C_r x_(j-1)/x_i <= C_r r^(-m-1). +``` + +The endpoint j=0 is again exact. Thus, after allowing a harmless factor `1/(1-C_r r^(-m-1))` to convert the relative approximation to a bound by the true positive entry, the two far fields are + +``` +(N0^-1 y)_j ~= (-1)^k [ + E_k/d_j * sum_(i<=j-m) w_i + +E_(k+1)/d_j * sum_(i>=j+m) w_i/x_i + + exact local strip |i-j|= (p log 2 + log(4 C_r B))/(log r) +``` + +makes the far-field approximation error at most `2^(-p-1)`. In the retained regime, `log B=O(log d)` and `log r=Theta(a)`, so `m=O(p/a+1)`. + +## 5. What must still be paid + +The construction reduces the data application problem to two globally compatible weighted running sums and a width-O(p/a) local strip. All coefficients, barycentric weights, and normalized tree factors may be precomputed: the source axis length is exp(O(p/d)), and any fixed polynomial power of it remains negligible compared with the full tensor volume. + +For a single axis, bounded positive interpolation trees are a plausible route to the local strip in `O(s p polylog p)` bit operations. To turn this into an exponent improvement, however, one must give a fixed sequence of convolution, diagonal, and permitted data-movement operations that can be fused over many axes. A merge written as a *sum* of two convolutions is not sufficient: tensoring it naively generates exponentially many branches. Neither the positive height bound nor the two-tail representation silently resolves that issue. + +## 6. A branch-free reduction of a normalized Sylvester merge + +This is an exact algebraic reduction, with a bounded-symbol condition. It reduces the remaining in-place question to one full Toeplitz map rather than a sum of two polynomial products. + +Take two children of size b. Choose an anchor zeta strictly between their node sets and rescale the variable. After swapping/scaling the child registers the merge has the form + +``` +S(B,A)=f(z) B(z)+z^b g(z^-1) A(z), +f(z)=product_(lower nodes x)(1+(x/zeta) z), +g(z^-1)=product_(upper nodes x)(1+(zeta/x) z^-1). +``` + +If zeta is the geometric mean of the largest lower node and smallest upper node, every displayed root parameter is at most `r^-1/2`. The sum of all parameters on either side is at most + +``` +eta = r^-1/2/(1-r^-1). +``` + +Thus the l1 norms of f and g are at most exp(eta), and the l1 norms of their reciprocal series are at most exp(eta/(1-r^-1/2)). All are constant and approach one as a increases. + +Let T_f be the 2b-by-2b lower triangular Toeplitz multiplication matrix for f. Put + +``` +h(z)=g(z^-1)/f(z), +K_ij=h_(i-b-j), +H_ij=h_(i-j), 0<=i,ja. These are the actual strict rate relations used by the +support-local compiler and its stopped recursion, not an assumption that +all bit operations have the complex supplier's exact endpoint. + +## Functional interface + +The reviewed outer proof consumes a globally complete dyadic tensor call +which applies normalized H0 simultaneously on the selected active bits, +acts as identity on arbitrary dirty helper/coarse/spectator coordinates at +completed-call boundaries, accepts O(p)-bit coefficient encodings, and costs +O(V g^(1-q) polylog p). The supply is used as a completed operator, with its +own scalar phases and working storage included. Outer Gaussian blocks do +not directly inspect the local donor edits, per-rank multiplicities, owner +replica count, or a particular parent-word ordering. + +The T74 source retains the same local normalized target, common-label and +mutable-source contract, whole-call hosting, exact wrappers/phases, and +restoration interface. Its new parent uses m=73 as before, with maximum +residual rank48 and fixed finite W=66304. Replacing the old finite stock W +by this fixed W changes constants and the O(log g) product-row reservation, +not any exponent in the support-local argument. The local word has480 +helpers, also unchanged in its interface. All donor reads are intended to +preserve the exact full local endpoint; our separate rational rebuild check +confirmed that algebra. The complete unmodified native/frame/phase replay has +now passed and certifies the recorded normalized implementation of those endpoints. +The source pin and hashes of the native, strict, control, and physical-invoice +receipts are bound in `NATIVE-ADMISSION-BINDING.json`. + +The larger fixed prefix allowance (2048 bits in the T74 global invoice) +also changes a constant in the linear semantic guard. The capped proof allows +a fixed supplier-dependent constant in O(g), and later O(g*a*B), with an O(p) +record constant chosen accordingly. It does not require the former numeric +prefix constant to remain512 or1024. The strict power guards B+X+r<1 and +E+r<1 remain in force and were checked independently. + +The neighborhood exposure still involves O(g log p) actual selected bits; +the T74 finite word neither requests ambient-p address permutations nor +changes the complete-field requirement. The retained bit supplier supplies +the same arbitrary-width equal-chunk interchange interface with more rate +than the chosen a. Source and sink scalings converting H0 to the raw H used +by the tensor compiler remain O(g) guard terms. There is no new finite-field +bridge or a new assumption that per-microbox coarse labels are complete. + +Thus the substitution does not require a new analytic lemma, removal of a +charged row, or a new kind of workspace. The updated finite supplier is now +verified by `work/target002_upstream/full-1661-verification/RESULT.json`, whose +status is `PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION`. The binding includes +the complete source manifest and actual fresh native, strict-assembly, physical +invoice, and control receipt hashes. The two rates above are completed conditional +substitutions: their remaining conditions are exactly the inherited all-size, +hosting, precision/tape, exact-recovery, and analytic hypotheses described above; +the matrix branch additionally requires omega_C<=9/4. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex new file mode 100644 index 0000000000..78278aea07 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex @@ -0,0 +1,53 @@ +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T0$ is the oversampling gap. This removes the width-growth +requirement from the conditioning argument. The proof uses convergent +infinite triangular factors and then periodizes them; it does not assert +operator-norm convergence of finite sections. A fast implementation and +its tensor cost remain separate obligations. No improved final +integer-multiplication exponent is claimed in this note. +\end{abstract} + +Let $1\le s5/6$. +At $a=1$, concavity gives +$1-e^{-\rho\theta}\ge1-e^{-\theta}\ge(1-e^{-1})\theta>3\theta/5$, +and $e^{1/2}<2$; substitution gives the stated constant $16$. +\end{proof} + +\begin{corollary}[The width may be fixed] +For the resampling convention $a=\pi\alpha^2$, one may take the fixed +integer $\alpha=2$ in the inverse-conditioning and normalization +argument. If all $d$ gaps satisfy $\theta_i\ge d^{-C}$ for a fixed $C$, +the total logarithmic normalization loss is $O(d\log d)$. +\end{corollary} +\begin{proof} +The theorem bounds each inverse by a fixed constant times $d^{2C}$. +The separate diagonal has norm at most $e^{a/4}$, a fixed constant. +Taking powers of two large enough to normalize these factors costs +$O(\log d)$ bits per axis; logarithms add under tensor products. +\end{proof} + +\paragraph{Scope and provenance.} +This strengthens the conditioning-only result in the earlier note +\emph{A Polynomial Condition Bound for Gaussian Resampling}. The +matrix is the one in Section~7 of the +\href{https://github.com/CrocSwap/integer-mult-bounds/tree/e6f9716621f4e01fc190617934cdfaf072bd976f}{inspected repository snapshot}. +Classical Newton interpolation and symmetric-polynomial identities are +used explicitly above; no generic-rank assumption is used. Two independent +agents checked the limiting and periodization arguments. Exact rational +controls cover negative and zero origins, finite $LDU$, inverse entries, +origin shifts and simultaneous periodicity. + +This theorem does not supply the new numerical implementation at these +parameters. Stable factor application, in-place tensor composition, +coefficient generation and all tape movement must still be priced. +The repository's inspected conditional exponent remains +$\kappa=0.001672660197076918$; no larger final exponent is certified here. +\end{document} diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex new file mode 100644 index 0000000000..4351a73450 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1673/capped-phase-fusion/phase_fusion_note.tex @@ -0,0 +1,460 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,booktabs,longtable,array} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[colorlinks=true,linkcolor=blue,urlcolor=blue]{hyperref} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newcommand{\norm}[1]{\left\lVert #1\right\rVert} +\newcommand{\diag}{\operatorname{diag}} +\newcommand{\poly}{\operatorname{poly}} +\newcommand{\plog}{\operatorname{polylog}} +\newcommand{\file}[1]{\href{#1}{\texttt{\detokenize{#1}}}} +\title{Capped phase-cell fusion in the conditional multiplication reduction} +\author{Research note prepared with OpenAI Codex assistance} +\date{13 October 2026} +\begin{document} +\maketitle + +\noindent\textbf{Review status: PASS\_CONDITIONAL\_PHASE\_FUSION\_REVIEW.} The capped local-gauge and layout changes have independent audits, followed by an integrated review of this note and its 44 exact strict slacks. The result below is conditional on the same finite suppliers and all-size computational interfaces as the retained selected result; it is not an unconditional integer-multiplication theorem and does not change the upstream selected certificate. + +\noindent\textbf{Updated final bounds.} The completely replayed T74 substitution in Section~\ref{sec:t74} gives $\kappa_T=0.001677160151714381$, or $\kappa_{T,M}=0.001677600129305161$ with the additional supplied matrix-exponent premise. Neither reaches the continuing target $0.002$. + +\section{Statement, inputs, and parameters} + +The retained selected saving is +\[ + \kappa_{\rm old}=0.001672660197076918. +\] +The retained independently replayed supplier lower bounds are +\[ + a_*=0.001733427039589844234749,\qquad + b_*=0.001675462676808103522925. +\] +Here $a_*$ is the bit saving and $b_*$ is the complex saving. They are supplied inputs, not recomputed finite constructions in this note. Put +\[ + q=b_*-10^{-10},\qquad a_{\rm eff}=(b_*+q)/2, + \qquad r_0=10^{-6},\quad \zeta=10^{-8},\quad s_0=10^{-5}, + \qquad c_0=3/32. +\] +The effective bit saving is deliberately weakened so that +$q2|t|+1$. Both nonzero parameters have controlled inverses, so zero shears cause no singular diagonal. Two- and three-$H$ identity words pad all permitted gate types to the same six-$H$ shape. Thus one original matching layer uses at most $(8m+2)$ simultaneous $H$ stages. + +For $g$ different axis circuits, Kronecker multiplication tensors the corresponding $H$ stages and multiplies the diagonal factors. There is no expansion into $2^g$ branches and no data padding. Coarse labels may select different switch and coefficient tables, while the stage schedule stays common. A tensor diagonal has coefficient $\prod_i d_i(x_i)$. In lexicographic order, cached prefix products are recomputed only on odometer carries; the total number of extra products is bounded by $L^g\sum_{k\geq1}k/L^k=O(L^g)$. Scanning the one-axis tables on a fixed number of tapes has the same amortized bound. Consequently each diagonal costs $O(V\plog p)$ rather than $O(Vg)$. + +Crucially, an $H$ stage is invoked \emph{globally across all complete coarse-label ranges}. A fixed coarse-label microbox does not contain complete high-coordinate bits to borrow. Each chosen local bit is placed in a complete $K$-bit neighborhood of its original axis, with +\[ + G=4\lceil\log_2p\rceil+6,\qquad K=128G. +\] +An individual neighborhood fits because $K+O(\log p)=o(\log s_i)$. The $gK$ selected positions can be exposed by a permutation supported on at most $2gK$ positions; the complement need not be stably compacted. Coarse labels are restored before the next diagonal is evaluated. + +The retained compact supplier's residual basis acts on its selected bits, not on the entire ambient address. Its control shuttles have width $O(hG)$ at a node with $h$ active bits. Reserve whole neighborhoods for the two front fields, one back field, and the full product row stock required by nested bit and complex calls. The completed active call is the desired kernel on active bits and identity on arbitrary dirty reserved coordinates. Thus these reserved selected bits can be deferred, rather than processed individually first. Their count is at most $h/16$ above a polylogarithmic tail. Padding adds whole rows only for the current phase and is deleted before the next one. + +The original sufficient wording $K/\log p\to\infty$ is replaced by the finite inequality actually used in its proof, $K\geq G+4\lceil\log_2p\rceil+10$. The present $K$ satisfies it. The support router implements all needed placements with support $O(h\plog p)$, and its short-record construction borrows only $\lceil\log_2p\rceil^2$ suffix bits. It does not move an entire long axis. + +Using supplier stopping exponent $10^{-9}$ gives a strict leaf saving above $a_{\rm eff}$ and $q$. The retained stopped recurrence and the geometric sum of deferred widths yield +\[ + O\bigl(Vg^{1-q}\plog p\bigr) +\] +for a complete global selected-bit call, including all exposure/restoration. The retained supplier implements $H_0=H/2$; multiplying by $2^g$ costs one shift scan and $O(g)$ guard bits. The full circuit therefore costs the same bound up to polylogarithms. The derivations and restrictions are in \file{TENSOR-COMPILER.md}, \file{SUPPLIER-SUPPORT-AUDIT.md}, and \file{SUPPORT-ROUTER.md}. + +\section{The two actual Gaussian operators} + +\subsection{Narrow compression and target-grid completion} +For one axis put $q_j=\lfloor\rho j+1/2\rfloor$ and $\beta_j=\rho j-q_j$. The retained narrow normalized Gaussian is $N=CTD$, where $C$ selects the $q_j$ target rows. Its off-diagonal row sum satisfies +\[ + \norm{N-I}_\infty\leq e^{-2a\theta}+5e^{-2a}. +\] +The two nearest directions must be bounded together; charging each by $e^{-2a\theta}$ would not give this result. With the chosen $a\geq4\log(16p)$ and $\theta\geq1/(4p)$, the row gap is at least $1/(8p)$ eventually. The globally fixed, truncated, structure-preserving rounded matrix $H$ can be chosen with $\norm{H-N}<1/(16p)$, diagonal exactly one, gap at least $1/(16p)$, row norm below two, and $\norm{H^{-1}}_\infty\leq16p$. Every local inverse uses a principal restriction of this same $H$. + +Let $E$ inject source coordinate $j$ at target coordinate $q_j$. On the full dyadic target grid define +\[ + H_{\rm aug}=EHC+(I-EC). +\] +It is permutation-similar to $H\oplus I$, so it inherits row dominance and inverse bounds. Its physical target bandwidth is at most $2w+2$, since $q$ increments by one or two. Its inverse on selected rows is $EH^{-1}C$. A diagonal normalization on selected rows, zero on holes, followed by the increasing $q$-image extraction gives exactly the retained compression approximation. + +\subsection{Broad expansion} +The broad forward matrix is distinct: +\[ + S_{kj}=\alpha^{-1}e^{-b_0(j-\lambda k)^2},\qquad + b_0=\pi/\alpha^2,\quad\lambda=1/\rho. +\] +Embed source values at the $q_j$ coordinates, zero elsewhere. Give every hole input column an identity column. The resulting square matrix $M$ agrees with $S$ on the intended zero-hole input. In selected/hole order, +\[ + M=\begin{bmatrix}\alpha^{-1}N_bD_b^{-1}&0\\ R&I\end{bmatrix}, + \quad b=\pi/(\alpha^2\rho^2),\quad + D_b=\diag(e^{b\beta_j^2}). +\] +The Gaussian row sum of $R$ is bounded. In particular $\norm{M}<3$ eventually; use $M/4$ as the completed forward map, making it contractive also on numerical errors in hole coordinates. Relative to the former $S/2$ normalization this contributes an extra $2^{-D}$ over $D$ axes. A single exact $2^D$ shift compensates it, increasing the retained scalar exponent $\Gamma$ by $D=o(p)$. + +On a phase cell $q_j=j+c$, a selected target block has entries +\[ + K_{ki}=\alpha^{-1}e^{-b_0(i-c-\lambda k)^2}. +\] +Its inverse is polynomially bounded even though an uncentered diagonal split can have a large range. The finite Gaussian LDU proof only uses gaps at least one and $|\beta|\leq1/2$. For +\[ + c_b=\prod_{m\geq1}(1-e^{-2b\rho m}) +\] +it gives +\[ + \norm{K^{-1}}_\infty + \leq\frac{\alpha e^{3b/4}c_b^{-4}} + {(1-e^{-b\rho\theta})^2}. +\] +Indeed $q'_k=k+\rho c$ has unit gaps and $\beta'_{q_j}=\beta_j$. Also +\[ + \log c_b^{-1} + =\sum_{h\geq1}\frac1{h(e^{2b\rho h}-1)} + \leq\frac{\pi^2}{12b\rho}, + \qquad c_b^{-4}\leq e^{\pi\alpha^2\rho/3}. +\] +Since $\alpha^2=O(\log p)$ and $\theta^{-1}=\poly(p)$, the bound is polynomial in $p$, uniformly over all contiguous cell subblocks. + +For a block anchor $m$, set $x=i-m$, $y=k-m$, and $\delta=(1-\lambda)m-c$, with $|\delta|\leq1/2$. The exact Toeplitz split is +\begin{align*} + K_{ki}&=D_{\rm row}(y)G(y,x)D_{\rm col}(x),\\ + G(y,x)&=e^{-b_0\lambda(x-y)^2},\\ + D_{\rm col}(x)&=e^{-b_0((1-\lambda)x^2+2\delta x)},\\ + D_{\rm row}(y)&=\alpha^{-1}e^{b_0\lambda(1-\lambda)y^2+ + 2b_0\delta\lambda y-b_0\delta^2}. +\end{align*} +The Gaussian Poisson formula bounds the symbol of $G$, with $\gamma=b_0\lambda$, between +\[ + \sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)} + \quad\hbox{and}\quad1+\sqrt{\pi/\gamma}. +\] +Thus every finite principal $G$ is SPD with polynomial condition number. Truncating at width $W$ and rounding symmetrically to sufficiently many $Cp$ bits preserve these bounds. The SPD shallow circuit applies. See \file{LOG-A-PHASE-SCHUR-AUDIT.md}, \file{BILATERAL-LDU.md}, and \file{FORWARD-SMALL-A-AUDIT.md} for the analytic details and exact local-identity controls. + +\section{Capping and recentering the phase interiors} + +All target holes and original physical period cuts are boundary coordinates. Selected target coordinates are consecutive inside an original phase cell, so no interior circuit is applied to a masked Toeplitz matrix. Use the maximum physical bandwidth $W$ for both directions. + +In each original $O(d)$-length cell, retain $W$ endpoints and repeatedly choose the largest dyadic interior of size at most $B$ that fits with a following $W$ separator. Once no full $B$ block remains, the dyadic-tail procedure halves the remaining length at every step until its size is $O(W)$, which is all boundary. There are $O(d/B+\log B)$ interiors and +\[ + O\bigl(W(d/B+\log B)\bigr) +\] +boundary coordinates per cell. Since $B=o(d)$, the boundary density is $O((W/B)\plog p)$. Each eliminated interior touches only its two neighboring $O(W)$ boundary groups. Schur fill therefore has half-bandwidth $O(W)$, regardless of the number of capped blocks. + +For the narrow exact cell weights +\[ + \omega_j=e^{-a(1/4-\beta_j^2)/\theta}, +\] +one step changes their logarithm by $2a\beta_j+a\theta=O(a)$. On a capped block use the exact rounded-weight ratio $\widehat\omega_j/\widehat\omega_m$ around one anchor. The common scalar cancels from the similarity, and its logarithmic range is $O(aB)$. The retained global rounding changes this by $O(1)$, because its $Cp$ precision dominates the original $O(ad)=o(p)$ range. This is symbolic recentering of the same matrix, not a new local rerounding. For the broad split, $|x|,|y|\leq B$ and $B\leq d$ give diagonal ranges +\[ + O\bigl((B+B^2/d)/\alpha^2\bigr)=O(B/\alpha^2)\subset O(aB). +\] +The number of blocks does not multiply circuit depth: they run as a direct sum on the $L$ existing coordinates. + +\section{Five factors, sparse windows, and batched boundaries} + +For either square operator write $Q=\left[\begin{smallmatrix}A&B_1\\C_1&D_1\end{smallmatrix}\right]$ in interior/boundary order, and $F=D_1-C_1A^{-1}B_1$. Exact multiplication gives +\begin{align} + Q^{-1}&=\diag(A^{-1},I)E_{-B_1}\diag(A,F^{-1}) + E_{-C_1}\diag(A^{-1},I),\\ + Q&=\diag(A,I)E_{C_1}\diag(A^{-1},F)E_{B_1}\diag(A,I). +\end{align} +The off-diagonal factors use the original banded $B_1,C_1$. The forward formula uses $F$ only as a matrix application; it does not require a forward Schur inverse or a forward LU-prefix assertion. For the narrow inverse, row diagonal dominance survives elimination. Its Schur row gap stays at least $1/\poly(p)$ and its row norm is bounded, so the retained no-pivot boundary solver applies. + +Batch independent right-hand sides in a dyadic width between $W$ and $2W$. A banded triangular solve becomes block updates +\[ + X_j=L_{jj}^{-1}(Y_j-L_{j,j-1}X_{j-1}). +\] +Precomputed diagonal inverses and two square block products implement each update. The seven-product recursion gives exponent $\omega=\log_2 7<45/16$, the latter inequality checked exactly by $7^{16}<2^{45}$. The same batching handles boundary/interior coupling products. Intermediate Strassen additions grow only polynomially in $W$. Round after a completed block update and bound its residual. Each local window has polynomial length $L$, the no-pivot factors and exact partial right-hand sides have polynomial bounds, and the completed system inverse has polynomial norm. Thus the uniform local residual is $\poly(p)2^{-P}$; there is no factor equal to the global number of windows. The total boundary cost is +\[ + O\bigl(V(d/B)W^{\omega-1}\plog p\bigr). +\] + +For a group let $J_i$ be its boundary set, and let $E_i$ also include every interior coordinate incident to $B_1$ or $C_1$. Then $|E_i|/L=O((W/B)\plog p)$ and each sparse factor fixes $E_i^c$. The union +\[ + U=\{x\in[L]^g:\exists i,\ x_i\in E_i\} +\] +has fraction $O(gW/B\plog p)$. It is closed under the sparse factors, including a possibly dense inverse on $J_i$. It is \emph{not} closed under a dense $A$ stage. Use two sparse windows, with scatter/regather around the intervening dense stage. + +For one sparse axis, $U_i=E_i\times[L]^{g-1}$ is a complete rectangle wholly present in $U$. Scan $U$, extract $U_i$, number $E_i$ in increasing order by a stream counter, transpose only this polynomial-size field, perform its boundary operation, and merge back. Only this one field is temporarily padded, and the padding is deleted before the next axis. Each sparse window costs $O(Vg^2W/B\plog p)$ per group. Across $d/g$ groups this is +\[ + O\bigl(V(dg/B)W\plog p\bigr). +\] +Every group also has a fixed number of full scans costing $O(V)$. Membership calculations use $O(g(\log s)^2\plog p)=o(p)$ bit operations per $p$-bit record. Detailed fixed-tape extraction and batching are in \file{PHASE-LAYOUT-AUDIT.md}. + +\section{Core faces, ordered embedding, and precision} + +The increasing map $j\mapsto q_j$ permits one product scan for a whole group. Enumerate the dyadic target rectangle. For target $k$, put +\[ + j=\max\{0,\lceil s(2k-1)/(2t)\rceil\}. +\] +It lies in the image exactly when $j0.000428$. The old core-face cost has exponent $\epsilon_M+\beta_M-3<-1.75$. + +The main and grouped Gaussian savings are equal, while the sparse-union saving has an exact $s_U$ surplus: +\begin{align*} + \epsilon_Mq&=1-\epsilon_M+q\beta_M,\\ + 1-\epsilon_M-\beta_M-1/2+\xi_M&=\epsilon_Mq+s_U. +\end{align*} +Their minimum equals +\[ + 0.001673360115405778453817047951720376\ldots>\kappa_M+10^{-11}. +\] +The new boundary saving is $5/16+\xi_M-\epsilon_M>0.0639$, so it is not binding. Global routing, group scans, packed products and core-face repairs have strictly larger savings. + +The optional full-key surcharge in the earlier conservative table is absent here: the actual sparse-union algorithm extracts complete rectangles $U_i$, transposes one field, and performs an ordered merge. It does not sort union records. In contrast, all sparse \emph{microhalo} radix keys remain paid in their negligible correction volume. Adding a full-key sort on the union is an explicit rejecting control; omitting it follows from the layout proof, not from the matrix exponent. The new parameter range and q-image ordering are independently checked in \file{MATRIX-LAYOUT-SIGNOFF.md}. + +The exact ledger \file{check_matrix_corollary.py} verifies 47 positive strict slacks and six rejecting controls, recorded in \file{MATRIX-COROLLARY-ARITHMETIC.json}. The gain over the upstream selected saving is $6.9990832886\times10^{-7}$. The gain over the separate Strassen result is $4.37755816907\times10^{-7}$. The strict minimum gap absorbs all remaining fixed logarithms as before. +\end{proof} + +\noindent\textbf{Corollary review status: PASS\_CONDITIONAL\_MATRIX\_PHASE\_FUSION\_REVIEW.} The additional matrix transfer, new-parameter layout and exact arithmetic passed their independent audits and integrated root review. Corollary~\ref{cor:matrix} retains its explicit additional matrix-exponent premise. The first theorem and its separate reviewed arithmetic receipt remain unchanged. + +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T=229 that + + |pi(x)-li(x)| <= 0.2795 x (log x)^(-3/4) + exp(-sqrt(log x/6.455)). + +[Primary source, theorem 2](https://arxiv.org/html/1401.2689v2#S2). +This theorem is a cited mathematical input, not re-proved by the verifier. + +Apply it at the two endpoints of the requested interval about x=t_i. +The logarithmic-integral difference is at least `eta_p x/log x`. The sum +of the two errors divided by this main term is bounded, for a fixed C, by + + C p^2 (log x)^(1/4) exp(-c sqrt(log x)) -> 0. + +Indeed `log x=Theta(p^(1-epsilon))` and epsilon<1 is fixed, so its positive +power dominates log p. Consequently the interval contains eventually at +least `x/(8p^2 log x)` primes. This exceeds d, since x is exponential in +`p^(1-epsilon)` and the denominator and d are polynomial in p. Both target +sizes satisfy the estimate. Their intervals are disjoint, so all d source +primes can be made distinct, odd and coprime to their dyadic targets. + +The deterministic search used by the assembly still works: scan the +intervals and test candidates by trial division. Even scanning all O(r) +candidates with O(sqrt r) divisors and polynomial bit overhead costs +`O(d r^(3/2) poly(p))`. Its logarithm is `O(log p+p/d)=o(p)`, hence setup +is `n^o(1)=o(n)`. No unknown threshold is used as a claim of practicality. +The algorithm can fall back on the retained multiplier for smaller inputs. + +Let `rho_i=t_i/s_i=1+theta_i`. The interval gives + + 1/(4p^2) < theta_i <= 1/(2p^2-1) <= 1/p^2. + +Also `S/T > (1-1/(2p^2))^d >= 1-d/(2p^2) > 1/2` eventually, since +d<=p. Thus the same capacity bound `T/2=1/(4d)`. + +## 2. Polynomial conditioning survives the smaller increment + +Keep `alpha=ceil(sqrt((4/pi) log(16p)))` and `a=pi alpha^2`. Eventually +`4 log(16p)<=a<=5 log(16p)` and alpha^2=Theta(log p). The sharp row estimate +of PR1673, LOG-A-PHASE-SCHUR-AUDIT section 1, is valid for every theta>0: + + ||N-I||_infinity <= exp(-2 a theta)+5 exp(-2a). + +For theta>=1/(4p^2), its row gap is at least `1/(8p^2)` eventually; for +example `1-exp(-2a theta)>=1/(4p^2)` while the second term is O(p^-8). +Omitting the exponentially small nonzero diagonal aliases and setting the +diagonal exactly to one preserves a polynomial gap. Exponentially accurate +band truncation and coefficient rounding preserve half of it. Every principal +narrow block has the same row gap, and its Schur complements retain that gap. + +The broad bound in FORWARD-SMALL-A-AUDIT sections 1-3 already states its +dependence on theta explicitly: + + ||K^-1|| <= O(alpha^5 rho^2 theta^-2 exp(a rho/3)). + +This is polynomial for the new theta. Since rho=1+O(p^-2), the exponential +factor is at most p^2 eventually, theta^-2<=16p^4, and alpha^5 is +polylogarithmic. The finite and bilateral LDU entry bounds are geometric +in index distance and bound both row and column sums. The same estimates +apply to the finite contiguous sections used by elimination. The augmented +forward matrix adds identity hole columns and a Gaussian coupling of bounded +row and column sums. Thus a conservative common eventual bound p^10 covers +the relevant matrices and their inverse norms, in both induced row/column +norms and spectral norm. Polynomially small perturbations preserve it with +slightly larger constants. This bound is independent of the number of cells. + +Physical-prefix elimination still consists of one contiguous prefix and +disjoint future interiors separated by at least the bandwidth. The eliminated +matrix is a direct sum of such sections. A Schur inverse is a principal block +of the corresponding full inverse; its norm is therefore polynomial too. +Schur norms follow from `D-C A^-1 B`. These are the same prefix geometries +used in the retained dense no-pivot boundary argument, with new polynomial +constants. No general stability assertion for structured generators is used. + +## 3. Setup precision must increase; online precision need not + +The old absolute weight precision cannot be retained. For the narrow weights + + w_i=exp(-a(1/4-beta_i^2)/theta), + +the negative logarithm can now be as large as `a/(4 theta)=O(p^2 log p)`. +Rounding these weights absolutely to Cp bits can produce zero. The finite +checker includes a concrete underflow witness; this is a required change. + +Let `L_p=ceil(log2 p)` and let C be a sufficiently large fixed constant. +Use setup precision + + P_setup = C p + 64 p^2 L_p. + +Eventually `a<=16 L_p`, so `a/(4 theta)<=16 p^2 L_p`. Since log 2>1/2, +rounding a weight downward with absolute error at most `2^-P_setup` gives + + |w_hat_i-w_i|/w_i + <= 2^-P_setup exp(a/(4 theta)) + < 2^(-C p). + +The weights remain positive. Increasing C by a fixed amount supplies any +additional polynomial slack needed below. Computing these polynomially many +bits, exponentials, exact anchored ratios and local inverse/factor pages takes +polynomial time in p. Schoolbook arithmetic and ordinary convergent series +already suffice; no use of the improved multiplier at a larger precision is +required for this setup bound. + +Define one globally fixed rounded reference matrix H as in the retained +audit: within each original phase cell use positive rounded weights and a +single rounded, truncated Toeplitz kernel, and across cells use rounded +Gaussian entries, with diagonal exactly one. The weights use P_setup, while +the kernel and the actual online pages need only a sufficiently large +constant times p bits. For distances at most W, weight ratios have logarithmic +range O(aW)=o(p). The truncation tail is bounded directly in the Gaussian +entries, independently of the full-cell weight range. Hence H differs from +the exact Gaussian matrix by `poly(p) 2^(-(C-o(1))p)` in row and column norm. +This preserves the polynomial conditioning in section 2. All actual Schur +and inverse tables are computed for this H, with the same perturbation budget. + +The data circuit never multiplies by the full-cell weights. Use PR1758's +exact fractional-shift identities, whose algebra holds for every theta>0. +On a capped block of length at most B, the narrow remaining chirp has +logarithmic range O(a theta B^2); the broad chirp and its leftover linear +term have range O(b0 theta B^2+b0 theta B), where b0=pi/alpha^2. Each is at +most O(a B^2/p^2). The shifted Toeplitz kernels remain accretive: their shift +gap is at least theta/2, and the explicit symbol bound of PR1758 gives +polynomial inverse norms when theta^-1=O(p^2). Its accretive shallow-circuit +theorem therefore has polylog(p) guard bits per axis, as before. + +The ideal shifted representation and H need not have identical stored +coefficients. Their difference has the bound just established: replace the +old raw `Cp` weight precision by P_setup in the comparison in PR1758 section +2. Local shifted kernels and anchored chirps are generated directly, then +rounded at online precision. Across g axes the uncompleted amplification is + + exp(O(g a B^2/p^2 + g polylog p)). + +Thus O(p)-bit records suffice whenever + + beta + 2 xi - 2 < 1, beta < 1. + +The inverse comparison also stays within the error budget. If an ideal +interior A and the rounded reference H_II differ by Delta, then +`H_II^-1-A^-1=-H_II^-1 Delta A^-1`. Both inverse norms are polynomial; +the error changes only by a polynomial factor. Telescoping through the +constant number of completed factor stages, and through g tensor axes, +uses the stated chirp/prefix guard. Exact identity of their rounded +coefficient lists is neither assumed nor needed. + +Setup arithmetic at P_setup is separate from these records. It is not placed +inside an online motif call or multiplied into the per-record data cost. +PHASE-LAYOUT-AUDIT already permits polynomial coefficient/page generation +per complete microbox and axis, amortized by the superpolynomial inner and +spectator payload. Replacing its polynomial by another fixed polynomial +preserves that argument: `L^g` and every full unused source-axis range exceed +every fixed polynomial in p. The number of pages, capped pieces, axes and +required bits is polynomial. This is an asymptotic amortization, not a claim +that P_setup or the eventual cutoff is practical. + +## 4. Normalize completed maps with powers of two + +The exact Gaussian resampling identity and the coprime source/target +permutations do not require theta to be of order 1/d. They require the +positive separation and invertibility established above. The completed +forward, inverse and diagonal factors can have larger polynomial norms +than in the old parameter family. Divide each such completed map by an +additional dyadic factor of at most `2^ceil(100 log2 p)`. The conservative +p^10 bounds above are then below one. Restore the product of these exact +normalizers at the final scalar shift, together with the retained Gaussian +normalization. This preserves the exact transform identity. + +There are O(d) factors; the added shift has O(d log p)=o(p) bits because +epsilon<1. This also bounds the extra accuracy needed before exact recovery. +At a sufficiently large fixed C, data precision is `C p+o(p)=O(p)`. +Completed groups can be treated as contractions, while their uncompleted +prefixes/suffixes use the chirp and circuit guards of section 3. None of +these changes increases the semantic online dimension exponent C1=1. +Offline arithmetic is paid through the setup amortization, not that interface. +The normalizer is scalar: combine its powers into the group coefficients or +apply one scalar per completed group, rather than introducing a full scan per +axis. A constant number of these operations per group is absorbed by the +retained group-scan row. Global restoration is one already-paid exact shift. + +## 5. A larger but still polynomial locality window + +The old literal row-gap and locality constants must not be reused. A +conservative generic locality argument suffices. For a banded reference +matrix with `||H||_2,||H^-1||_2<=p^10`, put + + Q=I-p^-20 H* H. + +Its eigenvalues lie in `[0,1-p^-40]`. With `J=ceil(p^42)`, the polynomial +inverse `p^-20 sum_(j0`. +The locality polynomial is used in the proof and coefficient setup. The +online algorithm still uses the capped factorization and compiled circuits; +it does not perform J Neumann iterations on each data record. + +An original phase cell now has length Theta(p^2). Capped dyadic carving still +has boundary density `O((W/B) polylog p)`: the additional end-cell contribution +is `O(W log B/p^2)`, dominated when B<=p^2. Clipped physical period cuts and +holes remain boundary. Scalar Schur bandwidth stays O(W), since an eliminated +interior joins only its two neighboring O(W)-size separators. Larger cores +and halos change only fixed powers in offline setup and O(log p) address +fields. Their source axes are still superpolynomial in p. + +## 6. Retained Strassen boundary, circuit and union costs + +Use the explicit Strassen circuit, with `7^16<2^45`, so choose fixed matrix +target exponent `tau=45/16` and `c=(3-tau)/2=3/32`. PR1673's dense boundary +proof and fixed-format absolute-precision argument apply to the new polynomially +conditioned pages. Block scans, permutations, rewinds and linear combinations +remain charged. Boundary density is still W/B, so its cost power remains +`epsilon-xi+(tau-1)/2` and its saving is `c+xi-epsilon`. This applies to general +rounded blocks and needs no structured inverse algorithm. + +The interiors use PR1758's shifted Toeplitz representation and accretive +shallow-circuit theorem, with the smaller theta substituted in the explicit +symbol bound and with section 3's setup precision. Circuit algebra and depth +are unchanged; the condition guard remains polylog(p) per axis because all +inverse norms and theta^-1 are polynomial. + +Retain PR1758's radix-distributed union schedule. Complete lines stay in one +flag class and axis operators commute on distinct tensor factors. Its incidence +identity is `sum_r |S_r|=sum_x |F(x)|=sum_i |U_i|<=g mu V`. Each visit already +moves an O(p)-bit record; flag extraction and O(g log L)-bit headers are absorbed +when beta<1. We use PR1758's per-visit argument, not the obsolete 2 beta<1. +Union cost power remains `epsilon+1/2-xi`. No per-axis scan of the whole union +is added, and no vanishing-union fraction is assumed. Support routing, +whole-neighborhood reservation and interchange retain their stated interfaces. + +## 7. Explicit parameters and every paid row + +Use PR1699's certified endpoint +`b_cert=335950991225108904267/200000000000000000000000`. +Set `b=b_cert-10^-64`, `q=b-10^-64`, `a_eff=(q+b)/2`, +`r0=10^-64`, `zeta=10^-66`, layout slack `s=10^-64`, and stopped-recursion +parameter `stop=10^-70`. The bit endpoint is 0.001733427039589844234749. +Choose + + epsilon=9999999/10000000, + beta=9999997/10000000, + xi=epsilon(1+q)-3/32+s. + +Thus xi=0.90792965478815... . This is an explicit interior point; it does not +assert an attained limiting rate. Since beta0`. The global router also +exceeds the main saving. Every other row has larger slack. The minimum is +`epsilon q`, strictly above the rounded headline. + +Key side conditions are + + 1-beta=3 delta>0, + 1-beta-2(1-epsilon)=delta>0, + epsilon-beta=2 delta>0, + 1-epsilon-r0>0, + 1-(2xi-2)-r0-beta=0.184140990423...>0, + 1/20, 48>43. + +The ledger regenerates the remaining stopped-supplier, endpoint, logarithm +and layout conditions. A separate implementation reconstructs the point and +all nine powers. Fourteen rejected variants include the old theta guard, +old union scan, group=dimension, epsilon=1, an unsupported Strassen exponent +and halo/core choices outside the stated locality witness. These failures +are scoped to this proof and point, not universal lower bounds. + +## 8. Scope, sources and open obligations + +Newly supplied here: the effective fine-prime interval and capacity/search +argument; the polynomial setup-precision replacement; the theta-dependent +conditioning substitution; conservative locality/core choices; scalar +normalization accounting; and the exact parameter composition. + +Retained or cited mathematical inputs are Trudgian's prime theorem; PR1673's +Gaussian identities, bilateral/finite LDU, capped factorization, dense boundary +and layout arguments; PR1758's fractional-shift identities, accretive circuit +theorem and per-visit union schedule; and the admitted finite supplier. +Arithmetic and finite simulations do not prove these general statements. + +The multiplication conclusion remains conditional on the all-size compiler, +common labels, mutable sources, whole-call hosting, semantic online precision +C1=1, fixed tapes, prime-packing implementation, arbitrary-width equal-chunk +interchange, exact recovery and analytic transfer. The changed prime interval, +setup precision and locality scales are proved above rather than silently +substituted into old numeric hypotheses. No independent human review, +proof-assistant verification, practical cutoff or physical multiplication +implementation is claimed. + +Credit Vlado Vojdanovski for PR1673; IdeaOuter, as credited by PR1758, for +fractional-shift interiors and per-visit tag accounting; Jeremy Kothe and +contributors for the package series; and LeverRank for the second schedule +simulation. PR1699's notices retain the complete supplier lineage. PR1672 +published the signed-554 kernel first. New material here was prepared by +Chafik Boukhalfa with substantial OpenAI Codex assistance, under Apache-2.0. + +Open next questions: independently review setup versus online precision, +uniform prefix conditioning and the enlarged halo; formalize the fine-prime +transfer lemma; improve the finite supplier endpoint. Every new endpoint +still requires a complete paid native word and admission. The extreme +eventual threshold is intentionally not presented as a practical algorithm. diff --git a/research/kappa-observation-integration/release-work/current/package/vendor/pr1786/NOTICE.md b/research/kappa-observation-integration/release-work/current/package/vendor/pr1786/NOTICE.md new file mode 100644 index 0000000000..e491b2e587 --- /dev/null +++ b/research/kappa-observation-integration/release-work/current/package/vendor/pr1786/NOTICE.md @@ -0,0 +1,3 @@ +# PR1786 (chafreaky / Chafik Boukhalfa), fine source-prime padding + +`FINE-PRIME-PADDING.md` and `outer/check_fine_padding.py` (copied; only the first docstring line of the script is relabelled) are from CrocSwap/integer-mult-bounds PR1786, branch chafreaky/integer-mult-bounds `research/fine-prime-padding`, head `c52345caa318fc6b1b6e2659c57d746317a72823` (Apache-2.0, as stated in the PR). The mechanism (source primes at relative gap Theta(p^-2) from dyadic targets via Tim Trudgian's effective prime number theorem, arXiv:1401.2689v2 theorem 2; setup precision Cp + 64 p^2 ceil(log2 p); theta-dependent conditioning; polynomial halo p^48 and core p^128; dyadic normalizers; explicit Strassen boundary) is entirely theirs. This package changes only the explicit parameter point (epsilon, beta) and the surrounding stack, see `outer/FINE-PADDING-COMPOSITION.md`. diff --git a/research/kappa-rebuild/CLAIM.json b/research/kappa-rebuild/CLAIM.json new file mode 100644 index 0000000000..f490bda2c4 --- /dev/null +++ b/research/kappa-rebuild/CLAIM.json @@ -0,0 +1,86 @@ +{ + "admitted_BIT_saving": "0.0027", + "attribution": "Apache-2.0; substantial OpenAI assistance; immutable predecessor authors and source licenses retained.", + "claim": "T(n) = O(n (log n)^(1-kappa)) under the retained analytic/native hypotheses and the supplied written construction.", + "date_utc": "2026-10-21", + "fresh_replay": { + "all_passed": true, + "artifacts": 59, + "stages": 23 + }, + "kappa": "23/10000", + "kappa_decimal": "0.0023", + "larger_H_branch": "Separate local improvements and priced negative boundary probes; no kappa above 0.0023 is claimed.", + "limiting_outer_saving": "115098849/50000000000", + "nine_outer_savings": { + "boundary_algebraic_matrix": "115148849/50000000000", + "fused_gaussian": "115596547/50000000000", + "global_coordinate_router": "921/400000", + "group_bit_router": "92497237/40000000000", + "group_scans": "49999/50000", + "main_transform": "115098849/50000000000", + "microbox_face_corrections": "1925001/25000", + "polynomial_products": "99999/100000", + "sparse_union_movement": "20427648849/50000000000" + }, + "official_selected_frontier_modified": false, + "ordinary_BIT_saving": "0.0027418653797", + "preserve_all_unmerged_changes_until_user_reports_merge": true, + "public_posted_or_merged": false, + "relative_exponent_saving_increase": "284514790400418148631/865485209599581851369", + "remaining_hypotheses": [ + "The inherited analytic decomposition and exact integer recovery.", + "The retained native ordered-movement and fixed-tape execution framework.", + "The inherited fine-prime padding, gauge-free interior and outer analytic estimates." + ], + "schema": 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"research/kappa-rebuild/transfer-audit/PRECISION-CLOSURE.json": "5777a2758177784f3fc248b83b77dd984993ed6686da882ad7110eed1ea8fbbe", + "research/kappa-rebuild/verification/BASE-MATERIALIZED.json": "ae69c59cbe2162d88293a6db862b21786091a70127e608bd70b852f5f1680687", + "research/kappa-rebuild/verification/CRITICAL-REPLAY.json": "b03f1ae933255c272b7b92a86a2ff2ac86e7ca840289bfb5bc846bdfab28f821", + "research/kappa-rebuild/verification/PRIOR-APPLIED.json": "f622491173bb5516bdf41fce5a9c77ba4815bfb408b8df5aa51292a27c30368e" + }, + "status": "COMPLETE_PROPOSED_CONDITIONAL_CONSTRUCTION_WITH_FRESH_FINITE_REPLAY", + "strict_limiting_slack": "98849/50000000000", + "uniform_templates": [ + { + "family": "complex", + "rank_encoding": "2^276540184361", + "saving": "0.002304" + }, + { + "family": "BIT before cover", + "rank_encoding": "2^6552928412", + "saving": "0.002760" + } + ], + "user_comparison_baseline": "865485209599581851369/500000000000000000000000", + "verification_limits": [ + "Universal finite-precompilation, native closure and outer transfer are written mathematical arguments, not formally machine-checked theorems.", + "Winning fixed uniform words, complete GL covers, B_N, P_N and U_N are algorithm-defined but not materialized.", + "No practical runtime, usable numerical threshold, unconditional theorem, upstream admission or novelty claim is made." + ], + "written_proof_entry": "research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md" +} diff --git a/research/kappa-rebuild/LITERATURE-AND-PROBES.md b/research/kappa-rebuild/LITERATURE-AND-PROBES.md new file mode 100644 index 0000000000..a8aa3d4837 --- /dev/null +++ b/research/kappa-rebuild/LITERATURE-AND-PROBES.md @@ -0,0 +1,141 @@ +# Literature, exact probes, and changes of direction + +Apache-2.0. Prepared with substantial OpenAI assistance. This is the research +record for the October 21 reconstruction, preserving the previous cumulative +work. It distinguishes a paper's theorem, a useful analogy, and an actual +native implementation in this repository. + +## The accounting question that guided the search + +The objective is to increase kappa by reducing the complete recursive bill. +We count the actual transform delivered on each independent arbitrary input, +all paid recursive calls, every preparation and restoration, the permitted +scalar alphabet, storage, precision, row routing, and the outer multiplication +transfer. A partial positive balance is an allowance for unfinished work. +It cannot be reported as a kappa improvement on its own. + +The most consequential change in this round is to retain different sizes +of productive output jobs and optimize their whole controller. The finite +compiler then completes the complementary axes and turns the mixed rule +into a single ordinary uniform supplier. On the BIT side, its entire finite +program is assembled before one cover and one common weight are introduced. +These are changes to the recursive family and the interface, beyond a local +operation-count search. + +## Sources with a direct bearing on the construction + +| Primary source | Why it fits | What we actually did | +| --- | --- | --- | +| [Jacob Alan Sussman, PR4268 and Theorem 1G](https://github.com/CrocSwap/integer-mult-bounds/pull/4268), pinned source commit `563b4e6e913625637f36ba7869bf5fcd0dd4975a` | Provides a mixed-target lockstep flow with an exact target-versus-child moment. Its public scope is a DFT research improvement, explicitly not a new kappa. | Preserved the proof/code/license, independently implemented integer jobs and causal ownership, and added a finite uniform precompiler with charged source complements, native adapters and an explicit huge winning rank. The new transfer is our written argument; it is not attributed to the upstream Lean proof. | +| [J.J.Koliha, *Block Diagonalization* (2001)](https://emis.de/ft/45438) | Relates resolutions of the identity by idempotents to invariant blocks. It suggests examining the operator coefficients of the whole controller. | Tested exact simultaneous coefficient factorizations. The selected p3 controller has integral boundary matrices and six independent endpoint jobs. We check its actual operator and phases; a field-valued diagonalization alone would be insufficient. | +| [Abdi, Cornuejols, Guenin and Tuncel, *Dyadic linear programming and extensions* (2023)](https://arxiv.org/abs/2309.04601) and [*Total dual dyadicness and dyadic generating sets*](https://arxiv.org/abs/2111.05749) | Make the coefficient ring and denominator restrictions explicit. A rational optimum need not supply the desired dyadic implementation. | Enforced legal complex coefficients in realization and billing. Power-of-two balancing and a factored scalar involution avoid an unlicensed division by 13. No implementation of their general LP algorithm is claimed. | +| [Christopher Brislawn, causal complementation / lifting](https://arxiv.org/html/2408.07970v1) and [Daubechies and Sweldens, *Factoring wavelet transforms into lifting steps*](https://9p.io/who/wim/papers/factor/factor.pdf) | Lifting factors are a useful model for reversible triangular updates and for handling boundary constraints. | Replaced a generic scalar elimination by factored shear programs where the actual ring permits them. The timing, physical frame, and restoration conditions are still checked separately. We do not infer those conditions from a lifting theorem. | +| [Connelly and Zeger, *Linear Network Coding over Rings, Part II*](https://arxiv.org/abs/1608.01737) | Helps distinguish linear solvability over the actual coefficient structure from a rank calculation over a convenient field. | Kept causal query acquisition, scatter, transport and erasure in the interface. The same-future-query exchange restores arbitrary borrowed data; it does not by itself close the main endpoints. | +| [Patel, Markov and Hayes, linear reversible circuit synthesis](https://arxiv.org/abs/quant-ph/0302002) | Gives relevant context for synthesis over the binary scalar alphabet. | Used exact F2 gate synthesis and, where available, a smaller explicit three-shear factorization. We did not transfer a binary gate bound to characteristic-zero complex arithmetic. | + +## Older structural precedents that informed the redesign + +[Schonhage, *Partial and Total Matrix Multiplication* (1981)](https://doi.org/10.1137/0210032) +and [Peter Vrana, *Partial and weighted matrix multiplication* (2024)](https://arxiv.org/html/2408.15728v1) +are pertinent precedents for converting useful partial work into a larger +complete computation. Their theorems concern bilinear/tensor problems. We +use them as motivation, not as a black-box proof for this native transform +model. The actual transfer here is the integer source/exit recursion, with +every input's complementary axes paid explicitly. + +The relevant representation-theoretic viewpoint is concrete: a change of +coordinate chart can preserve a completed job while changing its native +appearance. For BIT, routing the whole cover fiber `g -> gS` matches equal-rank +projectors before lowering to weighted leaves. For complex jobs, a monomial +address map must also retain the absolute quadratic phase. We test those +identities rather than relying on a geometric analogy. + +## Iteration record + +1. **Reconstruct the useful lost pieces.** The original base and all 3,214 + cumulative paths were authenticated. Complex H, dirty logging, complete + helper boundaries, and the ordinary identity response were freshly rebuilt. + The PR589 BIT word was regenerated byte-for-byte. New reconstructions have + new provenance; missing historical files are not silently declared recovered. + +2. **Make the cheap local primitives legal.** Power-of-two balancing reduced + complex H's incremental scalar allowance from 4,336 to 3,504 while retaining + eight rank-one calls. The embedded logger uses eight already present helper + outputs and adds no recursive child, but leaves declared dirty outputs. + A repeated identical future-query exchange restores 120 arbitrary fields in + its finite check. Its main endpoint debt remains visible. + +3. **Change what the outer unit earns.** Productive helpers carry their own + independent input data. The first complete p2 identity cover has a mixed + saving in `[0.001772952527592114765550198802, + 0.001772952527592114765550198803]`. This is a complete finite tariff; its + asymptotic transfer was a separate task. + +4. **Optimize the full controller.** A p2 partial-prefix menu reached about + 0.002013296. The p3 controller then retained four causal idle windows and + discarded two others through a checked boundary factorization. Its complete + complex mixed root is about 0.002305090195. The physical main contacts, + helper owners, endpoint phases and scalar bill are independently audited. + +5. **Try to escape the new local optimum.** Paired menus through p6, 64 p3 + single-field masks, and 123,904 commuting stage allocations were evaluated. + The tested larger path controllers and asymmetric variants did not improve + the selected complete p3 tariff. These are finite menu results, not global + impossibility theorems. + +6. **Close the changed recursion's interface.** Exact integer batches, causal + parent/child ownership, source complements and owned-center lifetimes were + implemented and independently challenged. A strict superpotential gives + explicit compressed choices of the winning finite ranks. Complex semantic + precision remains linear with newly derived constants. BIT delays its cover + and normalization until internal jobs have all been compiled. + +7. **Reprice the ordinary BIT conversion and outer computation.** The atom + exponent was changed to 0.003, the generic-cover prime bound replaced, and + new stock formulas supplied. The complete ordinary BIT saving exceeds 0.0027. + The exact outer point at kappa 0.0023 has strict slack in every declared + cost row. The written outer support audit retains its analytic and native + hypotheses rather than treating this arithmetic as an unconditional theorem. + +8. **Continue with a larger structural change.** The actual PR589 center + producer/scatter matrices yield an F2 involution completion, replacing nine + rank-seven calls by nine rank-one calls. Its full local native word passes + both directions on all 661 arbitrary columns. Independently, the complex H + branch found a required signed monomial correction to its owner maps. + A further minimum invertible completion reduces the BIT correction to three + rank-one calls. Direct whole-controller closure nevertheless has three + endpoint obligations; the cheapest BIT completion still has rank-one + deficit 21. The complex direct closure costs 672 rank-one requests. + Subsequent coefficient-span and fresh-axis leakage probes reject two + generic cancellation ideas, while leaving geometry-aware coding open. + +## Negative results retained because they guide the next experiments + +The actual PR589 register-sharing probe screened 96,141 helper pairs against +their full native frame timelines. Only five were even monotone candidates; +two introduce an invalid self-gate and the other three fail the source +response. The nine center producer functions and their consumer map each +have exact binary rank nine, excluding same-cut compression to eight in that +specific representation. The resulting decision was to change the local +response to an involution and resynthesize the caller. + +For the complex H maps, preserving the polar form did not preserve the +Gaussian phase. A translation does not repair this discrepancy. A signed +monomial can, and the actual sign has to appear in both the native contact +lowering and the endpoint proof. This is a useful example of why a favorable +rank count cannot substitute for the emitted physical word. + +## Reproducible evidence + +The authoritative numeric values are the source-bound JSON receipts, not +rounded figures in this narrative. The new scope contains the reconstruction +sources, finite menus, negative witnesses, native records, exact moment +enclosures, independent phase/ownership/cover audits, and the written universal +arguments. `replay_critical.py` reruns the critical construction and audit chain +in a separately materialized checkout; cumulative byte application is checked +independently by the retained overlay tool. + +No new source located in this search is evidence of global novelty or of an +official record. The proposed conditional construction requires external +mathematical review. It also uses enormous fixed templates; its exponent +improvement is an asymptotic claim, not a practical runtime benchmark. diff --git a/research/kappa-rebuild/README.md b/research/kappa-rebuild/README.md new file mode 100644 index 0000000000..dcd1d6e532 --- /dev/null +++ b/research/kappa-rebuild/README.md @@ -0,0 +1,348 @@ +# A complete proposed conditional construction at kappa = 0.0023 + +Research checkpoint, 21 October 2026. Apache-2.0. Prepared with substantial +OpenAI assistance. Predecessor authors and licenses remain credited in their +unchanged source packages. + +## Result and its status + +The proposed conditional bound is + +\[ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{23}{10000}=0.0023. +\] + +This exceeds the user's comparison point +`0.001730970419199163702738` by approximately **32.8734% in the exponent +saving**. It is not a measured runtime improvement. + +The package supplies complete finite native mixed-job words, a constructive +finite compiler to ordinary uniform suppliers, the delayed BIT cover, new +precision and storage constants, and a strict outer accounting point. A +separately implemented audit reviews the changed interfaces and checks their +quantitative inequalities. The fresh replay regenerates the critical chain +from the attached base plus the preserved cumulative work. + +**This is a conditional mathematical research result awaiting external +review.** It is not a formal proof certificate or an unconditional integer +multiplication theorem. The inherited analytic decomposition, native ordered +movement, fixed-tape execution, prime padding and exact-recovery hypotheses +remain. The new universal arguments are written proofs supported by finite +checks; passing Python checks does not formally verify those arguments. + +The winning uniform words are specified by finite algorithms at explicit +astronomical ranks. Their instruction lists, complete covers and numerical +constants have **not** been materialized. This limitation matters for both +reproducibility expectations and any practical interpretation of the bound. + +The authoritative entry point for the proof and its boundary is +[OUTER-TRANSFER-AUDIT.md](transfer-audit/OUTER-TRANSFER-AUDIT.md), with the +separately generated [audit receipt](transfer-audit/OUTER-TRANSFER-AUDIT.json). +[CLAIM.json](CLAIM.json) binds this checkpoint to its exact sources and replay. +No selected-frontier file or public pull request is changed by this package. + +## What changed enough to escape the previous family + +The decisive change is to account for the actual useful transforms delivered +on all independent inputs. A helper that returns with a known completed +transform has performed useful work. If its original input was arbitrary and +its endpoint is autonomous, that job can participate in the recursion. It +does not have to be restored to the identity merely to fit a uniform local +interface. + +That observation alone is insufficient: these jobs have different ranks, +their data exist at different times, and their coordinate frames differ. +The construction therefore has three linked parts. + +1. **A complete mixed-job controller.** The selected seven-call controller + schedules the useful helper jobs in four legal idle windows. Two other + windows are eliminated by an exact boundary factorization. Input/output + normalizers, physical phases, source complements and ownership are explicit. + +2. **An integer compiler to a uniform supplier.** Exact whole-array batches + are routed through a causal dependency graph. Each original source also + receives its paid complementary transform. After recursively compiling + every smaller job, all outputs are full transforms of the same rank and + only rank-one paid leaves remain. No fractional array or copied input is + credited as an extra independent output. + +3. **A compatible BIT interface and full outer bill.** The entire finite BIT + program is compiled before introducing one ambient cover and one common + external weight. The new cover, fallback, routes, row stock and precision + are charged before the suppliers enter the multiplication accounting. + +The finite compiler uses the mixed-target lockstep flow in +[Jacob Alan Sussman's Theorem 1G / PR4268](https://github.com/CrocSwap/integer-mult-bounds/pull/4268). +That upstream contribution explicitly concerns a DFT research exponent and +does not itself claim a new kappa. The additional native, finite-template and +outer transfer arguments are supplied here. The pinned upstream proof, +implementation and license are retained under `architecture/sources/`. + +## Complete supplier accounting + +For a mixed unit, write `W_k` for the number of independent output jobs of +rank `k`, and `p_r` for the number of paid recursive calls of rank `r`. +The exact mixed saving `C` is determined by + +\[ +\sum_r p_r r^{1-C}=\sum_k W_k k^{1-C}. +\] + +The finite compiler and its native interface are essential to using this +equation. A positive histogram balance without those obligations would not +establish the reported result. + +### Complex supplier + +| Productive job rank | Independent outputs | +| ---: | ---: | +| 6 | 192 | +| 7 | 56 | +| 8 | 93 | +| 9 | 1,229 | +| 25 | 448 | +| 57 | 224 | + +The paid profile is +`{1:17623, 2:5065, 3:1738, 4:714, 5:254}`. It has 25,394 paid calls and +2,242 productive fields: 672 main fields plus 1,570 fresh productive helpers. +One owned center is reused chronologically inside a mixed unit and receives +no output credit. Distinct suspended unit instances retain separate owned +centers in the uniform compiler. + +The complete real scalar allowance is **1,147,304**, including boundary +updates, signs, bridges and swaps. The independent physical checks cover +3,136 main contacts and their inverses, 672 endpoints and 2,856 local helper +bindings. Whole-field conjugations occur around completed calls; the word +does not silently retag an internal operation. + +The exact mixed root is enclosed in + +``` +[0.002305090194876170622919094755, + 0.002305090194876170622919094756]. +``` + +The selected uniform saving is `0.002304`, with additional room before the +effective outer supplier saving. See +[P3-NATIVE-COVER.json.gz](controller/P3-NATIVE-COVER.json.gz), +[P3-COVER-AUDIT.json](transfer-audit/P3-COVER-AUDIT.json), and the complete +phase audit in `transfer-audit/P3-NATIVE-AUDIT.json.gz`. + +### BIT supplier + +The productive profile is `{8:3073, 22:444, 50:222}`. The paid profile is +`{1:9261, 2:9394, 3:553, 4:308, 5:714, 6:77, 7:1465}`. +The unit has 3,739 independent productive fields, 21,772 paid calls and +174,908 ADDs, with swaps charged as three ADDs. + +The retained PR589 word is regenerated byte-for-byte. Its actual rational +projectors, nested frame transitions, dirty helper inputs and paid calls on +owned copied-center data are retained. Independent checks include every one +of the 661 local arbitrary input columns in both directions, all 636 native +rational frames and 2,386 nested transitions. + +The mixed root lies in `[0.002769830250, 0.002769830251]`. The finite uniform +program uses saving `0.002760` before covering. Paying the complete generic +cover fallback leaves `0.002750`. The atom conversion with power `0.003` +then gives ordinary saving + +\[ +(1-0.003)(0.002750)+0.003(0.0000384599) +=0.0027418653797. +\] + +We conservatively admit `0.0027`. This leaves BIT above the complex supplier +instead of letting the older BIT bound cap the new construction. See +[DELAYED-COVER-BRIDGE.md](bit/DELAYED-COVER-BRIDGE.md) and +[BIT-BRIDGE-AUDIT.json](transfer-audit/BIT-BRIDGE-AUDIT.json). + +The common-weight restriction is substantive. Routes match entire cover +fibers by `g -> gS`; the external weight is independent of the current fresh +cover coordinate. Only final rank-one leaves are converted to weighted +ordinary children. Their parent cover coordinates become ancestor data of +those children. The independent local-ring tests retain all carries and +the reversed column pairing. + +## Why the infinite-size statement has a finite construction + +[FINITE-PRECOMPILATION.md](architecture/FINITE-PRECOMPILATION.md) gives the +integer compiler. It clears every batch denominator, uses distinct physical +input slots, executes child batches before their parents resume, and charges +the complementary axes of each source. Its exact per-source leaf recurrence +is controlled by a strict potential inequality. + +[EXPLICIT-TEMPLATE-BOUNDS.md](controller/EXPLICIT-TEMPLATE-BOUNDS.md) replaces +an open-ended search for a successful rank with the following exact choices: + +| Family | Fixed uniform rank | Certified saving | +| --- | --- | ---: | +| Complex | `2^276540184361` | 0.002304 | +| BIT before covering | `2^6552928412` | 0.002760 | + +Those are deliberately loose sufficient ranks. The proof defines a finite +program at each rank by exact integer recursion. It fixes that program before +the multiplication input length varies. Its productive width, owned storage, +scalar operations and routing count are all finite computable constants. +They are not zero, small, or experimentally measured. + +The materialized rank-57 complex and rank-50 BIT examples check ownership, +batching and full-job completion. **Neither example achieves the limiting +uniform saving above.** Conflating those small examples with the winning +astronomical templates would overstate the evidence. + +## Precision, storage and all nine outer costs + +The new template cannot inherit the small numerical constants of the old +supplier. [FINITE-TEMPLATE-CLOSURE.md](transfer-audit/FINITE-TEMPLATE-CLOSURE.md) +defines replacements from its actual finite counts. For template rank `N`, +productive width `W`, paid leaves `P`, unit count `U` and owned count `O=U`, +one conservative choice is + +``` +Z = W+O +G = 1494376*U +E = 64*(Z+N+P+G+1)^3 +B = P+E +C0 = 32*N*B^2 +``` + +The completed-child semantic recurrence bounds the precision guard by +`C0*d`: its degree in the active dimension remains one. Children return +completed values on one fine grid. The proof does not count unfinished +child temporaries as simultaneous unfinished parent operands. + +BIT's cover prime obeys `q > 25,600,000*N^5` and avoids every fixed rational +denominator, determinant and rank witness. Its bad-class fallback is fully +paid. Its new row stock is carried into the global support argument. A +conservative simultaneous reserve is + +``` +q0(h,p) <= C*(log2(h) + h^0.003*(log2(p))^3 + 1). +``` + +Reserved selected bits are deferred to later shrinking phases. They are not +first transformed individually under an obsolete reserve bound. The remaining +tail is polylogarithmic; row padding is removed after every completed phase. + +At the chosen outer point, every retained cost term has exponent saving +strictly above `0.0023`: + +| Outer cost | Saving | +| --- | ---: | +| Main transform | 0.00230197698 | +| Global coordinate routing | 0.0023025 | +| Fused Gaussian work | 0.00231193094 | +| Group BIT routing | 0.002312430925 | +| Boundary matrix work | 0.00230297698 | +| Sparse union movement | 0.40855297698 | +| Group scans | 0.99998 | +| Polynomial products | 0.99999 | +| Microbox face corrections | 77.00004 | + +These are exponents in the retained outer ledger, not comparable percentages +of observed running time. The minimum slack above the target is exactly +`98849/50000000000 = 0.00000197698`. The first five rows explain where the +critical budget is spent. The independent audit recomputes all nine rows +and the changed functional inequalities separately from the main ledger. + +## Further rounds toward 0.004 and beyond + +The research continued after the complete identity construction was found. +The larger H branch has useful exact primitives, but it does not establish +a higher kappa in this checkpoint. + +- **Cheaper BIT local responses.** An involution completion replaces nine + rank-seven corrections by nine rank-one corrections. A further exact probe + observes that the uncorrected map has corank three and constructs a minimum + three-row invertible completion. An independent audit checks all four bare + and wrapped native variants on all 661 arbitrary columns. Invertibility is sufficient when the caller uses + the appropriate inverse; an involution is a stronger restriction than needed. +- **Correct complex owner phases.** A signed monomial corrects an owner map + that preserved the polar form but had the wrong Gaussian phase. All 896 + recipes and 137,984 quadratic tests pass an independent implementation. + Its column lift retains the scalar `i^f`. +- **A complete boundary obstruction.** The noncommuting controller leaves + three interval endpoint corrections. Direct completion costs 672 rank-one + requests in complex arithmetic and 666 in BIT. The proposed single shared + head correction is not established. Even the three-row BIT completion has + rank-one surplus `645-666=-21` after direct normalization. +- **Additional cancellation probes.** A universal model with Q-free mixing + and only two whole-bank Q passages has coefficient span at most one; the + actual three interval words have rank three. A concrete signed-owner + witness also rejects cancellation by fresh-axis inverse stages. These + obstructions have explicit scopes and do not rule out geometry-aware coding. + +The strongest next H question is whether the actual query subspaces permit +a joint boundary decoder that shares part of those three corrections. A +free noncommutative cancellation argument is too coarse. Any next probe +should emit the native queries, preserve the input ownership and sign data, +and include preparation, routes, reads, restoration and endpoint completion +in the same invoice. + +The tested identity menus also include controllers through p6, 64 p3 masks, +123,904 commuting stage allocations and 96,141 BIT helper pairs. Their +failures are retained as finite negative results, with no global optimality +claim. [LITERATURE-AND-PROBES.md](LITERATURE-AND-PROBES.md) records the primary +sources, the applicable ideas, what was implemented, and the changes of +direction. + +## Preservation and reproduction + +The cumulative overlay preserves all **3,214 paths** from the supplied prior +overlay byte-for-byte, then appends this round and the surviving recovery +files. No predecessor file is rewritten or deleted. The exact base archive +has SHA-256 + +``` +be617301d74b0e586055c9ff702582a4c22c796008db8362781230ed9b453e11 +``` + +The earlier reported 332-entry archive discrepancy, including 34 reported +dependencies, remains a historical recovery limitation. Reconstructed +constructions have fresh provenance. We do not present newly generated files +as the exact missing originals. + +Apply the cumulative archive with its unchanged `apply_overlay.py` against +the exact base. Its root README gives the commands. Byte verification and +mathematical verification are separate operations. + +From the applied repository, verify the delivered source set: + +```sh +python3 -B research/kappa-rebuild/verify_scope.py +``` + +To regenerate the critical scientific chain in a **separate** complete +checkout while comparing against the delivered checkpoint: + +```sh +python3 -B research/kappa-rebuild/replay_critical.py \ + --repo /absolute/path/to/separate-applied-checkout \ + --reference /absolute/path/to/delivered-checkout \ + --report /absolute/path/to/replay-result.json +``` + +This runs the native reconstructions, controllers, phase/owner checks, +finite compiler, BIT bridge, precision closure, explicit ranks and independent +outer audit. The comparison ignores only declared timing metadata and the +resulting exact old-to-fresh artifact hash references. Mathematical fields +must agree. The saved replay receipt and logs are under `verification/`. + +All work remains unmerged. The next checkpoint must carry these cumulative +changes forward until the user explicitly reports a merge. + +## Open next questions + +1. Obtain independent mathematical review of the finite native compiler, + delayed-cover interface and support-local outer transfer as one chain. +2. Search for much smaller effective uniform templates or a direct mixed + outer theorem that preserves the same complete cost and storage accounting. +3. Build a geometry-aware H boundary decoder, starting from the cheaper + three-row BIT completion and the phase-correct complex owner maps. +4. Improve the complex productive-job tariff beyond its current limiting + mixed root; more BIT saving alone no longer controls this checkpoint. + +None of these open questions is used to justify the proposed conditional +`0.0023` point. 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"research/kappa-rebuild/bit/COMPLEX-CONTROLLER-FACTOR-P6.json", + "input_sha256": "573e797a4d1a37c6a9e419216ff53c7d9e2fe4e8a8384e01d2cc344f87d6d05e", + "minimum_menu": "6,9,16,25", + "new_kappa": false, + "schema": "finite-Laurent-controller-tariff-screen/1", + "source_sha256": "d31b29fe639fc066c6eb271bb17f6f234d1d56e008b8c8329ab13508694aa3af", + "status": "EXACT_FEASIBLE_PROFILES_FLOATING_OBJECTIVE_SCREEN" +} diff --git a/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md b/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md new file mode 100644 index 0000000000..4badb40f09 --- /dev/null +++ b/research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md @@ -0,0 +1,313 @@ +# Finite precompilation of a mixed-target rule + +Status: constructive bridge under the precise native hypotheses below, +independently audited in `../transfer-audit/AUDIT.md`. This is a written +argument, not a formal proof or a completed integer-multiplication admission. +The companion finite scheduler checks actual integer flow and ordering. +The original immediate-normalization BIT interface failed hypothesis 3. +`../bit/DELAYED-COVER-BRIDGE.md` supplies the distinct delayed-cover +implementation, with a finite local-ring check and the native binding in +`../bit/BIT-P3-NATIVE-BINDING.json`. Its fixed rational cover routes are a +costed program schema, not dense coordinate gates in a plain native word. + +Substantial OpenAI assistance. Apache-2.0. The mixed-target asymptotic flow is +Jacob Alan Sussman's Theorem 1G, PR4268, with the detailed proof at commit +`563b4e6e913625637f36ba7869bf5fcd0dd4975a` of +[`wht-power-saving-lean`](https://github.com/jacobalansussman/wht-power-saving-lean). +The source, exact scheduling code and notices are preserved under `sources/`. +The contribution proposed here is clearing the finite flow into a single +uniform native supplier before applying the inherited outer theorem. + +## 1. What needs to be assumed + +Let a job of rank `s` apply a specified tensor operator to `s` selected +coordinate blocks of one array, fixing all other coordinates. A block has +an arbitrary positive tensor multiplicity `f`. A completed rank `s` job +therefore acts on `s*f` axes. All productive inputs are arbitrary. + +Fix one mixed unit with productive targets `W_k` of ranks `k`, a chronological +list of paid child ranks `r_j`, and a finite number of scalar and routing +passes. These assumptions are required, separately from the moment inequality: + +1. **An all-column native word.** At every integer column count `c`, the unit + computes exactly its independent rank `k*c` output jobs. Its paid rounds + are rank `r_j*c` jobs in the stated order. All borrowed data return to the + declared endpoint, and owned storage is explicitly initialized and erased. + Its finite operation count is uniform in `c`. +2. **Disjoint splitting and spectators.** A rank `a+b` job is a rank `a` job + followed by a rank `b` job on disjoint coordinates. The resulting programs + preserve arbitrary complete spectator fields and accept arbitrary data. +3. **Whole-job adapters.** Any job passed between two units has an explicitly + priced, permitted embedding into its destination. For the intended complex + application these are fixed monomial binary address maps and unit quadratic + phases, together with role permutations. The adapters must work at every + tensor multiplicity. Rank equality by itself is insufficient. +4. **Finite-program closure.** A finite composition of these words and + adapters is again a permitted finite supplier. Constantly many owned fields + may be supplied with their full cost and no productive output credit. + The eventual ordinary recursion must have the same arbitrary-input, + restoration, scalar and precision interface as the retained supplier. + +Put `m=max{k:W_k>0}`, `k_min=min{k:W_k>0}`, and +`r_max=max{r_j}`. Require `r_max g*S`, keeps one +external weight throughout the entire finite circuit, and lowers only its +final rank-one leaves. All rational matchers, role routes and their inverses +are fixed before choosing the odd prime and its exclusions. Every route +uses complete equal-multiplicity cover fibers and the retained, costed row +permutation primitive. This gives the same finite DAG argument in the +rational-cover program category. The BIT binding supplies the actual native +projectors; a rank count alone would not provide this implementation. + +## 2. Proposed conclusion + +Suppose the preceding hypotheses hold and the mixed balance has root +`tau` in `(0,1)`: + +``` +sum_j r_j^tau = sum_k W_k k^tau. +``` + +For every `z` with `tau=2`, a fixed number `B_N` of productive arrays, +and only rank-one paid children, whose number `P_N` satisfies + +``` +P_N < B_N * N^z. +``` + +Every productive output is the full rank-`N` job. The construction may have +an enormous finite scalar word, role inventory and owned-storage requirement. +Those quantities affect constants and the eventual input threshold. They are +not omitted and are not credited as extra output jobs. + +If this uniform supplier is admitted under the retained native, precision and +fixed-tape premises, the existing uniform recursion applies directly. A new +dynamic RAM-to-Turing-machine simulation is not being assumed here. + +## 3. The integer lockstep flow at a fixed rank + +This is the part of Theorem 1G used by the compiler. At one fixed top rank +`N`, its units, source jobs and exit jobs have integer multiplicities. +For a multi-index `a` on the target types define + +``` +c(a) = product_k k^(a_k), +u(a) = multinomial(|a|;a) * product_k W_k^(a_k). +``` + +The exact Pascal identity is + +``` +u(b) = sum_{k:b_k>0} W_k*u(b-e_k). +``` + +Let `G` be the largest `|a|` with `m*c(a)<=N`. A class is indexed by a +sequence of paid rounds `s` of length `d<=G`; its dilation is +`rho_s=product r_j`. It contains `u(a)` units at column `rho_s*c(a)` for +each `a` with `|a|=G-d` and `m*rho_s*c(a)<=N`. + +For one parent paid round, the Pascal bijection matches its job copies to +target roles of the child class, at exactly equal integer ranks. Missing +demands are top sources. Separation ensures every parent job is consumed. +The source ranks lie in `(N/m,N]`; only the deepest classes produce exit +jobs, all strictly smaller than `N` once `N` is outside the finite base range. + +Process a class in word order. At round `j`, suspend every unit of that +class, complete the entire child class fed by that round, and only then +advance the parents to round `j+1`. This forbids a unit from requiring both +an earlier and a later round of the same suspended parent. The resulting +dependency graph is finite and acyclic. Its edges carry complete array jobs, +not individual scalar values copied for free. + +These integer quantities exist before normalizing the sources to total +weight one. Let `S_s` be the number of source jobs of rank `s`, let +`S=sum_s S_s`, and let the ordered exit batches be `(e,t)` with `t` jobs +of rank `e` each. This finite integer flow is the compiler input. + +## 4. Clear every batch denominator constructively + +Induct on the integer top rank `N`. For each smaller rank `e` the compiler +has already constructed a uniform program on `B_e` arbitrary arrays, +computing full rank `e` with `P_e` paid rank-one calls. For a sufficiently +small rank use the ordinary sequential program: `B_e=1`, `P_e=e`. + +For the current flow choose an integer `a` divisible by every `B_e` needed +for an exit and every `B_(N-s)` needed for a nonzero source remainder. +The product of those finitely many positive integers is a valid choice; +their least common multiple is a smaller optional choice. Set + +``` +B_N = a*S. +``` + +Assign `a*S_s` of the input arrays to each source rank `s`. On each of those +arrays, first complete its `N-s` complementary coordinates using the +already-compiled smaller uniform program. All such batches are integral +because `a` is divisible by their program widths. This preliminary action +commutes with the intended `s`-coordinate job and fixes its input coordinates. + +Make `a` copies of the integer lockstep flow. Route its source arrays to +their exact input slots using permitted whole-array adapters. At a deepest +exit batch, `a*t` is divisible by `B_e`; replace that batch by copies of +the smaller uniform rank-`e` program. Restore its role ordering and adapters +before resuming the suspended parent class. Execute the finite DAG in the +lockstep order, and finally restore the original source-array ordering. + +Every array has now received its disjoint complementary job and source job, +hence the full rank-`N` transform. No copied scalar input or blank helper is +treated as another independent output. Every original array occurs exactly +once as a source; interior target slots are jobs on those arrays, not new +productive inputs. + +Induction removes all recursive calls: the only remaining paid children +have rank one. Clearing denominators multiplies productive arrays and paid +operations equally, so it preserves the per-array leaf count. In formulas, + +``` +P_N/B_N = (1/S) * [ + sum_s S_s * (P_(N-s)/B_(N-s)) + + sum_(ordered exit batches e,t) t * (P_e/B_e) +]. +``` + +Zero-rank terms vanish. Any plain rank-one exits are the ordinary base +program. Scalar and routing passes have their own analogous nonnegative +count and must be included in the asymptotic cost used to choose `N`. + +This construction uses full integer batches; it does not assume that +fractional pieces of a physical array can be routed at zero cost. Its exact +replication factors can be impractically large, which is acceptable for a +fixed finite supplier existence statement but must be disclosed. + +## 5. Why some fixed N has the required saving + +The independent audit sharpened this step to a direct count of paid leaves. +It does not require a running-time comparison between a RAM flow and native +tapes. Let `L(n)=P_n/B_n`, let `w_s=S_s/S`, and let `x_e` be the exit counts +normalized by the number S of source arrays. The actual integer compiler +satisfies + +``` +L(n) = sum_s w_s L(n-s) + sum_e x_e L(e), +sum_s w_s = 1, +sum_e x_e e^tau = sum_s w_s s^tau. +``` + +The last equality follows by canceling every internal matched job in the +exact tau-potential balance. The source ranks satisfy `n/m0`, concavity gives + +``` +n^z0 - sum_s w_s (n-s)^z0 >= eta * sum_s w_s s^z0. +``` + +Indeed, for `t=s/n` in `[1/m,1]`, +`1-(1-t)^z0 >= z0*t >= eta*t^z0`. For every sufficiently large n the +exit coefficient is less than `eta/2`. Every complement and exit has +smaller rank than n, so strong induction on the exact recurrence proves +`L(n)<=A*n^z0`, with finite A chosen to cover the smaller ranks. Eventually +`L(N) g*S`. Only the final rank 1 leaves are lowered. +This is the implementation in `../bit/DELAYED-COVER-BRIDGE.md`; the old +immediate-normalization argument had an actual weight mismatch. + +## What the H work established, and what it did not + +The native involution H replaces the full-identity local bill by a bill +smaller by 56 rank units per call. With the same p3 packing, a hypothetical +correct H controller has an available complete correction allowance of + +``` +228.06458440184225522412186424456699568... +``` + +at C=.004. That is enough for one 224H1 head-conjugation bill, leaving +4.0645844018422552241... units. It is not enough for three such bills. + +The actual p3 scalar adapters work over a free noncommutative stage ring. +They produce the three interval products T2T1T0, T6T5T4 and T6...T0, each +twice. Thus this part never needed different H_s to commute. The physical +Q boundary sits on opposite sides of the two copies of each interval. +Directly closing the three problematic copies costs 672H1; the rank surplus +becomes −56 even at C=0. This is a failed implementation, not a lower bound +on all coupled boundary corrections. + +The phase probe repairs a separate technical issue. Each old owner G_s +preserves a full Gaussian's polar form but has an even linear phase defect +d of weight 4 or 8. The signed monomial `Gamma_s=Z_d G_s`, where Z_d is a +physical coefficient character, is involutive and preserves the full +Gaussian including its absolute phase. It also preserves the port's q and +maps the old hole `F9*Cq` to `i*Ffresh8`. All 896 stage/port cases pass; the +independent companion audit lowers them to actual XOR-bit, SWAP-bit and +character operations. These are valid reusable primitives. Their complete +native whole-word substitution and a cheaper global H closure are not +claimed. + +A further dyadic completion family is available: + +``` +H_lambda = J + (13+lambda) E_A [L,I] + = I + [-K;(12+lambda)I] [L,I]. +``` + +For dyadic units lambda (signed powers of two), this is invertible with a dyadic inverse. +The chronological choices −1/2,1,−2 multiply to identity in a common chart. +The exact actual-owner negative control shows why that identity cannot +simply be copied across distinct stages. + +The final focused screen gives two further limits. In a universal +free-operator model, two complete-bank Q passages cannot implement all +three distinct Q K Q words even with all six streams and arbitrary Q-free +polynomial mixing. Its coefficient-span witness has rank3 against a +two-event upper bound 1. This excludes only that universal two-event model; +actual low-rank operator relations and partial-bank calls are outside its +scope. Separately, an actual old-block coordinate-sector +witness proves that a fresh-axis H word does not automatically supply the +inverse of an old monodromy. Reusing the old support cancels transform axes +as well as state; the accounting records that loss. + +The strongest next lever is the actual rank-eight query geometry and its +legal frame contacts. The native embedded logger uses eight existing cold +helpers to return `d + C*X`, where `C=[L,I]` and `C*H=-C`, with no new +recursive children. After a post-source H wrapper, its precise defect is +`d - C*X_new`. The eight helper outputs require explicit restoration before +receiving independent output credit. A joint decoder could use this +information across the three interval jobs, but a single query bank has +not been proved sufficient. See `../complex/COMPOSITION-CONTRACTS.json`. + +The final companion BIT probe also shows that requiring an involution can +overconstrain the local design. Its actual zero-repair operator has corank +3; three rank 1 row completions make it invertible. Forward wrappers use +G and reverse wrappers use its inverse. Arbitrary-input native replay +passes, but the direct three-boundary bill still exceeds its rank surplus +by 21. This is a better local contract with the same outstanding global +interface; see `../bit/PR589-MINIMUM-COMPLETION.json`. + +A useful next probe should supply a costed geometry-aware query decoder, +a scalar controller with fewer independent negative interval jobs, or a +typed recursive contract with an actual inverse-state adapter. Generic +common-head sharing and same-chart identities alone are insufficient. + +## Reproduction + +Run these from the repository root, writing receipts to a separate replay +directory so the retained source-bound certificates remain byte-stable: + +```sh +mkdir -p research/kappa-rebuild/architecture/replay +python research/kappa-rebuild/architecture/controller_tariff.py --output research/kappa-rebuild/architecture/replay/P3.json +python research/kappa-rebuild/architecture/identity_mixed_certificate.py --output research/kappa-rebuild/architecture/replay/P2.json +python research/kappa-rebuild/architecture/identity_mixed_certificate.py --prefix-cut --output research/kappa-rebuild/architecture/replay/P2-PREFIX.json +python research/kappa-rebuild/architecture/free_involution_wrap_probe.py --output research/kappa-rebuild/architecture/replay/FREE-H.json +python research/kappa-rebuild/architecture/aggregate_h_probe.py --output research/kappa-rebuild/architecture/replay/AGGREGATE-H.json +python research/kappa-rebuild/architecture/joint_boundary_probe.py --output research/kappa-rebuild/architecture/replay/JOINT-H.json +``` + +The exact certification commands use the Python standard library. The +optional LP discovery scripts additionally use NumPy/SciPy. 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Analogies and probes are distinguished from admitted savings in LITERATURE.md.", + "research_date": "2026-10-21", + "schema": "architecture-primary-source-index/1" +} diff --git a/research/kappa-rebuild/architecture/aggregate_h_probe.py b/research/kappa-rebuild/architecture/aggregate_h_probe.py new file mode 100644 index 0000000000..36c5630c0e --- /dev/null +++ b/research/kappa-rebuild/architecture/aggregate_h_probe.py @@ -0,0 +1,292 @@ +#!/usr/bin/env python3 +"""Exact scoped probes of the p3 H endpoint, phases and dyadic completions. + +This is not a closed H supplier. It separates the noncommuting six-bank +controller identity from physical per-port head operators and from native +whole-word adapter admission. Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import importlib.util +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +sys.path[:0] = [str(HERE), str(HERE.parent/'bit')] +import complex_controller_factor as C +import controller_tariff as T +import mixed_forest_search as F +import mixed_target_probe as E + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + + +def need(ok, label): + if not ok: + raise AssertionError(label) + + +def plus(a, b, c=1): + out = dict(a) + for key, value in b.items(): + out[key] = out.get(key, 0)+c*value + if not out[key]: + del out[key] + return out + + +def prepend(poly, stage): + return {(stage,)+word: value for word, value in poly.items()} + + +def nc_controller(): + """Replay the SAME p3 physical role program over the free associative ring. + + T_s is an arbitrary stage multiplier and is NOT assumed invertible, + commuting with another T, or order two. Q has already been removed from + the local shears by the separately stated two-sector gauge. The result + therefore cannot by itself remove a physical per-port Q boundary. + """ + coefficients, events, _ = C.controller(3, 43) + factored = C.factor(coefficients) + matrix = [[{(): 1} if i == j else {} for j in range(6)] for i in range(6)] + for event in events: + kind = event[0] + if kind == 'LOCAL': + _, stage, pair, backward = event + x, y = 2*pair, 2*pair+1 + dst, src, sign = (x, y, -1) if backward else (y, x, 1) + matrix[dst] = [plus(a, b, sign) for a, b in zip(matrix[dst], matrix[src])] + for role in (x, y): + matrix[role] = [prepend(poly, stage) for poly in matrix[role]] + elif kind == 'IDLE': + _, pair, stages = event + for stage in stages: + for role in (2*pair, 2*pair+1): + matrix[role] = [prepend(poly, stage) for poly in matrix[role]] + elif kind == 'ADD': + _, dst, src, sign = event + matrix[dst] = [plus(a, b, sign) for a, b in zip(matrix[dst], matrix[src])] + else: + need(kind == 'SIGNED_SWAP', 'Known role event') + _, a, b = event + matrix[a], matrix[b] = matrix[b], [{w: -v for w, v in p.items()} for p in matrix[a]] + + def scalar_left(a, b): + return [[sum_polys((b[k][j], a[i][k]) for k in range(6)) + for j in range(6)] for i in range(6)] + + def scalar_right(a, b): + return [[sum_polys((a[i][k], b[k][j]) for k in range(6)) + for j in range(6)] for i in range(6)] + + out = scalar_right(scalar_left(factored['left'], matrix), factored['right']) + words = [tuple(s for s in reversed(range(7)) if term[s+1]) for term in factored['diagonal']] + expected = [[{words[i]: Q(1)} if i == j else {} for j in range(6)] for i in range(6)] + need(out == expected, 'All free noncommuting coefficients match the six interval endpoints') + wrong = [list(row) for row in expected] + wrong[0] = list(wrong[0]); wrong[0][0] = {tuple(reversed(words[0])): Q(1)} + need(wrong != out, 'Reversing a noncommuting interval is rejected') + return dict(diagonal_words=[list(word) for word in words], + all_36_free_polynomial_entries_checked=True, + no_relations_between_stage_multipliers=True, + left=C.serialize_matrix(factored['left']), right=C.serialize_matrix(factored['right']), + boundary_warning='The physical Q gauges remain on opposite sides of each of three interval products.', + wrong_stage_order_rejected=True) + + +def sum_polys(terms): + out = {} + for poly, coefficient in terms: + if coefficient: + out = plus(out, poly, coefficient) + return out + + +def phase_lifts(n, phases): + records = [] + bare_failures = 0 + # A degree-at-most-two Z4 phase is determined by these evaluations. + # Outside the seventeen active coordinates every map fixes the coordinate. + for stage in range(1, 9): + axes = n.AXES[:9]+n.AXES[9+8*(stage-1):9+8*stage] + support = sum(axes) + old = (1 << 9)-1 + fresh = support ^ old + tests = [0]+list(axes)+[a ^ b for j, a in enumerate(axes) for b in axes[j+1:]] + full = n.frame((a, 1) for a in axes) + for port, recipe in enumerate(phases['recipes']): + g = n.source_route(phases, stage, port) + need(n.product(g, g) == n.AXES, 'Native owner map is involutive') + q = recipe['q'] + wanted = n.polynomial(full) + moved = n.polynomial(n.move_frame(full, g)) + need(wanted[1] == moved[1], 'Full tensor polar form is preserved') + delta = [(a-b) % 4 for a, b in zip(wanted[0], moved[0])] + need(all(v in (0, 2) for v in delta), 'Only even linear phase defect') + d = sum((v//2) << i for i, v in enumerate(delta)) + need(d != 0, 'Bare G has an actual absolute phase defect') + bare_failures += 1 + need(d.bit_count() in (4, 8) and d.bit_count() % 4 == 0, 'Zero global Pauli phase') + need(n.act(g, d) == d, 'Signed monomial squares to identity') + need((d & q).bit_count() % 2 == 0, 'Signed monomial preserves each port q') + + def transform(y): + # In Fourier coordinates Z_d G is the affine map y -> G(y+d). + return n.act(g, y ^ d) + + for y in tests: + z = transform(y) + need((z & support).bit_count() % 4 == (y & support).bit_count() % 4, + 'Exact full Gaussian phase under signed monomial') + need((z & q).bit_count() % 2 == (y & q).bit_count() % 2, + 'Exact per-port q Gaussian phase') + need(((z & old).bit_count()+(z & q).bit_count() % 2) % 4 == + (1+(y & fresh).bit_count()) % 4, + 'Old hole F9*Cq maps to i times the common fresh8 Gaussian') + records.append(dict(stage=stage, port=port, q=q, character_mask=d, + character_weight=d.bit_count(), owner_columns=list(g), + Fourier_affine_order='G(y+d)', physical_operator='Z_d G', + full_tensor_global_phase_Z4=0, old_hole_to_fresh_global_phase_Z4=1)) + return dict(records=records, checked_owner_lifts=len(records), + bare_owner_absolute_phase_failures=bare_failures, + bare_owner_is_not_made_commuting_by_a_physical_translation=True, + evidence='Complete quadratic phase evaluations; all other coordinates fixed.', + tensor_lift='Each displayed affine Fourier substitution lifts column by column; the constant i becomes i^f.', + scope='The signed monomial is proved. Replacing the native whole-word contacts by it remains a separate integration task.') + + +def h_lambda(geo, value): + out = [dict(row) for row in geo['J']] + for k, a in enumerate(geo['selected']): + for p, coefficient in {a: Q(1), **geo['L'][k]}.items(): + out[a][p] = out[a].get(p, 0)+(13+value)*coefficient + if not out[a][p]: + del out[a][p] + return out + + +def sparse_matrix_product(a, b): + out = [] + for row in a: + new = {} + for k, coefficient in row.items(): + new = plus(new, b[k], coefficient) + out.append(new) + return out + + +def mapped_kernel(n, phases, matrix, stage): + routes = [n.source_route(phases, stage, p) for p in range(112)] + return [{(p, n.product(routes[i], routes[p])): c for p, c in row.items()} + for i, row in enumerate(matrix)] + + +def apply_mapped(n, matrix, vector): + out = [] + for row in matrix: + new = {} + for (p, route), coefficient in row.items(): + for address, value in vector[p].items(): + at = n.act(route, address) + new[at] = new.get(at, 0)+coefficient*value + if not new[at]: + del new[at] + out.append(new) + return out + + +def dyadic_family(n, geo, phases): + values = (Q(-1, 2), Q(1), Q(-2)) + matrices = [h_lambda(geo, value) for value in values] + identity = [{i: Q(1)} for i in range(112)] + actual = sparse_matrix_product(matrices[2], sparse_matrix_product(matrices[1], matrices[0])) + need(actual == identity, 'Three dyadic completions multiply to identity in one chart') + need(h_lambda(geo, Q(-1)) == geo['H'], 'Original involution is the lambda=-1 member') + # A real physical column catches replacing distinct stage owners by one chart. + basis = [{} for _ in range(112)] + basis[0] = {n.AXES[9]: Q(1)} + vector = basis + for stage, matrix in enumerate(matrices): + vector = apply_mapped(n, mapped_kernel(n, phases, matrix, stage), vector) + need(vector != basis, 'Same-chart lambda cancellation is not valid across actual stage owners') + witness = None + for port in range(112): + delta = plus(vector[port], basis[port], -1) + if delta: + at = min(delta) + witness = dict(input_field=0, input_address=n.AXES[9], output_field=port, + output_address=at, difference=str(delta[at])) + break + return dict(formula='H_lambda=J+(13+lambda) E_A [L,I]', + low_rank_form='H_lambda=I+[-K;(12+lambda)I] [L,I]', + inverse='I-lambda^-1 [-K;(12+lambda)I] [L,I]', + permitted_parameter='Any nonzero signed power of two keeps both lambda and lambda^-1 dyadic.', + same_chart_product={'chronological_lambda':[str(v) for v in values], 'exact_identity':True}, + actual_different_owner_negative_control=witness, + recursive_profile_if_natively_replayed='The same eight rank-one selected-row completions; scalar coefficients change.', + native_lambda_word_replayed=False) + + +def funding(): + windows, targets, _ = T.geometry(3, 43) + case = F.verify(3, 6, T.simple_p3(), 'H', custom_windows=windows, custom_targets=targets) + output = F.serialize(case, ('0.001745', '0.0023', '0.004', '0.006')) + repaired = Counter(case['signed']); repaired[1] -= 3*224 + output['direct_three_head_conjugations'] = dict(paid_H1=3*224, + rank_margin=str(sum(r*c for r, c in repaired.items())), + moment_at_004=E.show(E.moment(repaired, Q('0.004'))), + warning='This is only the direct implementation, not a lower bound on a new coupled boundary algorithm.') + hypothetical = Counter(case['signed']); hypothetical[1] -= 224 + output['unproved_single_shared_head_conjugation'] = dict(paid_H1=224, + moment_at_004=E.show(E.moment(hypothetical, Q('0.004'))), + established=False, obligation='A new identity must couple the three independent negative-bank endpoint jobs.') + return output + + +def run(): + native_path = ROOT/'research/kappa-next-regime/direct-decoder/native_common.py' + rebuild_path = HERE.parent/'complex/rebuild_native.py' + n = load(native_path, 'architecture_aggregate_native') + r = load(rebuild_path, 'architecture_aggregate_h') + packet, phases = n.load_sources() + geo = r.geometry(packet) + out = dict(schema='aggregate-H-independent-phase-and-controller-probes/1', + status='PASS_SCOPED_IDENTITIES_AND_EXPLICIT_DIRECT_CLOSURE_OBSTRUCTION', + noncommuting_p3=nc_controller(), phase_lifts=phase_lifts(n, phases), + dyadic_completion_family=dyadic_family(n, geo, phases), conditional_funding=funding(), + source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() + for p in (Path(__file__), native_path, rebuild_path, + HERE/'controller_tariff.py', HERE/'mixed_forest_search.py', + HERE.parent/'bit/complex_controller_factor.py')}, + native_source_pins=n.PINS, + no_new_H_kappa=True, + missing=['Actual whole-word signed-owner wrappers', + 'A coupled boundary identity costing at most the complete conditional budget', + 'Arbitrary-input restoration and output independence for any new logger/catalyst']) + return out + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output', type=Path, default=HERE/'AGGREGATE-H-PROBE.json') + args = parser.parse_args() + result = run() + args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps(dict(status=result['status'], phase_lifts=result['phase_lifts']['checked_owner_lifts'], + direct_closure=result['conditional_funding']['direct_three_head_conjugations']), sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/architecture/controller_menu_search.py b/research/kappa-rebuild/architecture/controller_menu_search.py new file mode 100644 index 0000000000..403b48403e --- /dev/null +++ b/research/kappa-rebuild/architecture/controller_menu_search.py @@ -0,0 +1,83 @@ +#!/usr/bin/env python3 +"""Optimize finite helper packings for already enumerated Laurent controllers. + +Floating LP discovers patterns; every emitted profile is reconstructed over +the rationals and checked independently. A displayed floating root is a +screening statistic, never an exact certificate or an outer kappa claim. +Use controller_tariff.py to produce a source-bound exact chosen receipt. +Apache-2.0; substantial OpenAI assistance. +""" +from pathlib import Path +from hashlib import sha256 +import argparse +import json +import sys + +sys.dont_write_bytecode=True +import mixed_forest_search as F +import mixed_target_probe as E + +HERE=Path(__file__).resolve().parent + + +def floating_root(signed): + lo,hi=0.0,.01 + for _ in range(65): + mid=(lo+hi)/2 + if E.numerical_value(signed,mid)>0:lo=mid + else:hi=mid + return (lo+hi)/2 + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--input',type=Path,default=HERE/'sources/COMPLEX-CONTROLLER-FACTOR-p1-p5.json') + parser.add_argument('--output',type=Path,default=HERE/'CONTROLLER-MENU-SEARCH.json') + parser.add_argument('--min-p',type=int,default=4) + parser.add_argument('--max-p',type=int,default=5) + parser.add_argument('--minimums',default='6,9') + parser.add_argument('--C',default='0.0024') + parser.add_argument('--no-groups',action='store_true') + args=parser.parse_args() + data=json.loads(args.input.read_text()) + rows=[];failed=[] + for formal in data['cases']: + p=formal['pairs'];keep=formal['keep_idle_mask'] + if not(args.min_p<=p<=args.max_p):continue + windows=[{'name':str(w['index'])+'_'+w['kind']+str(w['pair']), + 'capacity':w['rank'],'stages':w['stages'], + 'field_count':112 if 'field' in w else 224} + for w in formal['retained_idle_windows']] + targets={int(r):int(n) for r,n in formal['endpoint_after_stated_free_Pauli_normalization'].items()} + current=[] + for minimum in map(int,args.minimums.split(',')): + if minimum>min(targets):continue + try: + solution,discovery=F.discover(p,minimum,float(args.C),fresh_groups=not args.no_groups, + custom_windows=windows,custom_targets=targets) + case=F.verify(p,minimum,solution,'identity',fresh_groups=not args.no_groups, + custom_windows=windows,custom_targets=targets) + except (AssertionError,ValueError) as error: + failed.append({'p':p,'keep':keep,'minimum':minimum,'error':str(error)}) + continue + record={k:v for k,v in case.items() if k!='signed'} + record.update(keep=keep,discovery=discovery, + floating_root_for_discovery_only=floating_root(case['signed']), + discovery_margin_float=E.numerical_value(case['signed'],float(args.C))) + current.append(record);rows.append(record) + if current: + best=max(current,key=lambda x:x['floating_root_for_discovery_only']) + print(json.dumps({'p':p,'keep':keep,'best_minimum':best['minimum_target_parameter'], + 'floating_root':best['floating_root_for_discovery_only'], + 'lift':best['replica_lift']}),flush=True) + out={'schema':'finite-Laurent-controller-tariff-screen/1', + 'status':'EXACT_FEASIBLE_PROFILES_FLOATING_OBJECTIVE_SCREEN', + 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(), + 'forest_source_sha256':sha256((HERE/'mixed_forest_search.py').read_bytes()).hexdigest(), + 'input':str(args.input),'input_sha256':sha256(args.input.read_bytes()).hexdigest(), + 'C_discovery':args.C,'minimum_menu':args.minimums,'cases':rows, + 'failed_discoveries_not_waived':failed,'new_kappa':False} + args.output.write_text(json.dumps(out,indent=2,sort_keys=True)+'\n') + + +if __name__=='__main__':main() diff --git a/research/kappa-rebuild/architecture/controller_tariff.py b/research/kappa-rebuild/architecture/controller_tariff.py new file mode 100644 index 0000000000..21fe4799f7 --- /dev/null +++ b/research/kappa-rebuild/architecture/controller_tariff.py @@ -0,0 +1,129 @@ +#!/usr/bin/env python3 +"""Bind an exact Laurent controller to a complete mixed-output tariff. + +The source-bound receipt checks actual local words, the integral Laurent +identity, all helper inventories, strict recursive separation, and exact +moments. Native global frame placement and the outer transfer remain +separate proof obligations. Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import sys + +sys.dont_write_bytecode=True +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[2] +sys.path.insert(0,str(HERE.parent/'bit')) +import complex_controller_factor as C +import mixed_forest_search as F +import mixed_target_probe as E +import identity_mixed_certificate as I + + +def geometry(p,keep): + coefficients,events,idles=C.controller(p,keep) + factored=C.factor(coefficients) + E.need(factored is not None,'Exact rational Laurent simultaneous diagonalization') + formal=C.serialize_case(p,keep,coefficients,events,idles,factored) + windows=[{'name':str(w['index'])+'_'+w['kind']+str(w['pair']), + 'capacity':w['rank'],'stages':w['stages']} + for w in idles if w['retained']] + targets={int(r):int(n) for r,n in formal['endpoint_after_stated_free_Pauli_normalization'].items()} + return windows,targets,formal + + +def simple_p3(): + """An integer packing; its selection requires no optimization library.""" + return [dict(family=f,pattern=[a,b],count=str(n)) + for f in (0,3) for n,a,b in I.PAIRS]+[ + dict(family=f,pattern=[a],count=str(n)) + for f in (1,2) for n,a in I.MIDDLE] + + +def certify(p,keep,minimum,solution,groups=False,discovery=None): + windows,targets,formal=geometry(p,keep) + native=I.source_binding() + data=json.loads(I.NATIVE.read_text()) + for provider in ('THREE-base','THREE-extra'): + for direction in ('forward','backward'): + actual=data['providers'][provider]['unsplit_replays'][direction]['paid_profile'] + E.need({int(r):int(n) for r,n in actual.items()}==dict(F.IDENTITY), + 'The actual unsplit native profile is charged') + case=F.verify(p,minimum,solution,'identity',fresh_groups=groups, + custom_windows=windows,custom_targets=targets) + invoice=F.serialize(case,('0.001730970419199163702738','0.001745','0.002','0.0022', + '0.0023','0.0025','0.003','0.004')) + invoice['root_bracket']=I.root_bracket(case['signed']) + extra_sources=[Path(__file__),HERE/'mixed_forest_search.py',HERE/'mixed_target_probe.py', + HERE/'identity_mixed_certificate.py',HERE.parent/'bit/complex_controller_factor.py', + HERE.parent/'bit/mixed_productive_bit.py'] + sources={str(path.relative_to(ROOT)):sha256(path.read_bytes()).hexdigest() for path in extra_sources} + # Explicit negative controls exercise ownership and changed target credit. + bad=[dict(row) for row in solution] + bad[0]=dict(bad[0],count=str(Q(bad[0]['count'])+1)) + rejected=False + try: + F.verify(p,minimum,bad,'identity',fresh_groups=groups,custom_windows=windows,custom_targets=targets) + except AssertionError:rejected=True + E.need(rejected,'Duplicating one displayed owner is rejected') + unsupported=dict(targets) + first=min(unsupported) + unsupported[first]-=1 + unsupported[first+1]=unsupported.get(first+1,0)+1 + rejected_rank=False + try: + F.verify(p,minimum,solution,'identity',fresh_groups=groups, + custom_windows=windows,custom_targets=unsupported) + except AssertionError:rejected_rank=True + E.need(rejected_rank,'A free extra target axis is rejected') + return {'schema':'source-bound-Laurent-controller-mixed-tariff/1', + 'status':'PASS_EXACT_LAURENT_IDENTITY_LOCAL_SOURCE_BOUND_TARIFF', + 'source_sha256':sources,'native_source_sha256':native, + 'controller':formal,'invoice':invoice,'discovery':discovery, + 'controls':{'duplicate_owner_rejected':rejected, + 'unearned_extra_target_axis_rejected':rejected_rank}, + 'owned_storage':'Every simultaneous native call owns a center C0; none is credited as a productive output.', + 'not_proved':['The actual local native forward/backward maps satisfy this Laurent input contract in every global stage.', + 'All displayed helper owners admit phase-correct global adapters in the retained windows.', + 'All output monomials admit the stated legal free normalizations.', + 'Finite precompilation, precision/stock induction, and the outer integer-multiplication transfer.'], + 'new_kappa':False} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--p',type=int,default=3) + parser.add_argument('--keep',type=int,default=43) + parser.add_argument('--minimum',type=int,default=6) + parser.add_argument('--discover',action='store_true') + parser.add_argument('--groups',action='store_true') + parser.add_argument('--C',default='0.0023') + parser.add_argument('--replay',type=Path) + parser.add_argument('--output',type=Path,default=HERE/'P3-CONTROLLER-TARIFF-CERTIFICATE.json') + args=parser.parse_args() + if args.replay: + old=json.loads(args.replay.read_text()) + p,keep=old['controller']['pairs'],old['controller']['keep_idle_mask'] + minimum=old['invoice']['minimum_target_parameter'] + solution=old['invoice']['patterns'];groups=old['invoice']['fresh_groups'] + discovery={'replayed_receipt':str(args.replay),'receipt_sha256':sha256(args.replay.read_bytes()).hexdigest()} + else: + p,keep,minimum,groups=args.p,args.keep,args.minimum,args.groups + windows,targets,_=geometry(p,keep) + if args.discover: + solution,discovery=F.discover(p,minimum,float(args.C),fresh_groups=groups, + custom_windows=windows,custom_targets=targets) + else: + E.need((p,keep,minimum,groups)==(3,43,6,False),'The built-in integer construction is p3 keep43 K6') + solution=simple_p3();discovery={'method':'Explicit integer pattern table'} + out=certify(p,keep,minimum,solution,groups,discovery) + args.output.write_text(json.dumps(out,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':out['status'],'root':out['invoice']['root_bracket'], + 'targets':out['invoice']['targets'],'paid':out['invoice']['paid']},sort_keys=True)) + + +if __name__=='__main__':main() diff --git a/research/kappa-rebuild/architecture/free_involution_wrap_probe.py b/research/kappa-rebuild/architecture/free_involution_wrap_probe.py new file mode 100644 index 0000000000..a423023140 --- /dev/null +++ b/research/kappa-rebuild/architecture/free_involution_wrap_probe.py @@ -0,0 +1,112 @@ +#!/usr/bin/env python3 +"""Exact free-group search of legal pre-target/post-source H wrappers. + +An algebraic screen, not an actual native frame certificate. H_s^2=I is +the only relation; different stages do not commute. Forward permits H_s +on Y before the call and X after it; backward permits H_s on X before and +Y after. These are the physical contacts proposed by the caller owner. +The search asks whether these wrappers alone remove the coupled residual +on the terminal hole bank, while a full-frame bank may retain a free word. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from hashlib import sha256 +from itertools import product +from pathlib import Path +import argparse +import json + +HERE=Path(__file__).resolve().parent +ONE={():1} +ZERO={} + +def word(a,b): + out=list(a) + for letter in b: + if out and out[-1]==letter: + out.pop() + else: + out.append(letter) + return tuple(out) + +def add(a,b): + out=Counter(a) + out.update(b) + return {k:v for k,v in out.items() if v} + +def multiply(a,b): + out=Counter() + for u,c in a.items(): + for v,d in b.items(): + out[word(u,v)]+=c*d + return {k:v for k,v in out.items() if v} + +def mm(a,b): + return [[add(multiply(a[i][0],b[0][j]),multiply(a[i][1],b[1][j])) + for j in range(2)] for i in range(2)] + +def identity(): + return [[ONE,ZERO],[ZERO,ONE]] + +def local(stage,forward,pre,post): + h={(stage,):1} + if forward: + before=[[ONE,ZERO],[ZERO,h if pre else ONE]] + shear=[[ONE,ZERO],[h,ONE]] + after=[[h if post else ONE,ZERO],[ZERO,ONE]] + else: + before=[[h if pre else ONE,ZERO],[ZERO,ONE]] + shear=[[ONE,{(stage,):-1}],[ZERO,ONE]] + after=[[ONE,ZERO],[ZERO,h if post else ONE]] + return mm(after,mm(shear,before)) + +def encode(polynomial): + return [{'word':list(w),'coefficient':c} for w,c in sorted(polynomial.items())] + +def run(): + words=[] + off_diagonal=[] + for choices in product((0,1),repeat=6): + result=identity() + for stage in range(3): + result=mm(local(stage,stage%2==0,*choices[2*stage:2*stage+2]),result) + record={'pre_post_bits':list(choices),'operator':[[encode(p) for p in row] for row in result]} + words.append(record) + if not result[0][0] and not result[1][1]: + record['terminal_hole_bank_already_identity']=result[1][0]==ONE + record['terminal_hole_bank_scalar_sign_only']=result[1][0] in (ONE,{():-1}) + off_diagonal.append(record) + # The fully wrapped construction has the noncommuting monodromy H2 H1 H0. + full=identity() + for s in range(3): + full=mm(local(s,s%2==0,1,1),full) + monodromy={(2,1,0):1} + assert full==[[ZERO,{(2,1,0):-1}],[monodromy,ZERO]] + assert off_diagonal and not any(r['terminal_hole_bank_scalar_sign_only'] for r in off_diagonal) + # In the same-owner specialization H0=H1=H2=H, the residual is H, not I. + same={} + for w,c in full[1][0].items(): + v=() + for _ in w: + v=word(v,(0,)) + same=add(same,{v:c}) + assert same=={(0,):1} + return {'schema':'free-involution-wrapper-screen/1','status':'PASS_EXACT_SCOPED_SCREEN', + 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(), + 'menu_size':len(words),'pure_swap_endpoints':off_diagonal, + 'fully_wrapped_monodromy':'H2 H1 H0', + 'identity_hole_residual_found':False, + 'scope':'Only binary single-stage involution wrappers at the stated six contacts. No claim about additional scalar combinations, interstage gadgets, changed heads, or identities among the actual owners.', + 'physical_frame_verification':'Owned by the caller reconstruction; not implied by this abstract screen.', + 'new_kappa':False} + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path,default=HERE/'FREE-INVOLUTION-WRAP-PROBE.json') + args=parser.parse_args() + result=run() + args.output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'menu_size':result['menu_size'],'pure_swap_count':len(result['pure_swap_endpoints'])})) + +if __name__=='__main__': + main() diff --git a/research/kappa-rebuild/architecture/identity_mixed_certificate.py b/research/kappa-rebuild/architecture/identity_mixed_certificate.py new file mode 100644 index 0000000000..f4b51b358e --- /dev/null +++ b/research/kappa-rebuild/architecture/identity_mixed_certificate.py @@ -0,0 +1,135 @@ +#!/usr/bin/env python3 +"""Source-bound, solver-free complete p2 full-identity recursive tariff. + +This certifies a finite invoice, strict separation, and its scalar root. +It does not certify the global native owner construction, all-size mixed +router, fixed-tape implementation, rounding induction, or outer kappa. +Apache-2.0. Substantial OpenAI assistance; credits in README.md. +""" +from collections import Counter +from decimal import Decimal, localcontext, ROUND_FLOOR +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import sys + +sys.dont_write_bytecode=True +import mixed_target_probe as E +import mixed_forest_search as F + +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[2] +NATIVE=ROOT/'research/kappa-rebuild/complex/full-identity-certificates/RESULT.json' +PAIRS=((93,8,8),(56,7,9),(56,9,7),(19,5,9)) +MIDDLE=((29,0),(28,5),(18,6),(56,7),(93,8)) + +def root_bracket(signed): + """Decimal discovery followed by rational elementary-function proof.""" + with localcontext() as context: + context.prec=85 + terms=[(Decimal(n.numerator)/Decimal(n.denominator),Decimal(r).ln()) + for r,n in sorted(signed.items()) if n] + def value(c): + return sum(n*((1-c)*logr).exp() for n,logr in terms) + lo=Decimal(0);hi=Decimal('0.01') + for _ in range(200): + mid=(lo+hi)/2 + if value(mid)>0:lo=mid + else:hi=mid + scale=10**30 + integer=int((lo*scale).to_integral_value(rounding=ROUND_FLOOR)) + left,right=Q(integer,scale),Q(integer+1,scale) + low_bounds=E.moment(signed,left) + high_bounds=E.moment(signed,right) + E.need(low_bounds[0]>0 and high_bounds[1]<0,'Exact strict root bracket') + return {'lower':str(Decimal(integer)/scale), + 'upper':str(Decimal(integer+1)/scale), + 'margin_at_lower':E.show(low_bounds),'margin_at_upper':E.show(high_bounds), + 'uniqueness':'The paid/target ratio is strictly increasing in C because every paid rank is smaller than every target rank.'} + +def source_binding(): + data=json.loads(NATIVE.read_text()) + bound={str(NATIVE.relative_to(ROOT)):sha256(NATIVE.read_bytes()).hexdigest()} + for name in ('THREE-base','THREE-extra'): + row=data['providers'][name] + E.need(row['all_408_helpers_autonomous'] and row['all_632_persistent_inputs_arbitrary'], + 'Authenticated arbitrary helper/data input contract') + E.need(row['external_cleanup_fields']==0 and row['owned_C_fields']==1, + 'Zero external bank and one explicitly owned center') + for orientation in ('forward','backward'): + replay=row['split_replays'][orientation]['independent_native_replay'] + profile={int(r):int(n) for r,n in replay['paid_profile'].items()} + E.need(profile==dict(E.split_profile(F.IDENTITY,5)),'Actual native split profile') + helpers={int(r):int(n) for r,n in row['helper_boundary']['widths'].items()} + E.need(helpers==dict(E.HELPERS),'Actual THREE helper inventory') + for name2,wanted in row['artifacts'].items(): + path=NATIVE.parent/name2 + E.need(sha256(path.read_bytes()).hexdigest()==wanted,'Native bytes '+name2) + bound[str(path.relative_to(ROOT))]=wanted + boundary=ROOT/row['helper_boundary']['certificate'] + wanted=row['helper_boundary']['certificate_sha256'] + E.need(sha256(boundary.read_bytes()).hexdigest()==wanted,'Helper boundary bytes') + bound[str(boundary.relative_to(ROOT))]=wanted + return bound + +def run(prefix_cut=False): + sources=source_binding() + solution=[] + for f in ((1,) if prefix_cut else (0,2)): + solution.extend({'family':f,'pattern':[a,b],'count':str(n)} for n,a,b in PAIRS) + solution.extend({'family':0 if prefix_cut else 1,'pattern':[a],'count':str(n)} for n,a in MIDDLE) + case=F.verify(2,6,solution,'identity',prefix_cut=prefix_cut) + wanted_targets=({'6':'165','7':'112','8':'186','9':'934','25':'224','41':'224'} + if prefix_cut else {'6':'146','9':'803','41':'448'}) + wanted_paid=({'1':'12584','2':'3602','3':'1217','4':'510','5':'169'} + if prefix_cut else {'1':'12565','2':'3621','3':'1217','4':'510','5':'169'}) + E.need(case['targets']==wanted_targets,'Literal output census') + E.need(case['paid']==wanted_paid,'Literal paid census') + out=F.serialize(case,('0.001730970419199163702738','0.001745','0.00175','0.00177','0.002','0.004')) + out['root_bracket']=root_bracket(case['signed']) + bad=[dict(x) for x in solution] + bad[0]=dict(bad[0],count=str(int(bad[0]['count'])+1)) + rejected=False + try:F.verify(2,6,bad,'identity',prefix_cut=prefix_cut) + except AssertionError:rejected=True + E.need(rejected,'Duplicated owner is rejected') + illegal=Counter(case['signed']);illegal[1]+=1 + E.need(sum(r*n for r,n in illegal.items())!=128,'Unfunded output rejected by rank budget') + return {'schema':'source-bound-identity-mixed-tariff/1', + 'status':'PASS_EXACT_NATIVE_BOUND_PROFILE_AND_FINITE_RECURSIVE_INVOICE', + 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(), + 'native_sources':sources, + 'enclosure_source_sha256':sha256((HERE/'mixed_target_probe.py').read_bytes()).hexdigest(), + 'invoice':out, + 'controls':{'duplicate_original_idle_owner_rejected':rejected, + 'unfunded_output_credit_rejected':True, + 'rank5_target_with_rank5_child_rejected':not(5<5)}, + 'explicitly_charged':['all five native full-identity calls', + 'all forty rank8 centers split into rank5 plus rank3', + 'each unused original idle portion', + ('rank1 completions for all ninety-three width5 fresh helpers' if prefix_cut else + 'rank1 completions for all seventy-four width5 fresh helpers'), + 'every productive output field at its actual target rank'], + 'prefix_cut':prefix_cut, + 'not_a_recursive_output':['one owned center field, erased after use'], + 'not_proved':['physical global helper placement and controller binding', + 'all-size mixed-target causal routing on fixed tapes', + 'rounded whole-caller precision and stock conversion', + 'outer integer-multiplication transfer'], + 'new_kappa':False} + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output',type=Path) + parser.add_argument('--prefix-cut',action='store_true') + args=parser.parse_args() + result=run(args.prefix_cut) + output=args.output or HERE/('IDENTITY-PREFIX-CUT-CERTIFICATE.json' if args.prefix_cut else + 'IDENTITY-MIXED-CERTIFICATE.json') + output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'root':result['invoice']['root_bracket'], + 'margin_001745':result['invoice']['invoices']['0.001745']},sort_keys=True)) + +if __name__=='__main__':main() diff --git a/research/kappa-rebuild/architecture/joint_boundary_probe.py b/research/kappa-rebuild/architecture/joint_boundary_probe.py new file mode 100644 index 0000000000..43f3ad3696 --- /dev/null +++ b/research/kappa-rebuild/architecture/joint_boundary_probe.py @@ -0,0 +1,168 @@ +#!/usr/bin/env python3 +"""Two focused joint-boundary/typed-state screens after the direct H failure. + +The two-query obstruction is a FREE-OPERATOR model, not a lower bound on +the actual native construction. The fresh-owner obstruction uses actual +phase-corrected native owners and an invariant old-coordinate sector. +Apache-2.0; substantial OpenAI assistance. +""" +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +import aggregate_h_probe as A + + +def rank(rows): + basis = {} + for row in rows: + row = list(map(Q, row)) + for pivot in sorted(basis): + if row[pivot]: + scale = row[pivot] + row = [x-scale*y for x, y in zip(row, basis[pivot])] + pivot = next((i for i, x in enumerate(row) if x), None) + if pivot is not None: + scale = row[pivot] + basis[pivot] = [x/scale for x in row] + return len(basis) + + +def universal_two_query_screen(): + """Coefficient matrices for Q K_a Q, Q K_b Q, Q K_f Q. + + K_a=H2 H1 H0, K_b=H6 H5 H4, K_f=K_b H3 K_a. The H letters + are free involutions and Q is a further free involution. The two-Q + normal forms remain different even after those stated relations. + In a circuit A E_i(Q) B E_j(Q) C with Q-free A,B,C, the coefficient + matrices of words with empty exterior and exactly two Q letters all + lie in span(A_empty[:,i] C_empty[j,:]); its dimension is at most one. + The extra three Q-free positive streams do not change that bound. + """ + words = [(2, 1, 0), (6, 5, 4), (6, 5, 4, 3, 2, 1, 0)] + matrices = [[int(i == j == k) for i in range(6) for j in range(6)] for k in range(3)] + dimension = rank(matrices) + A.need(dimension == 3, 'Three independent exact sandwich coefficient matrices') + minor = [[row[6*j+j] for j in range(3)] for row in matrices] + A.need(minor == [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 'Explicit unit determinant witness') + # Coalescing all three independent input/output roles would erase the gate. + fake = [[1] for _ in matrices] + A.need(rank(fake) == 1, 'Invalid merged-role model loses the obstruction') + return dict(middle_words=[list(w) for w in words], coefficient_span_rank=dimension, + explicit_minor=minor, two_complete_bank_Q_events_upper_bound=1, + six_stream_two_event_identity_possible=False, + scope='Only universal finite Q-free polynomial mixing and two complete-bank Q events. Actual low-rank native relations, partial-bank calls, additional transforms, or stateful recursion are not excluded.', + positive_partner_streams_included=3, + invalid_merged_independent_inputs_rank=1) + + +def phase_data(n, phases, stage, port): + g = n.source_route(phases, stage, port) + if stage == 0: + return g, 0 + axes = n.AXES[:9]+n.AXES[9+8*(stage-1):9+8*stage] + f = n.frame((a, 1) for a in axes) + old, new = n.polynomial(f)[0], n.polynomial(n.move_frame(f, g))[0] + d = sum((((x-y) % 4)//2) << i for i, (x, y) in enumerate(zip(old, new))) + A.need(n.act(g, d) == d, 'Signed owner involution') + return g, d + + +def signed_h_apply(n, phases, geo, stage, vector): + charts = [phase_data(n, phases, stage, p) for p in range(112)] + out = [] + for i, row in enumerate(geo['H']): + gi, di = charts[i] + result = {} + for p, coefficient in row.items(): + gp, dp = charts[p] + for address, value in vector[p].items(): + intermediate = n.act(gp, address) + final = n.act(gi, intermediate) + phase = ((dp & address).bit_count()+(di & intermediate).bit_count()) % 2 + result[final] = result.get(final, 0)+(-1 if phase else 1)*coefficient*value + if not result[final]: + del result[final] + out.append(result) + return out + + +def fresh_inverse_screen(n, phases, geo): + old_block = sum(n.AXES[9:17]) + incoming = [{} for _ in range(112)] + incoming[0] = {n.AXES[9]: Q(1)} + first = signed_h_apply(n, phases, geo, 0, incoming) + second = signed_h_apply(n, phases, geo, 1, first) + leakage = [(port, address, c) for port, row in enumerate(second) + for address, c in row.items() if address & old_block != n.AXES[9]] + A.need(bool(leakage), 'Old monodromy genuinely leaves the original old-block sector') + # H2 and every candidate fresh inverse H4,H5,H6 act only on original9 + # and their own disjoint block. They preserve every old-block1 sector. + preserved = [] + for stage in (2, 4, 5, 6): + for port in range(112): + g, d = phase_data(n, phases, stage, port) + A.need(all((g[i] & old_block) == (n.AXES[i] & old_block) for i in range(73)), + 'Later owner fixes the entire old-block sector label') + A.need(d & old_block == 0, 'Later character does not touch the old block') + preserved.append((stage, port)) + witness = min(leakage) + return dict(input_field=0, input_address=n.AXES[9], old_block_axes=list(range(9, 17)), + after_H1_H0_witness=dict(output_field=witness[0], output_address=witness[1], + coefficient=str(witness[2])), + nonzero_outside_sector_terms=len(leakage), later_signed_owners_checked=len(preserved), + later_operators_invertible=True, + conclusion='H2 preserves the nonzero leaked sector, as do H4,H5,H6 in every order. A fresh-axis inverse word cannot equal (H2 H1 H0)^-1.', + scope='Actual signed native H_s on these stages, without additional old-block mixing. Extra cross-boundary maps require a new exact and costed word.') + + +def overlap_accounting(): + first = set(range(25)) + same = set(first) + extension = set(range(49)) + disjoint = set(range(25, 50)) + # Two applications on the same coordinate square to a free Pauli for + # complex Gaussian factors (or identity for BIT). Only odd multiplicity + # coordinates retain a nontrivial full Gaussian target. + return dict(same_owner_forward_inverse_state_pair=dict( + first_rank=len(first), second_rank=len(same), independent_target_rank=len(first ^ same), + unearned_double_credit=len(first)+len(same)), + nested_full_union_pair=dict(first_rank=len(first), second_rank=len(extension), + independent_target_rank=len(first ^ extension), unearned_sum=len(first)+len(extension)), + disjoint_new_axes_pair=dict(first_rank=len(first), second_rank=len(disjoint), + independent_target_rank=len(first ^ disjoint), + obstruction='New native owners do not provide the same inverse monodromy; see the actual sector witness.'), + conclusion='State cancellation is useful only with a separately admitted inverse state on new axes. Reusing the old transform support does not produce two independent rank credits on one array.') + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output', type=Path, default=HERE/'JOINT-BOUNDARY-PROBE.json') + args = parser.parse_args() + native = A.ROOT/'research/kappa-next-regime/direct-decoder/native_common.py' + rebuild = HERE.parent/'complex/rebuild_native.py' + n = A.load(native, 'joint_boundary_native') + r = A.load(rebuild, 'joint_boundary_rebuild') + packet, phases = n.load_sources() + result = dict(schema='joint-H-boundary-and-stateful-inverse-screen/1', + status='PASS_TWO_SCOPED_OBSTRUCTIONS_AND_COMPLETE_OVERLAP_ACCOUNTING', + two_bank_query_screen=universal_two_query_screen(), + actual_fresh_inverse_screen=fresh_inverse_screen(n, phases, r.geometry(packet)), + typed_overlap_accounting=overlap_accounting(), + literature_inspiration='Noncommutative rational-series coefficient/Hankel ranks; the elementary two-query argument is proved in the module docstring.', + primary_source='https://fpsac-archive.github.io/FPSAC94/ARTICLES/45Reutenauer.pdf', + source_sha256={str(p.relative_to(A.ROOT)):sha256(p.read_bytes()).hexdigest() + for p in (Path(__file__), Path(A.__file__), native, rebuild)}, + no_general_lower_bound=True, no_new_H_kappa=True) + args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps(dict(status=result['status'], coefficient_rank=result['two_bank_query_screen']['coefficient_span_rank'], + fresh_inverse_witness=result['actual_fresh_inverse_screen']['after_H1_H0_witness']), sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/architecture/mixed_forest_search.py b/research/kappa-rebuild/architecture/mixed_forest_search.py new file mode 100644 index 0000000000..9b5213bf3f --- /dev/null +++ b/research/kappa-rebuild/architecture/mixed_forest_search.py @@ -0,0 +1,311 @@ +#!/usr/bin/env python3 +"""Column generation plus exact reconstruction for mixed-target idle forests. + +This searches p-pair controllers with two wide idle windows and one short +window for each j=1,...,p-1. A fresh productive job completes every unhosted +helper. The receipt checks exact rational multiplicities and the full +displayed child profile; numerical optimality is not a proof of a global +optimum. Native frame/emission and outer supplier obligations remain open. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from math import lcm +from pathlib import Path +import argparse +import json +import sys + +import mixed_target_probe as E +sys.dont_write_bytecode=True +HERE=Path(__file__).resolve().parent +IDENTITY=Counter({1:2487,2:713,3:224,4:102,5:20,8:8}) +WIDTHS=(0,5,6,7,8,9) + +def windows(p,prefix_cut=False): + """Literal pair-controller idle chronology, 224 fields per window.""" + out=[] + for j in range(1,p): + if not prefix_cut: + out.append({'name':f'early{j}','capacity':16*j,'stages':list(range(2*j))}) + out.append({'name':f'middle{j}','capacity':8,'stages':[2*j]}) + out.append({'name':f'late{j}','capacity':16*j,'stages':list(range(2*p+1-2*j,2*p+1))}) + expected=1792*(2*p+1)*(p-1)-(224*16*p*(p-1)//2 if prefix_cut else 0) + E.need(sum(224*w['capacity'] for w in out)==expected,'General idle rank identity') + return out + +def numerical_split(r,maximum,c): + return sum(n*k**(1-c) for k,n in E.split_rank(r,maximum).items()) + +def reconstruct_exact(numbers,columns,families,resource,inventory): + """Recover an exact feasible basic solution, not rounded inventories. + + Small denominators suffice for most screens. Degenerate larger masters + can have a denominator above that guess; recover their active integer + linear system and independently replay it instead of waiving an equality. + """ + support=[i for i,x in enumerate(numbers) if x>1e-7] + approximate=[Q(float(numbers[i])).limit_denominator(10**6) for i in support] + def equations(): + rows=[];rhs=[] + for f in range(len(families)): + if not families[f].get('unbounded'): + rows.append([int(columns[i][0]==f) for i in support]);rhs.append(families[f].get('field_count',224)) + for stage,r in resource: + row=[] + for i in support: + f,pattern=columns[i] + w=families[f] + row.append(sum(int(s==stage and v==r) for s,v in zip(w['stages'],pattern))) + usage=sum(x*float(numbers[i]) for x,i in zip(row,support)) + if abs(usage-inventory[r])<1e-7: + rows.append(row);rhs.append(inventory[r]) + return rows,rhs + rows,rhs=equations() + if any(sum(a*x for a,x in zip(row,approximate))!=b for row,b in zip(rows,rhs)): + n=len(support) + basis={} + for row,b in zip(rows,rhs): + vector=[Q(x) for x in row]+[Q(b)] + for pivot in sorted(basis): + factor=vector[pivot] + if factor: + old=basis[pivot] + for j in range(pivot,n+1):vector[j]-=factor*old[j] + pivot=next((j for j,x in enumerate(vector[:-1]) if x),None) + if pivot is None: + E.need(vector[-1]==0,'Active integer system consistent') + else: + factor=vector[pivot] + basis[pivot]=[x/factor for x in vector] + E.need(len(basis)==n,'Active support is uniquely reconstructed') + approximate=[Q(0)]*n + for pivot in sorted(basis,reverse=True): + row=basis[pivot] + approximate[pivot]=row[-1]-sum(row[j]*approximate[j] for j in range(pivot+1,n)) + E.need(all(abs(float(x)-numbers[i])<1e-4 for x,i in zip(approximate,support)), + 'Exact active solution agrees with numerical discovery') + E.need(all(x>=0 for x in approximate),'Reconstructed multiplicities nonnegative') + return [{'family':columns[i][0],'pattern':list(columns[i][1]),'count':str(x)} + for i,x in zip(support,approximate) if x] + +def discover(p,minimum,c,max_rounds=300,fresh_groups=False,prefix_cut=False, + custom_windows=None,custom_targets=None): + import numpy as np + from scipy.optimize import linprog + from scipy.sparse import coo_matrix + families=windows(p,prefix_cut) if custom_windows is None else [dict(w) for w in custom_windows] + calls=2*p+1 + if fresh_groups: + families.append({'name':'fresh_group','capacity':2*minimum+8, + 'stages':list(range(calls)),'unbounded':True}) + maximum=minimum-1 + sp=lambda r:numerical_split(r,maximum,c) + h=lambda r:r**(1-c) if r else 0 + v={r:h(max(r,minimum))-sp(max(0,minimum-r)) for r in WIDTHS[1:]} + v[0]=0 + def gain(f,pattern): + if families[f].get('unbounded'): + used=sum(pattern) + return (h(max(used,minimum))-sp(max(0,minimum-used)) if used else 0)-sum(v[r] for r in pattern) + return sp(families[f]['capacity'])-sp(families[f]['capacity']-sum(pattern))-sum(v[r] for r in pattern) + columns=[] + keys=set() + def add(f,pattern): + key=(f,tuple(pattern)) + if key not in keys: + keys.add(key) + columns.append(key) + return True + return False + for f,w in enumerate(families): + add(f,[0]*len(w['stages'])) + for r in WIDTHS[1:]: + # Seed every constant-width prefix and suffix that fits. + n=min(len(w['stages']),w['capacity']//r) + if n: + add(f,[r]*n+[0]*(len(w['stages'])-n)) + add(f,[0]*(len(w['stages'])-n)+[r]*n) + if not families: + return [],{'columns':0,'rounds':0,'discovery_converged':True} + resource=[(s,r) for s in range(calls) for r in WIDTHS[1:]] + rid={key:i for i,key in enumerate(resource)} + bounded=[f for f,w in enumerate(families) if not w.get('unbounded')] + bid={f:i for i,f in enumerate(bounded)} + done=False + largest_reduced=None + for iteration in range(max_rounds): + ar=[];ac=[];av=[] + er=[];ec=[];ev=[] + objectives=[] + for index,(f,pattern) in enumerate(columns): + if f in bid: + er.append(bid[f]);ec.append(index);ev.append(1) + for stage,r in zip(families[f]['stages'],pattern): + if r: + ar.append(rid[stage,r]);ac.append(index);av.append(1) + objectives.append(-gain(f,pattern)) + A=coo_matrix((av,(ar,ac)),shape=(len(resource),len(columns))).tocsr() + B=coo_matrix((ev,(er,ec)),shape=(len(bounded),len(columns))).tocsr() + result=linprog(objectives,A_ub=A,b_ub=[E.HELPERS[r] for s,r in resource], + A_eq=B if bounded else None,b_eq=[families[f].get('field_count',224) for f in bounded] if bounded else None,bounds=(0,None),method='highs', + options={'dual_feasibility_tolerance':1e-9,'primal_feasibility_tolerance':1e-9}) + E.need(result.success,'Restricted master feasible') + pi=result.ineqlin.marginals + lam=result.eqlin.marginals + new=0 + largest_reduced=0.0 + for f,w in enumerate(families): + dp={0:(0.0,())} + for stage in w['stages']: + nxt={} + for used,(score,pattern) in dp.items(): + for r in WIDTHS: + if used+r>w['capacity']: + continue + score2=score-v[r]+(pi[rid[stage,r]] if r else 0) + if used+r not in nxt or score2>nxt[used+r][0]+1e-13: + nxt[used+r]=(score2,pattern+(r,)) + dp=nxt + def terminal(used): + if w.get('unbounded'): + return h(max(used,minimum))-sp(max(0,minimum-used)) if used else 0 + return sp(w['capacity'])-sp(w['capacity']-used)+lam[bid[f]] + ranked=sorted(((score+terminal(used),pattern) + for used,(score,pattern) in dp.items()),reverse=True) + largest_reduced=max(largest_reduced,ranked[0][0]) + # Several best used-capacity patterns reduce degeneracy. + for improvement,pattern in ranked[:4]: + if improvement>1e-8: + new+=add(f,pattern) + if not new: + done=True + break + # Rational reconstruction checks feasibility independently of floating duals. + solution=reconstruct_exact(result.x,columns,families,resource,E.HELPERS) + verify(p,minimum,solution,'H',fresh_groups=fresh_groups,prefix_cut=prefix_cut, + custom_windows=custom_windows,custom_targets=custom_targets) + return solution,{'columns':len(columns),'rounds':iteration+1, + 'discovery_converged':done,'last_max_reduced_cost_float':largest_reduced, + 'optimality_scope':'Floating column-generation screen only; rational receipt certifies displayed construction, not global optimum.'} + +def verify(p,minimum,solution,kernel,fresh_groups=False,prefix_cut=False, + custom_windows=None,custom_targets=None): + E.need(kernel in ('H','identity'),'Known exact local invoice') + calls=2*p+1 + maximum=minimum-1 + families=windows(p,prefix_cut) if custom_windows is None else [dict(w) for w in custom_windows] + if fresh_groups: + families.append({'name':'fresh_group','capacity':2*minimum+8, + 'stages':list(range(calls)),'unbounded':True}) + counts=[Q(0) for _ in families] + inventory=[Counter({r:Q(n) for r,n in E.HELPERS.items()}) for _ in range(calls)] + source=E.NATIVE if kernel=='H' else IDENTITY + paid=E.split_profile(source,maximum) + paid=Counter({r:Q(calls*n) for r,n in paid.items()}) + if kernel=='H': + paid[1]+=8*calls + targets=(Counter({16*p+9-16*j:Q(224) for j in range(p)}) if prefix_cut + else Counter({16*p+9:Q(224*p)})) + if custom_targets is not None: + targets=Counter({int(r):Q(n) for r,n in custom_targets.items()}) + E.need(sum(targets.values())==224*p,'Changed main targets have the original data field count') + expected_total=sum(r*n for r,n in targets.items())-sum(w.get('field_count',224)*w['capacity'] for w in families if not w.get('unbounded')) + old_total=224*p*(16*p+9)-sum(224*w['capacity'] for w in windows(p)) + E.need(expected_total==old_total,'Literal removed windows match the changed main rank credit') + for row in solution: + f=row['family'];pattern=row['pattern'];count=Q(row['count']) + E.need(0<=f=0,'Pattern count') + w=families[f] + E.need(len(pattern)==len(w['stages']) and set(pattern)<=set(WIDTHS),'Window chronology') + E.need(sum(pattern)<=w['capacity'],'Window capacity') + counts[f]+=count + for stage,r in zip(w['stages'],pattern): + if r: + inventory[stage][r]-=count + if w.get('unbounded'): + E.need(sum(pattern)>0,'A fresh productive group consumes some actual helper') + targets[max(sum(pattern),minimum)]+=count + E.add_scaled(paid,E.split_rank(max(0,minimum-sum(pattern)),maximum),count) + else: + E.add_scaled(paid,E.split_rank(w['capacity']-sum(pattern),maximum),count) + E.need(all(n==w.get('field_count',224) for n,w in zip(counts,families) if not w.get('unbounded')), + 'Every idle window accounted') + global_roles=Counter() + for stage,stock in enumerate(inventory): + E.need(all(n>=0 for n in stock.values()),'Stage inventory '+str(stage)) + global_roles.update(stock) + for r,n in global_roles.items(): + if n: + targets[max(r,minimum)]+=n + E.add_scaled(paid,E.split_rank(max(0,minimum-r),maximum),n) + paid=+paid;targets=+targets + E.need(max(paid)= 0 else b) + high += n*(b if n >= 0 else a) + return low,high + +def show(bounds): + scale=10**35 + low,high=bounds + a=low.numerator*scale//low.denominator + b=-((-high.numerator*scale)//high.denominator) + def fixed(n): + sign='-' if n<0 else '' + s=str(abs(n)).rjust(36,'0') + return sign+s[:-35]+'.'+s[-35:] + return {'lower':fixed(a),'upper':fixed(b)} + +def split_rank(r,maximum): + """Least concave tariff partition with positive integers <= maximum.""" + need(maximum>=1 and r>=0, 'Legal split') + q,remainder=divmod(r,maximum) + result=Counter() + if q: + result[maximum]+=q + if remainder: + result[remainder]+=1 + return result + +def split_profile(profile,maximum): + result=Counter() + for r,n in profile.items(): + for child,count in split_rank(r,maximum).items(): + result[child]+=n*count + return result + +def add_scaled(target,source,multiple=1): + for key,value in source.items(): + target[key]+=multiple*value + +def base_case(p,minimum,pairs=None,middle=None): + """Expand all role inventories; one fresh global field per unhosted role.""" + need(p in (1,2), 'Only reconstructed p1 and p2 schedules are in scope') + need(2<=minimum<=41, 'Finite minimum-target menu') + calls=2*p+1 + main_fields=224*p + main_rank=16*p+9 + inventories=[Counter(HELPERS) for _ in range(calls)] + paid=split_profile(NATIVE,minimum-1) + paid=Counter({r:calls*n for r,n in paid.items()}) + paid[1]+=8*calls # same-width H completion, never waived + targets=Counter({main_rank:main_fields}) + if p==1: + need(pairs is None and middle is None, 'p1 has no original idle bins') + else: + pairs=pairs if pairs is not None else [(224,0,0)] + middle=middle if middle is not None else [(224,0)] + need(sum(n for n,a,b in pairs)==224, 'Each pair window has 224 bins') + need(sum(n for n,a in middle)==224, 'Middle window has 224 bins') + for stages in [(0,1),(3,4)]: + for n,a,b in pairs: + need(a+b<=16 and a in (0,5,6,7,8,9) and b in (0,5,6,7,8,9), 'Pair capacities') + for stage,r in zip(stages,(a,b)): + if r: + inventories[stage][r]-=n + add_scaled(paid,split_rank(16-a-b,minimum-1),n) + for n,r in middle: + need(r in (0,5,6,7,8), 'Middle capacity') + if r: + inventories[2][r]-=n + add_scaled(paid,split_rank(8-r,minimum-1),n) + global_roles=Counter() + for inventory in inventories: + need(all(n>=0 for n in inventory.values()), 'No helper role used twice') + global_roles.update(inventory) + for r,n in global_roles.items(): + if not n: + continue + target=max(r,minimum) + targets[target]+=n + add_scaled(paid,split_rank(target-r,minimum-1),n) + paid=+paid + targets=+targets + need(max(paid)0] + # Middle bins are independent single slots; choose positive-gain roles first. + candidates=sorted(((sp(8)-sp(8-r)-value(r),r) for r in (5,6,7,8)),reverse=True) + middle=[] + left=224 + for gain,r in candidates: + if gain>1e-12: + n=min(HELPERS[r],left) + if n: + middle.append((n,r)) + left-=n + if left: + middle.append((left,0)) + return pairs,middle,{'status':int(out.status),'message':out.message,'not_an_exact_optimality_claim':True} + +def run(discover=False): + candidates=[] + for p in (1,2): + for minimum in range(2,26): + candidates.append(base_case(p,minimum)) + candidates.append(base_case(2,9,OLD_PAIRS,OLD_MIDDLE)) + discovery=[] + if discover: + for c in (.001745,.004): + for minimum in range(5,18): + pairs,middle,meta=discover_pair_patterns(minimum,c) + case=base_case(2,minimum,pairs,middle) + case['discovery_target']=str(c) + case['solver']=meta + discovery.append(case) + candidates.extend(discovery) + best={} + for p in (1,2): + for text in ('0.001745','0.004'): + c=Q(text) + rows=[case for case in candidates if case['p']==p] + ranked=sorted(rows,key=lambda case:-numerical_value(case['signed'],float(c))) + winner=ranked[0] + # Certify that this displayed winner beats all numerically distinct + # profiles in the finite scanned menu, not all possible schedules. + win=moment(winner['signed'],c) + for case in ranked[1:]: + if dict(case['signed'])!=dict(winner['signed']): + need(win[0]>moment(case['signed'],c)[1], 'Exact finite-menu ordering') + best[f'p{p}_{text}']=serialize_case(winner) + controls={} + wrong=base_case(1,6) + wrong['targets'][6]+=1 + controls['unearned_output_credit_rejected']=(sum(r*n for r,n in wrong['targets'].items())-wrong['rank_bill']!=wrong['rank_margin']) + controls['rank5_child_with_job5_rejected']=not(5<5) + need(all(controls.values()), 'Corruption controls') + return {'schema':'productive-helper-mixed-target-tariff/1', + 'status':'PASS_CONDITIONAL_EXACT_ACCOUNTING', + 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(), + 'native_profile':dict(NATIVE),'helper_inventory':dict(HELPERS), + 'assumed_recursive_rule':'sum_paid n_r r^(1-C) < sum_targets W_k k^(1-C); max_paid_rank < min_target_rank', + 'best_in_finite_menu':best, + 'fixed_p2_minimum9':serialize_case(base_case(2,9,OLD_PAIRS,OLD_MIDDLE)), + 'discovery_candidates':[serialize_case(c) for c in discovery], + 'controls':controls, + 'not_proved':['native whole-helper adapters and arbitrary-input block diagonal contract', + 'correct main controller and all its heads/caches/cleanup', + 'literal eligibility of any displayed head-cut grant', + 'mixed-target all-size recursion in this native model', + 'finite-precision integer-multiplication transfer and scalar/stock costs'], + 'new_kappa':False} + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--discover',action='store_true') + parser.add_argument('--output',type=Path,default=HERE/'MIXED-TARGET-PROBE.json') + args=parser.parse_args() + result=run(args.discover) + args.output.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'best':{k:v['targets_at_C'][k.split('_',1)[1]]['after_native_completion_before_main_closure'] for k,v in result['best_in_finite_menu'].items()}},sort_keys=True)) + +if __name__=='__main__': + main() diff --git a/research/kappa-rebuild/architecture/sources/COMPLEX-CONTROLLER-FACTOR-p1-p5.json b/research/kappa-rebuild/architecture/sources/COMPLEX-CONTROLLER-FACTOR-p1-p5.json new file mode 100644 index 0000000000..7afc93ade4 --- /dev/null +++ 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Apache-2.0 ([NOTICE](NOTICE)). Built with AI agents (Claude); see *Credit* below. + +## 1. What this is + +Marius-Paul Dumitrean's #2979 (`research/rank7-joint-endframes`, on main) prices its words at a complex-side root of +b = 1.7290339826e-3. This package changes nothing in those words or their packing, and in its registered +instance 2979 nothing in their allocation; the headline A-fc (section 9) carries a re-solved allocation (table of +hosts) for the same words. It changes the +**recursion** that turns a word into a running time, and with that change the same words give + +| | figure | status | +|---|---|---| +| #2979 as it is on main (one job size, 73 axes) | 1.7290339826e-3 | the community's own checks; reproduced here to 24 digits | +| **#2979 with the lever (jobs of 73 and 49 axes)** | **1.7556473153e-3** | the registered check: price, heads and rate inequality checked by computer here; the DFT bound at 1.7556473e-3 is proved in Lean (section 5) | +| the same lever on Dugongue's #3945 | 1.7501109246e-3 | checked by computer with the same programs (section 6); proved in Lean at 1.7501109e-3 | +| **the lever on #3945 with every head kept** (#3945's words and price, a re-solved table of hosts) | **1.7581520168e-3** | checked by computer with the same programs and the table this package carries (section 7); the DFT bound at **1.7581520e-3** is proved in Lean (section 5) | +| **the full cut on #3945 with every head kept** (the same table; heads removed, job 25 on their arrays; the other rank-48 blocks split 24 + 24) | **1.7806935406e-3** | checked by computer with the same programs (section 8); the DFT bound at **1.7806935e-3** is proved in Lean (section 5), the highest of the five | +| the lever on #2979 with every head kept | 1.7564708e-3 | proved in Lean at 1.7564707e-3 (section 7); not replayed in this package | +| **the full cut on #2979's words with a re-solved table (A-fc)** (#2979's words, order and dispatch unchanged; a new table of hosts carried here, with its own paid profile; heads removed, job 25 on their arrays) | **1.7887045643e-3** | **the headline of this package**: checked by computer here, as registered commands (section 9); **not proved in Lean**. Its own figure with no lever is 1.6897030e-3, below #2979's, so the community's admission rejects it at its strict-gain steps (section 9) | + +The highest figure here, A-fc's 1.7887046e-3, is checked by computer and not proved in Lean; the highest figure proved +in Lean is the full cut on #3945 with every head kept, 1.7806935e-3. The figure is a saving for the **discrete Fourier transform** in OpenAI's exact-arithmetic machine model, the model +of OpenAI's "Exact Fourier transforms below n log n" (openai/math, family 130): the program computes the DFT of every +length n in O(n (log n)^(1-s)) operations with s = 1.7806935e-3. It is the same kind of number as the community's +complex side b, so the comparison above is b against b. It is not a κ for integer multiplication: whether the outer +step accepts a recursion with two job sizes is open (section 4), and no κ is claimed. + +## 2. The idea in one paragraph, and the numbers + +The community's recursion gives every array of a unit a full job of 73 axes. In #2979's unit, 16,086 of the 28,190 +rank-48 blocks per round are **literal heads**: each is the first non-identity operation on its own array, acting on +that array alone, with no helper hosted there before it. On such an array the unit can take a smaller job instead. +I cut each head by 24 of its 48 axes, so that its array takes a job of 49 axes and the head pays a block of rank 24; +every other array keeps the full job of 73. The recursion must then feed the word with jobs of two sizes in a fixed +ratio (66,304 arrays per round: 50,218 full jobs and 16,086 cut jobs). Theorem 1G ([PROOF.md](PROOF.md)) shows that +such a *mixed-target recursion* runs at the speed its balance predicts, provided every paid block is smaller than every +job (here 48 < 49). Its balance is the root b of + + sum_r n_r (r/73)^(1-b) = 50,218 + 16,086 (49/73)^(1-b), + +with the paid blocks n_r = {1: 1,645,680, 2: 470,196, 3: 159,468, 4: 71,928, 5: 13,716, 8: 5,328, 24: 16,086, +48: 12,104} per round. Cutting only half of each head (not all of it) is what keeps every paid block below the +smaller job here. I call this step *the lever*. Cutting the whole head (a job of 25 axes) needs every other paid block +below 25 as well: on #3945 with every head kept, splitting each remaining rank-48 block into two blocks of 24 on the +same role does that, and that *full cut* is the highest figure proved in Lean here (section 8). On #2979's words a +table of hosts re-solved for the full cut (A-fc) has no rank-48 block other than its 16,576 heads, so nothing needs +to be split, and the full cut prices at 1.7887045643e-3 (section 9, checked by computer, not proved in Lean). + +## 3. The registered check + + python3 -B research/lever-mixed-targets-2979/verify.py --part main # about 50 s on a laptop + python3 -B research/lever-mixed-targets-2979/verify.py --part controls # about 5 s + python3 -B research/lever-mixed-targets-2979/verify.py --instance 2979Fc --part main # A-fc, section 9 + python3 -B research/lever-mixed-targets-2979/verify.py --instance 2979Fc --part controls + +The last two run the same checks on the headline A-fc (section 9). Standard library only; it reads #2979's ten files on main (pinned in `SOURCES.json`, declared in `VERIFY.json`) and +writes only to a temporary folder. What it checks, in order: + +1. **Pins.** Every input and every data file of this package equals its recorded sha256; the stated local profile, + widths, zero buffers and complex root are read from #2979's `CLAIM.json` and `allocation/PROVIDERS.json`. +2. **My conversion of both words** (gauge44 and extra). `tools/transport.py` rebuilds the community's native + canonical form from the borrowed word (the PR does not ship one) and replays it exactly on all 632 columns, in + both orientations. `tools/lower_w.py` lowers it to my certificate format: frames nested at every climb, warps + carried by their maps, residual widths equal to #2979's widths helper by helper, the block histogram equal to + the local profile, the whole program replayed to the community's endpoint (91 zero buffers back at zero), and + seven wrong versions of each word rejected. For #3945 the same transport reproduces the community's own native + canonical words exactly (equal content hashes), which is how it was validated. +3. **The head test** (`tools/g0_staged.py`) in my network: the step order of my nine-stage hosted network, read as + data from its Lean source (`network/`; nothing is compiled), run on #2979's ORDER, DISPATCH, PLAN, TEMPLATE and all + 271,728 FILL rows. Every role lies in a window of my network, no bin exceeds its rank, every role is used once + per call, and the global paid profile equals `allocation/PROFILE.json` digit for digit. It finds **16,086 literal + heads** (8,064 in child 8, 8,022 in child 10) and 12,104 other rank-48 blocks (children 12 and 13) that are + neither heads nor tails. Its controls: moving gate A4 before its climb leaves no heads; deleting gate C2 turns + the 12,104 other blocks into tails. +4. **The price** (`tools/price_ours.py`, with `pricer.py`, an exact re-implementation of the community's moment + engine, and `mixed.py`): the uniform root of my profile equals #2979's `complex_lower` to all 24 digits; the half + cut gives 1.7556473153e-3; the full cut is priced as a what-if only. +5. **The rate certificate** (`tools/rate_cert.py`): on the half-cut profile derived from the measured heads (not + typed in), exact rational enclosures prove the inequality of section 2 at the figure 17,556,473/10^10 and + refute it one grid step higher. The derived profile equals the histogram of the Lean rate module. +6. **Controls** (second command): the head test on the earlier T74 circuit (`controls/t74`, from Jeremy Kothe's + #1839) finds the 11,010 heads (6,150 + 4,860) of the first head test; the figure is refused with a single head, + with one head fewer (16,085), for a profile that is not separated, and one grid step above. + +## 4. What changes from the community's model, and what does not + +- **Changed: the recursion.** The community's recursion is one call per job, and every paid block is an independent + call of the same job size; the ceiling tool assumes this. The mixed-target recursion needs several job sizes at + once, fractional portions of batches, a tree of classes run in lockstep, a top split and exit calls + ([PROOF.md](PROOF.md) section 3). As the card `complex-new-helper-inventory` allows when the coverage law changes, + Theorem 1G is given as the replacement bound, and the incumbent ceiling is not applied. +- **Unchanged: the words, the packing, the endpoint.** No gate moves; no axis moves between children. The cut changes + the job taken by an array whose first operation is the head; the rest of the word is the same. The allocation is + #2979's own in the registered instance 2979 and #3945's own in instance 3945; the instances 3945H and 3945F and the + headline A-fc (2979Fc) each carry a re-solved table of hosts (sections 7 and 9), checked as a table by the + community's own programs. +- **Not checked here: the community's eight-stream operator model.** The heads and the cut are checked in my + lowering and my network (and, for the DFT, proved in Lean, section 5), not in the community's exact operator + replay. The idle-axis moves retracted on 18 and 19 October changed that operator; this package does not exchange + idle blocks, but the check in that model has not been run (open question 2). +- **κ.** None is claimed. For orientation only: b = 1.7556e-3 is above the selected bit side c = 1.7334270e-3 + (#589), so if the outer step accepted this supplier the bit side would bind, at κ = 1.7334270e-3 in the newer + convention or 1.7304275e-3 in the older one. The halving guard m^2 > 2R^2 of the outer recipe, read per array, holds + for the half cut (on a head array the remaining paid block has 24 axes and 49^2 > 2 * 24^2; argued, not checked) + but fails for the full cut, the headline included (25^2 = 625 < 2 * 23^2 = 1,058; < 1,152 on #3945), so the outer + recipe as written does not take the full cut; one more reason no κ is claimed. + +## 5. The Lean proofs (outside this repository) + +The DFT bounds are proved in Lean 4 in my public repository +[jacobalansussman/wht-power-saving-lean](https://github.com/jacobalansussman/wht-power-saving-lean), commit +`563b4e6e913625637f36ba7869bf5fcd0dd4975a`, one package per instance: `package/leverF` (the full cut on #3945 with every head kept, the +highest), `package/leverB` (#3945 with every head kept, 1.7581520e-3), `package/leverA` (#2979 with every head kept, +1.7564707e-3), `package/lever2979` (#2979, 1.7556473e-3) and `package/lever` (#3945, 1.7501109e-3). For the highest, +the statement is OpenAI's `UniformFourier` statement with only the exponent (1 - 17806935/10^10) and the names changed +(statement file sha256 `2e32984d9d6a269b19d2fabafc9085360284d352d4bdccf90e4dff5786061f35`); the theorems are +`OAI.PowerSaving.transform_mainL3945F` and `OAI.PowerSaving.convolution_mainL3945F`, with only the axioms `propext`, +`Classical.choice` and `Quot.sound`. The other four are the same statement at their own exponents. The same statement one +grid step stronger is refused. The official comparator, with the Lean kernel and nanoda, accepts it against OpenAI's statement at this exponent and rejects the statement one step stronger. It is prepared for the Lean leaderboard +bigospeedrun.com/exact-dft, whose robot re-checks it independently: a pull request with it is meant to be opened alongside this package, and I will add its link and the robot's receipt as a comment once it is open. + +This repository cannot pin bytes outside its own Git history, so the link is by content: every program in `tools/` +(`*.py`) and every file under `network/` of this package equals, byte for byte, a file of that commit (`SOURCES.json` lists the pairs of +paths), and the Lean rate modules `Rate2979`, `Rate3945H` and `Rate3945F` there state the inequality of section 2 for +the profiles recorded in `instances/2979.json`, `instances/3945H.json` and `instances/3945F.json`. Lean is not run by +this check. + +## 6. The second instance: #3945 + +The same programs on Dugongue's #3945 (`research/staged544-terminal-gap-fill`, head `91f81d13`, archived on main on 20 October 2026 with the same files): +57,312 literal heads (28,672 + 28,640), the global profile equal to #3945's, the uniform root equal to its stated +root to 24 digits, and a half-cut price of **1.7501109246e-3**, certified at 17,501,109/10^10. It is not a registered +check here: when this package was made, #3945's files were on an open pull request. Its folder on main is the +default source of the instance, so it replays from the repository root with + + python3 -B research/lever-mixed-targets-2979/verify.py --instance 3945 + +(about 3 minutes; every input is checked against `instances/3945.json`; on a checkout of #3945's head, add +`--source `). The DFT bound at 1.7501109e-3 on #3945 is +proved in Lean too, in the same repository (`package/lever`). On #3945's published table the lever gives less than +on #2979's, from a unit five times the size (360 replicas against 74); with every head kept it gives more than on either published table +(section 7), and the full cut on that table gives the most (section 8). + +## 7. Every head kept: #3945 with a re-solved table of hosts + +The community's programs that decide where helper storage goes (the table of hosts, `owner/FILL.json.gz`) put each +empty rank-48 block either where it is a head or where it never is, at the same price, as their solvers happened to: +on #3945, 57,312 of the 80,640 blocks of children 8 and 10 end up as heads. I re-ran the community's own allocation +programs (Erwan Secousses' / ChaloupeMusic's, from #3815, as #3945 uses them) with one added condition, that every +block that can be a head stays empty. The result, `fills/3945H/FILL.json.gz`, keeps #3945's words, order, dispatch, +plan, template and global profile (`PROFILE.json` is unchanged), so the circuit's own price stays #3945's +1.7306632365e-3; #3945's complete stated verification (`owner/verify.py` and `admission/full_replay.py`) passes on +it from scratch, with all 119 of its controls rejected (checked by computer, the community's own programs). Then +**80,640** blocks are heads (40,320 in child 8, 40,320 in child 10), and the half cut prices at **1.7581520168e-3**: + + python3 -B research/lever-mixed-targets-2979/verify.py --instance 3945H + +(about 6 minutes; the same checks as section 3 on #3945's files and this package's table: 80,640 literal heads, +57,312 other rank-48 blocks, the global profile equal to #3945's, the uniform root equal to its stated root to 24 +digits, and the rate certificate at 17,581,520/10^10, refused one step higher). The DFT bound at 1.7581520e-3 is +proved in Lean (section 5). The commands that made the table are in the Lean repository, `package/leverB/fill/`. +On #2979 the same condition gives 16,576 heads instead of 16,086 and 1.7564708e-3 (proved in Lean at 1.7564707e-3, +one grid step lower, because the Lean rate enclosure stops just short of the rounded price); its table and the +community's complete replay of #2979 on it are in `package/leverA/fill/` of the Lean repository. + +## 8. The full cut on #3945 with every head kept + +On the same table, each of the 80,640 heads can be removed entirely instead of halved: its array then takes a job of +73 - 48 = 25 axes. The recursion needs every paid block smaller than every job, so the other 57,312 rank-48 blocks of +the unit (children 12 and 13, whole idle bins that host no role) are each split into two blocks of rank 24 on the same +role. A block of rank 24 + 24 is the product of its first 24 and its last 24 directions, so the split changes no +output; it only makes the recursion pay two blocks of 24 where it paid one of 48. Every paid block then has rank at +most 24 < 25. The balance becomes + + sum_r n_r (r/73)^(1-b) = 241,920 + 80,640 (25/73)^(1-b), + +with n_r = {1: 8,041,680, 2: 2,306,880, 3: 729,648, 4: 353,160, 5: 69,120, 8: 25,920, 24: 114,624} per unit, and its +root is **1.7806935406e-3**, 1.28 % above the half cut on the same table and 2.89 % above #3945's own figure. On #3945's +published table the full cut does not pay (1.7307680e-3, break-even 57,262 heads against 57,312); with every head +kept it does. + + python3 -B research/lever-mixed-targets-2979/verify.py --instance 3945F + +(about 3 minutes; the same checks as section 7, then the full-cut price, the profile derived from +the measured heads equal to the Lean rate module `Rate3945F`, the rate certificate at 17,806,935/10^10, refused one +step higher, and as controls the figure refused with a single head, with one head fewer, and without the split.) The +DFT bound at 1.7806935e-3 is proved in Lean (section 5, `package/leverF`); in Lean the split is a general identity +(a block of rank a + b is the product of two blocks of ranks a and b), with no hypothesis on the block. + +## 9. The headline: the full cut on #2979's words with a re-solved table (A-fc) + +**Status: checked by computer, not proved in Lean.** #2979's words, order, dispatch, plan and template are unchanged +and read in place from main; only the table of hosts is new (`fills/2979Fc/FILL.json.gz`, with its paid profile +`fills/2979Fc/PROFILE.json`). I re-solved it for the full cut: a class solution of a mixed-integer program on +#3815's Hall model, then a literal placement with #3815's own reverse-time placement (Erwan Secousses' / +ChaloupeMusic's `construction/build_fill.py`) run in a tolerant mode: a role with no planned slot left at its call +goes to the cheapest spare place, a bin open at that call that is not a head bin, not yet used at that call, and has +room in its final capacity. **Nine roles were placed that way** (8 of width 6 at call 553 and 1 of width 4 at call 0), +and 8 planned width-6 slots stayed idle. These are exactly the window, same-call and capacity rules that the +community's owner engine enforces. However the table was made, it is checked as a table: it passes #2979's unmodified +owner engine (PR2839, `allocation/vendor/independent_owners.py`, 271,728 roles, T = 74), and that engine rejects all +seven of its corruption controls. + + python3 -B research/lever-mixed-targets-2979/verify.py --instance 2979Fc --part main + python3 -B research/lever-mixed-targets-2979/verify.py --instance 2979Fc --part controls + +(the same checks as section 3, on #2979's files and this package's table: **16,576 literal heads** (8,288 in child 8, +8,288 in child 10) and **no other rank-48 block**, the global profile equal to `fills/2979Fc/PROFILE.json`, the +uniform root equal to the community's own root for this table to 24 digits, then the full-cut price, the full-cut +profile derived from the measured heads, and the rate certificate at 17,887,045/10^10, refused one step higher; as +controls, the figure is refused with a single head, with one head fewer, and with the heads kept as blocks of 48 +while their arrays take the job of 25.) Every head is removed, its array takes a job of 73 - 48 = 25 axes, and every +paid block has rank at most 23 < 25, so the recursion is separated. The balance is + + sum_r n_r (r/73)^(1-b) = 49,728 + 16,576 (25/73)^(1-b), + +with n_r = {1: 1,630,639, 2: 470,196, 3: 154,512, 4: 71,928, 5: 11,988, 6: 5, 7: 3, 8: 5,328, 17: 8, 19: 1, 23: 25,905} +per round, and its root is **b = 1.7887045643e-3** (three routes, exact rational enclosures on a 1e-24 grid, 60-digit +arithmetic and an independent 70-digit interval program, agree to 3e-60). + +**Why the community's admission rejects it.** The table's own figure with no lever (the uniform root of its paid +profile, as the community's exact moment auditor computes it) is **1.6897030e-3**, below #2979's 1.7290340e-3: this +table pays off only with the full cut. The community's checks of #2979's line, run unmodified on a fresh clone at the +base, reject it at every step that compares that lower figure with a higher one, by design, and nowhere else: + +| check of #2979's line (unmodified) | A-fc | +|---|---| +| `verify.py` (three-stage local replay of the words, 86 controls) | pass | +| `check_allocation.py`: pins, words, PROVIDERS, the PR2839 owner engine, the physical count 66,813, its 5 rejection controls | pass | +| `check_allocation.py`: the exact moment auditor (`tools/audit_allocation_moment.py`): rank mass, root bracket, exact bisection | pass, root 1.6897030e-3 | +| `check_allocation.py`: the auditor's strict-gain step (`audit_allocation_moment.py` line 65) | **rejected**: `strict frontier improvement and lower endpoint` | +| `check_cap.py`: cap certificate, rank mass, local profile paid, root at most the cap | pass | +| `check_cap.py`: `recorded gap` (line 68) | **rejected**: this is a near-optimality bound on #2979's own claim (the root within 1e-6 of the cap 1.7290358e-3), not literally a strict-gain check; A-fc's gap is 3.93e-5, the same lower figure measured against the cap | +| complete replay (`reproduce.py`): exports, fresh retained bootstrap, the native stages, owners, parent lift, rebind, discovery (7 steps), outer ledger (63 positive checks), independent check | pass | +| complete admission (`admit_extension.py` line 388) | **rejected**: `candidate must strictly improve full exact pinned frontier headline before full release` | +| the release stage, run past that step by my own driver: its `verify.py` (139 strict controls rejected) and outer checks | pass; then **rejected** at its strict-gain list over earlier packages, at `ledger kappa does not exceed PR2269` | + +The same harness passed #2979's published table and design A completely (positive controls). + +**Why the community's moment engine cannot price this figure.** The community's moment engine asserts the halving +constraint m^2 > 2 r_max^2. For the uniform model of this table it holds (73^2 = 5,329 > 2 * 48^2 = 4,608; its largest rank is the +48 of the 16,576 heads), and that is the 1.6897030e-3 above. With the cut arrays' job of 25 it fails +(25^2 = 625 < 2 * 23^2 = 1,058; after the cut the largest paid rank is 23), so the community's engine +cannot price A-fc's model with two job sizes at all. The figure 1.7887046e-3 comes from my own pricing under +Theorem 1G (`tools/price_ours.py`, `tools/mixed.py`, `tools/rate_cert.py`), which has no such condition: the full cut on +#3945 with every head kept, with 25^2 = 625 < 2 * 24^2 = 1,152, is proved in Lean the same way (section 8). + +**What is proved in Lean and what is not.** Proved in Lean, in the Lean repository of section 5: Theorem 1G, the +recursion with jobs of two sizes (`Eng.engine_seam_final`), and the general theorem of the full cut +(`cut_heads_full` in `Work/Mixed/Bridge/CoreF`), as used for the full cut on #3945 with every head kept at +1.7806935e-3. **Not in Lean: A-fc's own instance**, that is, its heads-first certificate, the cut fact on +#2979's words, and the rate fact at 17,887,045/10^10. Its heads-first certificate and the rate fact are checked by +computer (this section's commands); the cut fact on #2979's words is argued by hand from the template it shares with +#3945's (where it is proved in Lean), not checked. + +**What is claimed.** b, the complex side only, for the DFT in OpenAI's model, checked by computer. No κ: even if the +outer step accepted this supplier, κ would be capped by the selected bit side, about 1.7334e-3 (#589). The figure is +above every complex-side construction claimed on the hub by others; some research notes there print higher ceilings, +targets or hypothetical ledgers, which are not constructions. + +## Open next questions + +The structured list with first steps is `open_questions` in [TASK.json](TASK.json); in short: + +1. **The outer step with two job sizes.** Does the outer theorem accept a complex supplier whose recursion uses jobs + of 73 and 49 axes (the half cut), or of 73 and 25 axes (the full cut, the headline)? First step: write the mixed recursion as a supplier interface in the outer theorem's terms and + list the changed hypotheses (PROOF.md section 3). +2. **The cut in the community's operator model.** First step: replay the cut of a sample of heads with #3187's exact + eight-stream operator replay, with a control that a wrong cut changes the operator. +3. **A-fc in Lean.** The full cut on #2979's words is checked by computer with the re-solved table A-fc (section 9, + 1.7887046e-3), and proved in Lean on #3945 with every head kept (section 8). First step: build A-fc's instance in + Lean the way the full cut on #3945 was built (its heads-first certificate, the cut fact on #2979's words, the rate + module at 17,887,045/10^10). +4. **More heads.** Keeping every head that the table of hosts allows is done (section 7). Still open: whether some + of the remaining 'neither' blocks (children 12 and 13) can become heads by moving the first touch of their array. + First step: for each, find that touch before its climb and test whether it can be moved after it. +5. **Newer words** (for example #4088). First step: write `instances/.json` and run the head test. +6. **The log factor of Theorem 1G.** First step: search small separated rules for a flow whose exits carry less + than the sources (PROOF.md, remark after Theorem 1G). + +## Credit + +- **The words, allocation and checks this package reads** are Marius-Paul Dumitrean's #2979 (with substantial AI + assistance, Anthropic Claude via Claude Code), on Jeremy Kothe's #2957 / #2948 / #2966 rewrites, Mason Hale's #2917, + Jeremy Diviney's #2908, Dugongue's #2969, #2863 and #2839, Yiyang Lu's #2857 / #2848, RossRichardson's #2918, + Jeremy Kothe's #2894 / #2833, Tekalign's owner checker, icekylinx's auditor, themilkman216's integrality + observation and Gareth Davidson's #1995 / #2013, as #2979's NOTICE and CREDITS.md name them. +- **The borrowed helpers and their warps** that #2979 uses are Cinna Davis's (1337isnot1337, #2226), with Jeremy + Kothe's width exchange (#2312); the T74 layout underneath grows from the work of Rohan Arun, Marius-Paul Dumitrean, + Jeremy Kothe, LeverRank, Thomas DiFiore, Jeremy Diviney, Chafik Boukhalfa (chafreaky) and Vlado Vojdanovski. + The T74 control data come from Jeremy Kothe's #1839. +- **The headline's table** (`fills/2979Fc/`) was placed with the construction of Erwan Secousses (ChaloupeMusic, + #3815), its Hall model and reverse-time placement, run in a tolerant mode of mine, on #2979's own files. +- **The second instance** is Dugongue's #3945 (with substantial OpenAI Codex assistance, as its NOTICE says) and the + line of circuits it ends, as its NOTICE names them: dataisfire's #3734, zxv's #3682 and #3638, Ross Richardson's + #3605, Erwan Secousses' (ChaloupeMusic's) #3815, hexapode's #3802 and #3875, Dugongue's #3857 and #3929, and the + earlier line under them. I read and re-ran the checks of #3945 by others, among them Chris R's (Clear1nsight's) + audits #3969, #3988, #3994 and #4016. The table + with every head kept (`fills/3945H/`) was made with the allocation programs of Erwan Secousses (ChaloupeMusic, + #3815) that #3945 uses, from #3945's own files. +- **The moment engine** that `tools/pricer.py` re-implements is the community's + (`research/native-complex-moment552`, `research/gauge44-frames-x`). +- **The Lean route** rests on Ryan Shea's conversion pipeline, generators and Lean tree, Nil Mamano's gauged Lean + extension and Fourier chain templates, and OpenAI's model and statement. `network/Gates.lean` is a file of Ryan + Shea's tree; its header credits Dugongue (#401), Lydakis (#275), jacklightChen (#377) and DaysSky (#363). +- **New here** (mine): the lever (the head test, the half cut, the full cut with the split, the mixed-target + recursion and Theorem 1G), the tables with every head kept (the added condition), the table A-fc (its class solution and the tolerant placement), + my transport and lowering of these words, the rate + certificate, and the Lean proofs of section 5. The new files were + written by AI agents (Claude) under my direction; the Lean proof is checked by Lean's kernel, and no person outside + this project has reviewed this package. + +## Files + +| path | what | +|---|---| +| `verify.py` | the registered check (both parts) and the replays of the other instances | +| `instances/2979.json`, `instances/3945.json`, `instances/3945H.json`, `instances/3945F.json` | each instance's input paths and sha256, the stated values, and the recorded results (`3945F`: the full cut on the table of `3945H`) | +| `fills/3945H/FILL.json.gz` | the table of hosts of #3945 with every head kept (section 7) | +| `instances/2979Fc.json`, `fills/2979Fc/FILL.json.gz`, `fills/2979Fc/PROFILE.json` | the headline A-fc: its instance, its table of hosts and that table's paid profile (section 9) | +| `tools/` | `transport.py`, `lower_w.py` (my conversion), `g0_staged.py` (head test), `pricer.py`, `mixed.py`, `lever_price.py`, `price_ours.py` (price), `rate_cert.py` (rate certificate); [tools/README.md](tools/README.md) says where each comes from | +| `network/` | the Lean source of my nine-stage hosted network, read as data by the head test | +| `controls/t74/` | the T74 regression data | +| `PROOF.md` | Theorem 1G and what the mixed recursion needs | +| `RESULT.json` | the result in machine-readable form (no κ field) | +| `TASK.json`, `SOURCES.json`, `VERIFY.json`, `NOTICE`, `.gitattributes` | package metadata | + +## Sources + +- #2979: `research/rank7-joint-endframes` on main, head be191346391661f4d35fc0d1508d0a6029572576; the ten inputs + with their Git blobs and sha256 in `SOURCES.json`. #3945: head 91f81d131252db704f945891da1ea54e29d46b8b. The + table with every head kept: `fills/3945H/FILL.json.gz`, sha256 + 9c06677d66b04436b02ec8d88036327bdc8fb12e6fd4f71a1b9c7e915c10c2d3. The headline's table (A-fc): + `fills/2979Fc/FILL.json.gz`, sha256 7fc766c6c3e1561cee60b258138a918020b96d85ea782a2b01ad0ee035fd082e; the programs + that made it are in the Lean repository, `research/record-track/programs/a-fc/`, with the class solution and the + re-create commands in `research/record-track/data/a-fc/`. The community's checks of section 9 were run on a fresh + clone at the base on 20 October 2026. +- The package's base commit is `SOURCES.json` `base_commit`. Hub figures as their authors state them, read on + 20 October 2026, 03:00 EDT. +- The Lean repository: commit `563b4e6e913625637f36ba7869bf5fcd0dd4975a`; leaderboard submission a pull request to bigospeedrun/verifier adding submissions/dft/jacobalansussman-lever-3945-full-cut. diff --git a/research/kappa-rebuild/architecture/sources/PR4268/lean-tree.json b/research/kappa-rebuild/architecture/sources/PR4268/lean-tree.json new file mode 100644 index 0000000000..469e3493b3 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/PR4268/lean-tree.json @@ -0,0 +1 @@ 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\ No newline at end of file diff --git a/research/kappa-rebuild/architecture/sources/PR4268/tools/mixed.py b/research/kappa-rebuild/architecture/sources/PR4268/tools/mixed.py new file mode 100644 index 0000000000..c22ba942a5 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/PR4268/tools/mixed.py @@ -0,0 +1,24 @@ +# lever-reprice: mixed-target moment root, the same formula as rl-plan/tools/rescue.py (b = 1 - tau, tau the root of +# sum_children n_r r^tau = sum_arrays k_w^tau), with W and targets passed in. Pure-target {73: W} is the hub's root. +from decimal import Decimal as D, getcontext +getcontext().prec = 60 +_ln = {} +def _l(x): + if x not in _ln: _ln[x] = D(x).ln() + return _ln[x] +def root(children, targets, lo=D('0.97'), hi=D(1), iters=200): + ch = [(D(n), _l(r)) for r, n in children.items() if n] + tg = [(D(n), _l(k)) for k, n in targets.items() if n] + for _ in range(iters): + x = (lo + hi) / 2 + lhs = sum(n * (l * x).exp() for n, l in ch) + rhs = sum(n * (l * x).exp() for n, l in tg) + if lhs > rhs: lo = x + else: hi = x + return 1 - (lo + hi) / 2 +def add(d, r, n): + d = dict(d) + if n: d[r] = d.get(r, 0) + n + return d +def sep(children, targets): + return max(r for r, n in children.items() if n) < min(k for k, n in targets.items() if n) diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/LICENSE b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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OpenAI did not write, review or +endorse it. The package name "OAI" and the namespace "OAI.PowerSaving" are +those of openai/math and are kept only because the new files extend that +development. + +This repository is not a project of CrocSwap/integer-mult-bounds either. Its +authors did not write, review or endorse it. Sections 6, 9 and 10 say what +is taken from their repository. + +This repository contains files from, and files derived from, the repository +"openai/math" (https://github.com/openai/math), family 130, "Exact Fourier +transforms below n log n", at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. +That repository is published under the Apache License, Version 2.0. A copy of +the licence is in the file LICENSE. The upstream repository has no NOTICE file +and its licence text names no copyright holder; the files are attributed here +to the authors of openai/math (OpenAI). + +1. Unmodified upstream files. + The 89 files under OAI/Computability/ are byte-for-byte copies of the files + with the same paths under lean/ in openai/math at the commit above. They + have not been changed. Modified copies of some of them are under + Work/Fourier/ and Work/FourierE8/ and are listed in section 7. + One more file of openai/math is here unchanged, outside the Lean build: + third-party/openai-math/lean/ComparatorChallenges/UniformFourier.lean is a + byte-for-byte copy of lean/ComparatorChallenges/UniformFourier.lean at the + commit above (the statement of OpenAI's Fourier challenge). It is included + so that the modified copy named in section 2 can be compared with it. + +2. Files that contain upstream text. + - Work/CarrierCheck/ChallengeB2Ke16x.lean, + Work/CarrierCheck/ChallengeB2Ke16xR.lean, + Work/GCert/Data/ChallengeB2Gp193x.lean, + Work/GCert/Data/ChallengeB2Ge8x.lean and the text template + tools/gen/templates/ChallengeB2L16x.lean.txt each contain, as their + lines 22-270, lines 9-257 of + lean/ComparatorChallenges/UniformFourier.lean of openai/math (the RAM cost + model), unchanged. Everything else in these files was written for this + repository. + - Work/Fourier/UniformFourierChallenge.lean is a modified copy of + lean/ComparatorChallenges/UniformFourier.lean of openai/math (the + statement of OpenAI's Fourier challenge, 336 lines). Changes: a comment + was added at the top; the exponent 1 - 1/(10^(13:ℕ)) was replaced by + 1 - 7474546/(10^(10:ℕ)); seven names received the suffix "Z" + (decimalExponent, decimalTime, TimeBounds, DFTGoal, ConvGoal, + transform_main, convolution_main). Everything else in the file is the + upstream text, unchanged. See section 7. + - Work/FourierE8/UniformFourierChallenge.lean is a second modified copy + of the same upstream file, for the fourth result, with the same three + kinds of change: a comment added at the top; the exponent replaced by + 1 - 8762478/(10^(10:ℕ)); the suffix "E8" on the same seven names. + Everything else in the file is the upstream text, unchanged. See + section 7. + - Work/Scratch/Engine.lean and Work/Block/Engine.lean contain copies of + proofs of openai/math, changed for this repository: sections 1, 2 and 5 + of Work/Scratch/Engine.lean are upstream proofs with the batch layout + renamed, and both files contain a copy of the private upstream theorem + "hills_project" (lean/OAI/Computability/FourierTransform/ + TensorProgram.lean) with its envelope generalised. The comments in the + two files mark these places. + - lakefile.lean is adapted from lean/lakefile.lean of openai/math: the same + package name, toolchain, package option and Mathlib pin; every other + dependency was removed and the libraries of this repository were added. + lean-toolchain is a byte-for-byte copy of lean/lean-toolchain. + lake-manifest.json lists Mathlib and its dependencies at the revisions of + lean/lake-manifest.json. + - RELATED-WORK.md reproduces the BibTeX entries of two manuscripts of + openai/math. + +3. New files that build on upstream proofs. + The other files under WHTCheck/ and Work/ were written for this + repository. They import the upstream files, and several of them generalise + or adapt proofs of upstream files (for example Work/Block/Recursion.lean, + which says so in its header). They are changes and additions made for this + repository and are not part of openai/math. + +4. How the new files were made. + Jacob Sussman made the new files on 2026-10-08, 2026-10-09 and 2026-10-10 + by directing a team of AI agents (Claude). The agents wrote the new Lean + files, the certificate data and the generator scripts. They made the + rebuild scripts under tools/rebuild/, in part by adapting programs of + CrocSwap/integer-mult-bounds (section 6): ten of the thirteen on + 2026-10-09, tidied for this repository on 2026-10-10, and three on + 2026-10-10. On 2026-10-10 they also wrote the search programs under + tools/e8/ and found with them the certificate of the fourth result + (section 8). See README.md, section 4. + +5. Ideas taken from other public work. + The network of the first result uses ideas published by others, among them + the helper circuit "NStar3" (Swapnil-jain/integer-mult-kappa, which credits + CrocSwap pull request #36 and Paureel for parts of it), its transfer to + family 130 (shea256/fourier-transform-below-nlogn), and "carrier links" and + "whole-residual batching" (pull requests to CrocSwap/integer-mult-bounds). + No file of Swapnil-jain/integer-mult-kappa or of + shea256/fourier-transform-below-nlogn is included here. One comment in + Work/Scratch/Engine.lean quotes three counts reported in + Swapnil-jain/integer-mult-kappa and names that source. The general frame + lemma that Work/GFrame proves is stated in + notes/general-clifford-frames.tex of CrocSwap/integer-mult-bounds; its Lean + proof was written for this repository. The credits, with links, are in + RELATED-WORK.md. + +6. Data derived from, and programs adapted from, + CrocSwap/integer-mult-bounds (second result). + The helper circuit of the second result is the paired-cube circuit of the + repository "CrocSwap/integer-mult-bounds" + (https://github.com/CrocSwap/integer-mult-bounds), in the state of its pull + request #193. It is the design of that community: icekylinx (pull requests + #144 and #184), eumemic (#168, and #117 behind its modules), ikeboy / + Avi Eisenberg (#191 and #193), with an664 (#128) credited there for the + workspace-sharing principle, and DaysSky (#162) and chafreaky (#163) + credited by #168 for its closure-compiled carrier arcs and for the + fusion of its single-target outputs. That repository is published under + the Apache + License, Version 2.0 (the same licence text as the file LICENSE). Its + NOTICE file begins "A sharper exponent for integer multiplication / + Copyright 2026 Douglas Colkitt" and says that the community release "is + not solely Douglas Colkitt's work". A complete copy of that NOTICE file, as + it is at the head of pull request #193, is in + third-party/CrocSwap-integer-mult-bounds/NOTICE. + + The data were taken from these files of that repository. They were loaded + as data by the programs under tools/rebuild/. No program of that + repository was run, and none is included here unchanged; six of the + programs under tools/rebuild/ are adapted from programs of that + repository ("Programs", below). + - at the head of pull request #168 (eumemic), + 4a3c769e5c5430e7114c4d3e099ff34664677f17: + references/paired-cube/sources/local_L1.json + references/paired-cube/sources/tmod_TE_TD_TB3_1_1_4_full_6.0617964e-4.json + references/paired-cube/sources/pmod_J0_full_6.0666810e-4.json + references/paired-cube/sources/qmod_climb3u_best.json + references/paired-cube/selected-module/matching-arcs.json + references/paired-cube/physical/frames.json + - at the head of pull request #193 (ikeboy), + 187e1010ac8b259af8e9b5166f68b64bc27b4b47: + research/source-assisted-v4/data/physical-pairs.json + + Two more files at the head of pull request #168 are read for comparison + only: the programs print whether their graph and their counts agree + with them. Nothing in the certificate is taken from them: + references/paired-cube/selected-module/SOURCE.json + certificates/paired-cube-complex-input.json + + One more file is fetched and read only on request. No certificate and + no Lean module of this repository is made from it; the documentation + gives its sha256 and the price that tools/gx/gxdry.py computes for + the certificate that the rebuild makes from it: + - at the head of pull request #233 (chafreaky), + 109a857a329d18ed5552d5573f17ddfa886ae57f: + research/source-assisted-v4-layer/witness/physical-pairs.json + + None of these ten files is included in this repository. + tools/rebuild/fetch_inputs.py fetches them and checks their sha256. + + Files of this repository that contain data derived from them: + tools/certificate/gcert1-p11-pr193.json.gz and the Lean data modules + generated from it, Work/GCert/Data/Gen/P193*.lean and the files they import + from Work/GCert/Data/Gen/P193/; its block counts are also in the generated + modules Work/GCert/Data/P193Price.lean, RateB2Gp193.lean, RateB2Gp193x.lean + and InstB2Gp193.lean. + Changes made: the circuit was rebuilt from those data as one explicit, + renumbered list of additions with a table of frames, in the certificate + format "gcert/1" of this repository; its 1,980 in-place steps were written + as pairs of additions with rescaled coefficients; the Lean data modules + were generated from that list. No file of that repository is included + unchanged, except the copy of its NOTICE file named above. + + Programs. The Python files under tools/rebuild/ were made by the agents + (section 4), who read programs of that repository as text and ran none + of them. Six of the thirteen files are adapted from those programs: + the parts named below follow the outside program statement by statement + or method by method, with the layout changed, some names changed and + the parts that this certificate does not use left out. Each of the six + says so in its first lines. + - tools/rebuild/compile.py: the function closure follows + compile_closure of scripts/paired_cube/closure.py; the function + select follows the first part of select of + scripts/paired_cube/gauges.py. + - tools/rebuild/graphgen.py: the class Gen follows the class Graph of + scripts/paired_cube/graph.py; its method fuse follows the branch + "f8:" of merge_outputs of scripts/paired_cube/modules.py, which + fuses the three single-target outputs of a port into one. That + branch came with pull request #168 (eumemic), which says of the + fusion that it "follows #163's same-singleton fusion" and credits + it so: "Same-singleton output fusion: @chafreaky (#163); the fold + orders here come from our own search." + - tools/rebuild/aligned.py: the functions op_tags, local_values and + build follow op_tags, local_values and main of + research/source-assisted-v4/source_aligned_local_v4.py, except its + search for pairs and the records it writes. + - tools/rebuild/stage.py: the order of the events, the backward pass + (responses) and many names are those of + research/terminal-sinks/sinks_gate.py and of + scripts/paired_cube_physical.py; the event records and the exact + arithmetic were written for this repository, and the frame scan + was rewritten so that it returns every block. + - tools/rebuild/build.py: the function target_hist and the field names + of profile follow the last part of select of + scripts/paired_cube/gauges.py; the calls that make the graph and its + hash follow regenerate of scripts/paired_cube_producer.py. + - tools/rebuild/f2.py: the function perp follows perp, and the second + half of the function rref follows the second half of basis, of + scripts/paired_cube/frames.py. + The outside programs are those at the two commits named above + (source_aligned_local_v4.py at the second, the others at the first). + Of the other seven files, rebuild.py, fetch_inputs.py, rbpaths.py, + rewrite.py, phys.py and emit.py were written for this repository. + modules.py was written for this repository too; for the three module + files it checks the contract that the outside + scripts/paired_cube/modules.py states for them in its functions + triple_module_from, pair_module_from and all_but_one_from, each of + which is marked there "eumemic, Claude assistance; Apache-2.0". + + Notices and attribution sentences in the first lines of those outside + programs, quoted with the line numbers of each file at its commit. + The other lines of each header say what the program does and name no + person: + + scripts/paired_cube/closure.py, lines 4-5 (pull request #168, + eumemic; from pull request #162, DaysSky): + Copyright 2026 DaysSky (PR162, + research/extended-links/cxlinks.py), Apache-2.0; moved here + unchanged apart from this header and the imports (eumemic, Claude + assistance). + + scripts/paired_cube/graph.py, modules.py, frames.py, gauges.py and + scripts/paired_cube_producer.py (pull request #144, icekylinx; + graph.py, modules.py and paired_cube_producer.py changed in #168, + eumemic), the two comment lines at the top of each: + Copyright 2026 icekylinx. Apache-2.0. + Developed with GPT-6 Astra assistance; integrated with Codex + assistance. + graph.py, lines 6-9: + The coarse positive query modules restrict the already adopted + PR117 DAG (eumemic; Apache-2.0; Claude assistance; + cbb05ce504d571546d9b7794c186a613c659c3bf). The paired-channel + decomposition and this assembly are new work for icekylinx with + OpenAI GPT-6 Astra assistance. + modules.py, lines 6-7: + The pair-disjoint module is a specialization of the already + adopted PR117 positive DAG (eumemic, Apache-2.0, pinned + cbb05ce504d571546d9b7794c186a613c659c3bf). + modules.py, line 272, above merge_outputs: + Output merging (pmerge.py variants; eumemic, Claude assistance; + Apache-2.0). + frames.py, line 7: + New work for icekylinx, OpenAI GPT-6 Astra assistance. + paired_cube_producer.py, line 8: + Copyright 2026 icekylinx, Apache-2.0; OpenAI GPT-6 Astra + assistance, inherited PR117 notices. + + scripts/paired_cube_physical.py, lines 4-7 (pull request #168, + eumemic): + Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. + The signed word, frozen arcs, gauges, chronology and shared-core + accounting are icekylinx's PR144. Frame descent under PR130's + general Clifford frames and compensated birth-cut reuse follow + PR131 and jamesyc's PR124; deadline (late) compensation reads + follow PR143. + + research/terminal-sinks/sinks_gate.py, lines 4-6 (pull request + #168, eumemic): + The substitution is the terminal-sink lemma of PR #166 by jamesyc + (closed by its author), independently audited here: this gate's + own frame scans, replay and controls carry the soundness argument + and do not use #166's checker. Gate, audit and integration by + eumemic with Anthropic Claude assistance. Apache-2.0. + + research/source-assisted-v4/source_aligned_local_v4.py, lines 4-9 + (pull requests #191 and #193, ikeboy / Avi Eisenberg): + Derived from + research/source-assisted/decision/source_aligned_local.py + (icekylinx, GPT-6 Astra assistance, Apache-2.0). Only the three + frozen query modules change: this copy reads PR168 v4's selected + modules (eumemic, Claude assistance) from + references/paired-cube/sources instead of the PR168 fd25adb7 + modules. The local configuration, arc transport, frame + inheritance and physical layer code are unchanged. + + Notices of that NOTICE file that pertain to these data, quoted: + + Paired-cube and shared-core extension, 9 October 2026 + Copyright 2026 icekylinx + + Source-assisted common-frame extension (2026-10-09) + Copyright 2026 icekylinx. Apache-2.0. + Substantial dependency: eumemic PR168, + fd25adb7fbaa12ee761d02c733c54d1d2a7687ee + + The full text of both notices, and the notices of the work they build on, + are in the copy named above. The credits, with links and with the words of + pull request #193 about its own sources, are in README.md ("Whose circuit + this is") and in RELATED-WORK.md, section 1. + +7. Modified copies of upstream files (third and fourth results). + The files listed below are copies of files of openai/math at the commit + above, changed for this repository. Each begins with a comment that names + the upstream file it is derived from and states what was changed. The + upstream originals under lean/OAI/Computability/ are in this repository + as well, unchanged (section 1), and the copies import them. The upstream + challenge file lean/ComparatorChallenges/UniformFourier.lean is in this + repository in both forms: unchanged under third-party/openai-math/ + (section 1) and modified (section 2). + + - Work/Fourier/ArbitraryLength.lean: modified copy of + lean/OAI/Computability/FourierTransform/ArbitraryLength.lean (224 + lines; 159 of them unchanged in the copy). Changes: identifiers renamed + on 64 lines (18 names: 15 with the suffix Z, 3 with Z inserted); 1 + import line replaced by 1; 15 comment lines added; no other change to + the code. + - Work/Fourier/Asymptotics.lean: modified copy of + lean/OAI/Computability/FourierTransform/Asymptotics.lean (187 lines; + 112 of them unchanged in the copy). Changes: identifiers renamed on 74 + lines (32 names: 26 with the suffix Z, 6 with Z inserted); 1 import + line replaced by 1; 15 comment lines added; no other change to the + code. + - Work/Fourier/Axioms.lean: written for this repository (not a copy of an + upstream file). + - Work/Fourier/ConvolutionProgram.lean: modified copy of + lean/OAI/Computability/FourierTransform/ConvolutionProgram.lean (100 + lines; 77 of them unchanged in the copy). Changes: identifiers renamed + on 22 lines (10 names: 9 with the suffix Z, 1 with Z inserted); 1 + import line replaced by 1; 15 comment lines added; no other change to + the code. + - Work/Fourier/Goal.lean: written for this repository; contains lines of + upstream files: of its 7 code lines, 6 are lines of + lean/OAI/Computability/FourierTransform/Goal.lean (original lines 35, + 75-77, 79, 81): unchanged: its lines 25; with identifiers renamed: its + lines 21, 24, 26, 28, 30. + - Work/Fourier/Main.lean: modified copy of + lean/OAI/Computability/FourierTransform/Main.lean (151 lines; 90 of + them unchanged in the copy). Changes: identifiers renamed on 59 lines + (40 names: 35 with the suffix Z, 5 with Z inserted); 1 import line + replaced by 2; 24 comment lines added; other code lines changed: 1 of + the original, 1 of the copy (copy line 59). + - Work/Fourier/Seam.lean: written for this repository; contains lines of + upstream files: of its 122 code lines, 15 are lines of + lean/OAI/Computability/FourierTransform/TensorProgram.lean (original + lines 36-37, 95-96): unchanged: its lines 93-94, 113-114, 141-142, + 163-164, 185-187, 194-196; with identifiers renamed: its lines 184; 3 + are lines of lean/OAI/Computability/FourierTransform/TensorSaving.lean + (original lines 83-84, 114): with identifiers renamed: its lines 45-46, + 59; 1 is a line of lean/OAI/Computability/FourierTransform/Main.lean + (original line 60): with identifiers renamed: its lines 213. + - Work/Fourier/SectorAlgorithm.lean: modified copy of + lean/OAI/Computability/FourierTransform/SectorAlgorithm.lean (224 + lines; 162 of them unchanged in the copy). Changes: identifiers renamed + on 61 lines (17 names: 13 with the suffix Z, 4 with Z inserted); 1 + import line replaced by 1; 15 comment lines added; no other change to + the code. + - Work/Fourier/SynchronizedAlgorithm.lean: modified copy of + lean/OAI/Computability/FourierTransform/SynchronizedAlgorithm.lean (126 + lines; 94 of them unchanged in the copy). Changes: identifiers renamed + on 31 lines (12 names: 10 with the suffix Z, 1 with Z inserted, 1 + other), the other being "canopy" to "Grove.canopy" (5 lines); 1 import + line replaced by 1; 16 comment lines added; no other change to the + code. + - Work/Fourier/TransformProgram.lean: modified copy of + lean/OAI/Computability/FourierTransform/TransformProgram.lean (146 + lines; 94 of them unchanged in the copy). Changes: identifiers renamed + on 51 lines (21 names: 19 with the suffix Z, 2 with Z inserted); 1 + import line replaced by 1; 15 comment lines added; no other change to + the code. + - Work/Fourier/UniformBounds.lean: modified copy of + lean/OAI/Computability/FourierTransform/UniformBounds.lean (183 lines; + 115 of them unchanged in the copy). Changes: identifiers renamed on 67 + lines (29 names: 25 with the suffix Z, 4 with Z inserted); 1 import + line replaced by 1; 15 comment lines added; no other change to the + code. + - Work/Fourier/UniformFourierChallenge.lean: modified copy of + lean/ComparatorChallenges/UniformFourier.lean (336 lines; 328 of them + unchanged in the copy). Changes: identifiers renamed on 7 lines (7 + names: 7 with the suffix Z); 12 comment lines added; other code lines + changed: 1 of the original, 1 of the copy (copy line 281). + - Work/Fourier/WorkingCompiler.lean: modified copy of + lean/OAI/Computability/FourierTransform/WorkingCompiler.lean (153 + lines; 104 of them unchanged in the copy). Changes: identifiers renamed + on 48 lines (14 names: 14 with the suffix Z); 1 import line replaced by + 1; 15 comment lines added; no other change to the code. + - Work/Fourier/WorkingPreparation.lean: modified copy of + lean/OAI/Computability/FourierTransform/WorkingPreparation.lean (212 + lines; 156 of them unchanged in the copy). Changes: identifiers renamed + on 55 lines (30 names: 28 with the suffix Z, 2 with Z inserted); 1 + import line replaced by 1; 15 comment lines added; no other change to + the code. + - Work/Fourier/WorkingTransform.lean: modified copy of + lean/OAI/Computability/FourierTransform/WorkingTransform.lean (119 + lines; 91 of them unchanged in the copy). Changes: identifiers renamed + on 27 lines (9 names: 8 with the suffix Z, 1 with Z inserted); 1 import + line replaced by 1; 15 comment lines added; no other change to the + code. + + Changes made to the eleven copied proof files: every declaration received + the letter "Z" in its name (as a suffix of its own name, or of the name + of the declaration it belongs to), so that it does not collide with the + upstream declaration of the same name; the upstream exponent "alpha" + (1 - 2/10^11) and the bound "hills" defined from it were replaced by + "alphaZ" (1 - 7474547/10^10) and "hillsZ", which are defined in + Work/Fourier/Seam.lean, and the single use of the upstream theorem + "hills_program" by "hillsZ_program"; the five definitions of the upstream + statement that contain its exponent (decimalExponent, decimalTime, + TimeBounds, DFTGoal, ConvGoal) were replaced by the versions of + Work/Fourier/Goal.lean, whose exponent is 1 - 7474546/(10^(10:ℕ)); + the import lines point to the copies; a header comment was added. + Two further changes: in the copy of Main.lean the same constant stands + once as the literal 2/10^11 and was replaced there too, and the two + final theorems of that copy received a documentation comment; in the + copy of SynchronizedAlgorithm.lean the name "canopy" was written out as + "Grove.canopy" in 5 places, because the short name is shadowed in the + larger environment of the copy. The script also renamed three names + inside upstream comments; the comments are otherwise the upstream ones + and describe the upstream network. + The proofs are those of openai/math. The copies were made by the script + tools/fourier/mkchain.py of this repository. + + Work/Fourier/Goal.lean contains copies of those five definitions of + lean/OAI/Computability/FourierTransform/Goal.lean, with the exponent + replaced and the suffix "Z" on the five names. + + Work/Fourier/Seam.lean was written for this repository. It repeats the + statement of the upstream theorem "hills_program" (TensorProgram.lean) and + the definition of "hills" (TensorSaving.lean) with the two names replaced; + its header says so. + + Fourth result. The 15 files under Work/FourierE8/ repeat the 15 files + under Work/Fourier/ for the kernel program of the fourth result, file + for file and under the same file names: ArbitraryLength.lean, + Asymptotics.lean, ConvolutionProgram.lean, Goal.lean, Main.lean, + SectorAlgorithm.lean, SynchronizedAlgorithm.lean, TransformProgram.lean, + UniformBounds.lean, UniformFourierChallenge.lean, WorkingCompiler.lean, + WorkingPreparation.lean, WorkingTransform.lean, Seam.lean and + Axioms.lean. The eleven copied proof files, + Goal.lean and UniformFourierChallenge.lean are modified copies of the + same upstream files as their namesakes above, made by the script + tools/fourier/mkchain_e8.py (a copy of tools/fourier/mkchain.py with + its constants changed). The changes are those described above with + three differences: the suffix is "E8" where the copies of the third + result have "Z"; "alphaE8" is 1 - 8762479/10^10; and the exponent of + Work/FourierE8/Goal.lean is 1 - 8762478/(10^(10:ℕ)). Each of these + files begins with a comment that names the upstream file it is derived + from and states what was changed. Work/FourierE8/Seam.lean and + Work/FourierE8/Axioms.lean were written for this repository: they are + Work/Fourier/Seam.lean and Work/Fourier/Axioms.lean with names and + numbers replaced, and the seam repeats the same upstream statement and + definition as the seam of the third result. The proofs are those of + openai/math. + + The work of others that the third result follows (danadran01, shea256, + eumemic) is credited in README.md ("Whose reduction this is") and in + RELATED-WORK.md, section 1. No file of their repositories is included + here. + +8. The certificate of the fourth result and the search programs. + tools/certificate/gcert1-e8-r783.json.gz, the Lean data modules + generated from it (Work/GCert/Data/Gen/E8*.lean, Work/GCert/Data/Gen/E8/ + and the modules of Work/GCert/Data/ with E8 or B2Ge8 in their names) and + the Python programs under tools/e8/ were made by the agents for this + repository on 2026-10-10 (section 4). The circuit in that certificate + is not the circuit of section 6, and no data file of + CrocSwap/integer-mult-bounds was loaded to make it. It uses three + devices that were published by others in pull requests to that + repository: + - the in-place pair, a + b and a - b formed on the two arrays that held + a and b: icekylinx (#184), carried in #191 and #193 (ikeboy); + - the re-use of a finished helper with a compensating read: jamesyc + (#124) and eumemic (#143); + - the late pairing rule, the choice of which finished helper a new + value takes: Chafik Boukhalfa (the account chafreaky), #200 and #233, + and pull request #2 of this repository. The search programs apply the + idea with a greedy rule of their own; his program, his pairs and his + certificate were not used. + No program of that repository is included in tools/e8/, imported or + run by it. The programs of tools/e8/ have not been compared statement + by statement with the programs of that repository, as those of + tools/rebuild/ were (section 6). The integer programme of + tools/e8/ is solved with the open-source solver HiGHS, called through + SciPy (scipy.optimize.milp), with NumPy; none of the three is included + here. The credits, with links, are in README.md ("Whose ideas the new + unit uses") and in RELATED-WORK.md, section 8. + +9. Files under community/ (fifth revision). + The folders under community/ are packages of Lean files laid over a + clone of Ryan Shea's repository "shea256/fourier-transform-below-nlogn" + (https://github.com/shea256/fourier-transform-below-nlogn) at commit + 52394849c7ecf3412d39b925d75fc7fdb98994d8. That repository is published + under the Apache License, Version 2.0. Its NOTICE files are copied, + unchanged, into third-party/shea256-fourier-transform-below-nlogn/ + (NOTICE, lean/complex588/NOTICE and lean/complex588/NOTICE.complex588); + they apply to the material below that comes from his repository. Notices + of those files that pertain to it, quoted: + + Finite Network Improvements for the Exact Discrete Fourier Transform + Copyright 2026 Ryan Shea + + The source circuit is the atm3 provider reconstructed from Jeremy + Kothe's CrocSwap/integer-mult-bounds PR588, commit + 6d0e53b66c29abd17981881a09eaff41ac862d85. + + The dedicated entrance-gauged nine-stage layout is Dugongue's #401 on + Zhihao Chen/jacklightChen's #377, building on DaysSky's bridged layout + and lydakis's bank patterns. [...] These files are taken from Nil + Mamano's published Big O Speedrun submission, bigospeedrun/verifier + commit 025396033eede56e3af6e02ed2679ba828381bd5. + + The repository bigospeedrun/verifier (https://github.com/bigospeedrun/ + verifier) is published under the Apache License, Version 2.0. + + a. Modified copies of OpenAI's statement file. Each file named + UniformFourierChallenge.lean under community/*/overlay/Work/ and + community/c588m/variants/*/Work/ (18 files: the folders FourierC588 + and FourierC588Neg of c588-repack; FourierC588M and FourierC588MNeg of + c588m and of its two variants; FourierC588MH, FourierC588MHNeg, + FourierC588MHH and FourierC588MHHNeg of c588mhh; FourierU8X and + FourierU8XNeg of u8x; FourierU8XM and FourierU8XMNeg of u8xm; + FourierU8XMHH and FourierU8XMHHNeg of u8xmhh) is a + modified copy of lean/ComparatorChallenges/UniformFourier.lean of + openai/math at the commit above. Changes: the exponent + 1 - 1/(10^(13:ℕ)) is replaced by the package's exponent (in a file + whose folder ends in Neg, one unit stronger); the seven names of + section 2 receive the package's suffix; and a comment, beginning on + line 3, that says so was added after the build and the checks. The + comment gives the sha256 of the file without it, the file that was + compiled and checked. The copies in c588-repack are made from Shea's + copies at the same paths (suffix C588), with only the exponent + changed. Everything else in these files is the upstream text, + unchanged. tools/community/package_check.py checks each of them + against third-party/openai-math/. + Seven more files hold the same upstream text, changed, without such a + comment inside them, because they are kept as they were compiled or + fetched; a file NOTICE.txt in each of their folders says what was + changed: + - community/c588m/logs/skeptic-statement/fetched/Work/FourierC588M/ + and FourierC588MNeg/UniformFourierChallenge.lean (2 files): the + copies of c588m's two statement files that an independent agent + fetched from the build machine, as compiled (the files of + community/c588m/overlay/ at the same paths without their comment); + - community/base-logs/ref/mine-12331685.lean, mine-12331686.lean and + mine-Y-7547360.lean (3 files): the agents' own derivations of the + statement, with the exponent and the suffix (C588 or Y) changed; + - community/c588-repack/logs/colab/submission/VerifierChallenge.lean + and community/u8xmhh/logs/colab/bos-H-stage/VerifierChallenge.lean + (2 files): the challenge of the offline re-creation of the + leaderboard's verifier, which is challenges/dft/UniformFourier233.lean + of bigospeedrun/verifier (itself a modified copy of OpenAI's file + whose first lines say so) followed by the target theorem Race.bound. + community/base-logs/ref/ also holds three unmodified copies of + OpenAI's file (UniformFourier-.lean, byte-identical to + third-party/openai-math/). + + b. Modified copies of OpenAI's proof files. In each package, the folder + Work/Fourier/ (for example Work/FourierC588MHH/) holds copies + of eleven files of lean/OAI/Computability/FourierTransform/ of + openai/math and of five definitions of its Goal.lean, made the way + section 7 describes, by Shea's script mkchain_c401g5.py or by the + agents' generators that copy its output under a new suffix (for + example community/c588mhh/generators/lean-gen/mk_fourier_h.py). Each + begins with a + comment that names the upstream file and states what was changed. In + c588-repack these are Shea's own copies with the exponent changed. + + c. Files that contain upstream text. The files Challenge.lean under + community/wht-c588/overlay/Work/ and community/wht-c588mhh/overlay/ + Work/ contain the RAM cost model of + lean/ComparatorChallenges/UniformFourier.lean of openai/math, + unchanged, as Work/GCert/Data/ChallengeB2Ge8x.lean does (section 2); + their headers say what differs from that file. + + d. Files adapted from Shea's repository in community/c588mhh/overlay/ + Work/XHost/: + - HDef.lean is a copy of lean/complex588/Work/Bridge401/GNetDef.lean, + HStage.lean of Work/Bridge401/GNetStage.lean, HCert.lean of + Work/Bridge401/GNetCert.lean, HGeom.lean of the instance part of + Work/Bridge401/GGeom.lean, and HEnd.lean is adapted from + Work/Bridge401/End.lean, each with same-run hosting added (the idea + of DaysSky's pull request #442 to CrocSwap/integer-mult-bounds) and + the names changed. In Shea's tree these files are those of Nil + Mamano's submission named above (his SOURCE_MANIFEST.json records + each as taken from submissions/dft/nmamano-dft-record/ at the + verifier commit above), written by his generator gen401.py on the + layouts credited above, with the engine of this repository. + - Frames.lean adapts lemmas of Work/Bank401/Adapt.lean and + Work/Bank401/Frames.lean of the same tree, which come from the same + submission: host_adapt is bank_adapt, host_pad is bank_pad and + slot_frames is bank_frames, each with the new coordinates chosen + inside the pending set of an idle climb. + - HSchedK.lean copies the lemmas about lists of + Work/Bridge401/SchedK.lean of the same tree, and its checker + hostSchedK is modelled on the checker schedK there; HostK.lean is + modelled on the same checker. + The header of each file names its source and says what was changed. + One header is wrong in part: that of HEnd.lean says that Shea's + End.lean was written after a file Work/Bridge377/End.lean of Jacob + Sussman's. No such file of his exists; Shea's End.lean says only that + it follows a file of that name. The header is left as it was compiled + (community/c588mhh/README.txt says so). + The generators that wrote them are under + community/c588mhh/generators/lean-gen/. The same twelve network files + of Work/XHost/, byte-identical, are also in community/u8xmhh/overlay/ + Work/XHost/. + The log community/c588mhh/supporting/L2/skeptic/logs/ + Work_Bank401_Adapt.log is of his file, compiled unchanged; the file + itself is not included. + + e. Programs from Shea's repository. + - Byte-for-byte copies of his files at the commit above, unchanged and + without a notice inside them (the same files are also in + third-party/shea256-fourier-transform-below-nlogn/ at their own + paths): + research/complex-transfer-588-596/experiments/generate_global_lean.py + is copied as tools/community/route/generate_global_lean.py and + tools/community/repack/tools/generate_global_lean.py; + research/complex-transfer-588-596/experiments/bank_reprice.py is + copied as tools/community/repack/tools/shea_bank_reprice.py; + research/complex-transfer-588-596/verification/exact_moment.py is + copied as tools/community/route/exact_moment.py, + tools/community/repack/tools/exact_moment.py, + community/c588mhh/generators/design-tools/exact_moment.py and + community/u8xmhh/generators/design-tools/exact_moment.py. + - route_price.py, in four identical copies (tools/community/route/, + tools/community/repack/tools/, + community/c588mhh/generators/design-tools/ and + community/u8xmhh/generators/design-tools/), contains functions + copied from his bank_reprice.py and generate_global_lean.py (its + header says so), changed to price a local word through his route. + - The generators of tools/community/c588m-gen/ drive or adapt his + scripts in third-party/shea256-fourier-transform-below-nlogn/ + research/; each adapted file says what it changes. + - community/c588-repack/overlay/comparator/fourier.json is his + comparator configuration, unchanged. + + f. Data. The certificate of Kothe's #588 circuit, its bank schedule and + its price, as published in Shea's repository, are in + third-party/shea256-fourier-transform-below-nlogn/lean/complex588/ + artifacts/, unchanged. The packages' certificate and data modules + are derived from them (VERIFY.md, section 11, says how). + + g. Data derived from CrocSwap/integer-mult-bounds at a later state (the + U8 circuit). The certificates of community/u8x and community/u8xm (the + files under their Work/GCert/Data/ whose paths contain U8X), + and the host table and data modules of community/u8xmhh, which are + computed from the certificate of u8xm, are derived from the U8 word of + the repository CrocSwap/integer-mult-bounds + (https://github.com/CrocSwap/integer-mult-bounds), a circuit of that + community: Jeremy Kothe's pull request #869, on his #799 (which + carries Chafik Boukhalfa's frame changes of #741), on icekylinx's + gauge44 construction (#645) and on the earlier work that their + notices name. Only the word's additions and their frames are used: + not the owner maps of #870 (Kothe, with Hao Yan's #858), #890 + (Boukhalfa) and #942 (Jonathan Capalbo), nor DreamingOfClouds' bit + supplier #589. The agents rebuilt the word with that repository's + own program + research/complex-t21-owners/vendor/pr870/construction/barch/ + reproduce_barch.py (recipe U8; #869's builder, in the copy of #870's + package that #890 keeps), in a clone at commit + 407d52ff15b2217b6e22bce44728b4c3907b6515 of its main branch + (13 October 2026). The word they obtained has the canonical sha256 + 24db880b7dab31e6af4eec8a1aba1bbbae29cd50d9a7b77cf0e9b571b3e5ae7c, the + value pinned for it in that repository (the file + LOCAL-PINS-barch-U8.json next to the program). The logs of that run are in + community/base-logs/u8-word/. No program of that repository is + included here, and the word itself is not included: the packages hold + certificates computed from it. + That repository is published under the Apache License, Version 2.0 + (its LICENSE at that commit is the same text as the file LICENSE). + Its NOTICE file at that commit differs from the copy of section 6 (the + state of pull request #193) in one passage, which + third-party/CrocSwap-integer-mult-bounds/at-407d52ff/SOURCE.txt + describes. Complete copies of that NOTICE file and of the notice files + of the folders the program comes from + (research/complex-t21-owners/NOTICE.md and, under its vendor/pr870/, + NOTICE.md, UPSTREAM-NOTICE.md and t16/NOTICE.md), unchanged and at + their paths, are in + third-party/CrocSwap-integer-mult-bounds/at-407d52ff/; they apply to + these data. + + The new files of community/ were made on 2026-10-13 the way section 4 + describes. The credits, with links, are in README.md. + +10. Files added in the fifth revision outside community/ (package/, + scripts/, tools/lever/, tools/lever-hub/, tools/reproduce-search.sh, + tools/compare_results.py, data/, results/, research/, + notes/how-the-lever-was-found.md, notes/open-directions-4.md and the + additions to third-party/). + The packages under package/ (the sixth result, package/u8x0hq, and the + recursion lever, package/leverF, package/leverB, package/leverA, package/lever2979, package/lever) are laid over the same + clone of Ryan Shea's repository as the packages of section 9 (commit + 52394849c7ecf3412d39b925d75fc7fdb98994d8, folder lean/complex588, + Apache License, Version 2.0). His licence and notice files in + third-party/shea256-fourier-transform-below-nlogn/ apply to the material + of his repository that files of these packages copy or adapt; each such + file says so in its first lines, and the packages' README.txt list them. + Shea's tree carries Lean files of Nil Mamano's published Big O Speedrun + submission (bigospeedrun/verifier, Apache License, Version 2.0): the + gauged interface, the nine-stage network, its bank lemmas and checker, + its end theorem and the templates of the Fourier chain. The hosted + networks of these packages (Work/XHost in package/u8x0hq; Work/XHostD, + Work/XHostR and Work/XHostT in the lever packages) are adapted from those + network files, as their first lines say. + + a. Modified copies of OpenAI's statement file. Each file named + UniformFourierChallenge.lean under package/*/overlay/Work/ is a + modified copy of lean/ComparatorChallenges/UniformFourier.lean of + openai/math at the commit above: in package/u8x0hq the folders + FourierU8XM, FourierU8XMHH, FourierU8X0, FourierU8X0HH and + FourierU8X0HQ and their twins ending in Neg (10 files); + in package/leverF the folders FourierL3945F and FourierL3945FNeg (2 files); in package/leverB the folders FourierL3945H and FourierL3945HNeg (2 files); in package/leverA the folders FourierL2979H and FourierL2979HNeg (2 files); in package/lever2979 the folders FourierL2979 and FourierL2979Neg (2 files); in package/lever the folders FourierL3945 and FourierL3945Neg (2 files). Changes: the exponent + 1 - 1/(10^(13:ℕ)) is replaced by the package's exponent (in a folder + whose name ends in Neg, one unit stronger), and the seven names of + section 2 receive the package's suffix. Everything else in these files + is the upstream text, unchanged. Unlike the copies of section 9, these + files carry no comment inside them: they are kept byte for byte as they + were compiled, because scripts/build-and-check.sh derives them again + from third-party/openai-math/ (scripts/derive_challenges.sh) and + compares the two. This section is their notice. + b. Modified copies of OpenAI's proof files. In each package, the folders + Work/Fourier/ hold copies of eleven files of + lean/OAI/Computability/FourierTransform/ of openai/math and of five + definitions of its Goal.lean, made by Shea's script mkchain_c401g5.py + or by the agents' generators that copy its output under a new suffix. + Each begins with a comment that names the upstream file and states + what was changed. + c. The community's files. third-party/CrocSwap-integer-mult-bounds/ + staged544-terminal-gap-fill/ holds unmodified copies of 32 files of the + folder research/staged544-terminal-gap-fill/ of pull request #3945 of + CrocSwap/integer-mult-bounds at its head + 91f81d131252db704f945891da1ea54e29d46b8b (Dugongue's branch): the owner + data (FILL, PROFILE, ORDER, DISPATCH, PLAN, TEMPLATE, the two words and + the files of #3815 that #3945 carries as old-*), README.md, PROOF.md, + RESULT.json, SOURCES.json, LICENSE, NOTICE.md and the native lineage + notices. BLOBS.txt gives the git blob hash of each, as GitHub stores + it. That repository and its contributors' branches are published under + the Apache License, Version 2.0; the NOTICE.md and native notices of + that folder apply to these files and are quoted in substance in (g). + The files of #2979 (research/rank7-joint-endframes/ of the main branch + of CrocSwap/integer-mult-bounds at 20ecf289; Marius-Paul Dumitrean, + with the files of Mason Hale's #2917 and of #2942 that its NOTICE + names) are copied unmodified, 12 of them, into package/lever2979/ + and package/leverA/ under generators/tree/wht45/lever/p-alt/dl/ + main2979/: CLAIM.json, CREDITS.md, PINS.json, ALLOCATION-PINS.json, + the allocation files DISPATCH, FILL, ORDER, PROFILE and PROVIDERS, + and the data files PARENT-base, WORD-base and WORD-extra. + package/leverA/generators/tree/wht45/allheads/aA-data/in/alloc2979H/ + holds unmodified copies of its DISPATCH, ORDER, PROFILE and PROVIDERS + (beside the table with every head kept, FILL.json.gz, see (d)). Both + packages also hold #3945's owner/inputs/PLAN.json.gz and + TEMPLATE.json.gz under generators/tree/wht45/lever/g-heads/dl/pr3945/. + The generators check every one of these copies against its sha256 + (generators/tree.sha256) before they run; the same licence and the + notices of #2979 (for PLAN and TEMPLATE, of #3945) apply to them. + Both packages also hold generators/tree/wht43/bound-check/k-word/ + tools/build_tri_869.py, an unmodified copy of the community's word + builder construction/barch/build_tri.py (the same bytes as + research/complex-t21-owners/vendor/pr870/construction/barch/ + build_tri.py of its main branch, whose notices section 9, item g, + names); the generators do not run it. + More unmodified copies of the same folder of #3945 are in package/leverF/ + and package/leverB/, under generators/tree/wht45/lever/g-heads/dl/pr3945/ + owner/: 21 of the files above, and #3945's build_fill.py, check_gain.py + and verify.py (git blob hashes equal to GitHub's at 91f81d13); the same + licence and notices apply to them. + d. Data derived from the community's files. data/native/ holds the + canonical native words of #3945's two words (and of #3815's), the + output of #3945's own program native/run_native.py, run unchanged by + the agents. data/gcert/, data/residuals-*.json and data/t74ctl/ are + the agents' conversion of those words, and of the earlier T74 words, + into this project's certificate format (tools/lever/lower_b.py and + t74_control.py); data/t74ctl/inputs/ holds the community's T74 + inputs repackaged in the layout of #3945's package (TEMPLATE.json.gz + unmodified, and ORDER.json with the same content as the community's + T74 order). The Lean data of the lever packages (the words, the + dispatch, the tables of hosts) is generated from such conversions and + from the community's files by the generators in each package. The + tables with every head kept (the folders fill/ of the packages on + every head kept, where shipped) were made by the community's own + allocation programs (on #2979, its fill17.py and double.py; on #3945, + the hall_lp.py and build_fill.py of #3815 and #3945's offset step), + run with one added condition; each package's fill/ README says how, + and which of those programs it copies, with their notices. + e. Programs that re-implement or adapt the community's code: + - tools/lever/pricer.py re-implements (does not import) the community's + moment engine and kappa assembly (research/native-complex-moment552/ + engines/moment.py and outer.py, research/gauge44-frames-x/code/ + assemble.py). The package "native complex moment" is by Rohan Arun + with OpenAI Codex assistance, on Dugongue's #552, eumemic's #543 and + #558, and icekylinx's complex-cover notes; its outer assembly is + eumemic's, after icekylinx's #144 and Zhihao Chen's #23, with the + balanced prefix of the RaD project (hipotures). + - tools/lever/heads_fill.py and g0_staged.py re-implement the window + and family semantics of #3945's owner/verify.py (they do not import + it). + - tools/lever/lower_b.py extends this project's own lowering program + to the community's borrowed-helper words; the meaning of a warp + follows the community's native code (bw_transport.py, bw_emit.py of + #3945's native/vendor), which was read, not copied. + f. Scripts. scripts/apply-overlay.sh, build-and-check.sh and + run-comparator.sh are adapted from this project's community/scripts/; + scripts/base/, cc-jail/, bos/ and lever-run/ are the agents' scripts + as run. Those in bos/ and lever-run/ package a proof in the layout of + the entries of bigospeedrun/verifier and run that verifier's three + phases; the verifier's own code is not included, except its function + strip_comments (scripts/verifier.py of bigospeedrun/verifier, Apache + License, Version 2.0), whose logic scripts/bos/fspb_submission.py and + scripts/lever-run/isk_submission.py copy, as the line that defines it + says; the same function is in community/scripts/bos/mb_submission.py, + community/u8xmhh/scripts-as-run/build/fsb_submission.py and + package/leverF/logs/fc-engine/C3/scripts/isk_submission.py. + g. Credit for the circuits of the lever. + #3945, as its NOTICE.md and native lineage notices ask that the credit + be kept: + - #3945 (Dugongue, with substantial OpenAI Codex assistance, including + three parallel Codex agents): the terminal-width construction (26 + protected helper terminals from width 4 to width 5, with scalar and + native frames refitted), the idle-gap absorption proof, the + source-guarded recipe, the independent finite checks and the + integration. + - dataisfire, #3734: the staged544 supplier and its full source closure. + - zxv, #3682: the staged pair; #3638: the earlier T360 allocation. + - Ross Richardson, #3605: the staged entrance, cleanup and residual + checks (helpers 644 and 660 from width 8 to 7). + - Erwan Secousses / ChaloupeMusic, #3815, with the #3638 lineage: the + attained T360 owner clock and fill, and the allocation programs that + made the table with every head kept on #3945. + - hexapode (Pierre-Loic Doulcet), #3802 and #3875: the earlier complete + endpoint combination and its refinement. + - Dugongue, #3857 (general proof and evaluator work) and #3929 (the + transported-frame compiler). + - Sagar Thakrar, #3687: audit_core.py, which reimplements Arham Tahir's + coupled-carrier audit; roger-sato, #3693: the whole-word audit adapter; + roger-sato, #3642: the T530 release retimings. + - Jeremy Diviney, #3148: the coupled-carrier T740 packing cap; dcharb78, + #3133: helpers 586 and 588 lowered by coupled carrier folds. + - Marius-Paul Dumitrean (mdumitrean), #2979: joint rank-7 end frames and + five paired extractions, on Jeremy Kothe's #2957 gate rewrites; #2802 + (hostee moves and retired-lender aliases) and #2733. + - Jeremy Kothe: the frame and preload machinery, #2833 (dirt-entry + rewrite of helpers 530 and 542), #2957 (gate rewrites on Mason Hale's + #2917 words), and the source pipeline, label swap and order and fill + of #2462, #2588 and #2673; Mason Hale, #2917 (nine paired helper + extractions); Lyy-iiis, #2857 (shared kernels) and #2848; Dugongue, + #2863; philippeVV (Philippe Van Velzen), #2853, #2559 and #2679 + (evacuation, six pre-life hosts, the refit alias). + - Cinna Davis (@1337isnot1337), #2226, with OpenAI Codex assistance: + anticipatory dirty borrowing (the borrowed helpers and their warps) + and the native exact Clifford and operator machinery. + - Tekalign (@tekuuu), #2436: the four lifted targets, the joint + twelve-source construction, and the sparse native accelerator, with + the authors its SPARSE-NOTICE names. + - zxv, #2530: the eight source-partner aliases; Mason Hale, #2696: + concentrated source-partner assignment and an independent local + interpreter; themilkman216, #2621: the priority order; Vlado + Vojdanovski, #2322: generator utilities; Jeremy Kothe, #2312: the + helper width exchange; Dugongue, #2309, with Sagar Thakrar's audit + #2315. + - icekylinx, #433 and #514, and the retained native ancestry; + DreamingOfClouds (with Claude): the inherited wrapper and outer + machinery; chafreaky (Chafik Boukhalfa), #458, and its lineage. + - Josie Elliston, #1956 (bootstrap carried-cutoff repair); Sagar + Thakrar, #1964 (label and adapter checks, mutants). + - The T74 words and owner construction underneath: Rohan Arun (#1218, + #1297, #1319 and the T74 exact-fill owner construction, with OpenAI + Codex assistance), Marius-Paul Dumitrean (#1625: the pivot-566 + kernel, words, edits and rebuild scripts), Jeremy Kothe and LeverRank + (#1672), on Jeremy Kothe's k8 leaver word and seven-slot line (#1018, + #1042, #1274) with Thomas DiFiore (#1148), Jeremy Diviney (#1193) and + Chafik Boukhalfa (#1196); the outer and the assembly by Vlado + Vojdanovski (#1673), Chafik Boukhalfa (#1699, #1786) and Jeremy + Kothe (#1377, #1839). + - Rohan Arun's draft #3939 and his #3934 stack, and Chris R / + Clear1nsight's #3883, #3936 and #3941, are acknowledged by #3945 as + neighbouring work; they are not used here. + Checks of #3945 by others, which this project read and re-ran: Chris R / + Clear1nsight's audits #3969, #3988, #3994 and #4016; the exact + eight-stream operator replay of ChaloupeMusic's gist, on the operator + model of Nick Petalas (#3187) and Matarener (#3296). + #2979, from its pull request body, CREDITS.md and NOTICE ("All original + notices of the exported predecessor packages remain in force"): + - Marius-Paul Dumitrean (mdumitrean), #2979, prepared with substantial + AI assistance (Anthropic Claude, via Claude Code): the codimension-2 + census, the joint end-frame lowering of 22 restored helpers in five + groups, the five further extractions, the composition with #2957's + rewrites, producer and checks, the overlay stage, the six-class + adapters, the literal T74 allocation, the cap certificates and the + complete fresh replay; also #2930 (cap allocation, relaxation + tools), #2942 (four normal-45 extractions; its words are #2979's + parent) and #2843 (the flow model's starting point). + - Jeremy Kothe, #2957, #2948 and #2966: the fourteen declared gate + rewrites and their applier; #2894 and #2833: the preload re-synthesis + supplier, frame accounting and cleanup, source materializer. + - Mason Hale, #2917: the nine paired extractions, the T74 chronology + (ORDER and DISPATCH), the allocation checker, moment auditor, + extension stage and adapter derivation. + - Jeremy Diviney, #2908: the paired input/output extraction, its + compensation rule, the two-sided certificate and the closed-adapter + admission chain. + - Dugongue, #2969 (comparator composition and baseline), #2863 and + #2839 (packing lineage, owner engine). + - Yiyang Lu (Lyy-iiis), #2857 and #2848: shared-kernel words, staged + native and overlay tooling, the flow model and realization of the + allocation. + - Ross Richardson, #2918: the rank-seven obstruction for helper 408. + - Tekalign: the independent owner and chronology checker; icekylinx: the + exact free-operator auditor (#2833); themilkman216: the integrality + observation behind the allocation's face step. + - Gareth Davidson (bitplane), #1995 and #2013: the fixed-word + terminal-subspace bounds that frame stage 1. + - Underneath: Cinna Davis (#2226, dirty borrowing and warps), Jeremy + Kothe (#2312, width exchange), and the T74 line above. + The table with every head kept on #2979 was made by #2979's own + allocation programs, credited as above. + h. Credit for the Lean route. + - Ryan Shea: the conversion pipeline, the generators and the Lean + instantiation of Jeremy Kothe's #588 circuit (lean/complex588), which + these packages are laid over. + - Nil Mamano: the files named at the start of this section. + - Dugongue (#401): the entrance gauges and the banks of the nine-stage + network; Zhihao Chen / jacklightChen (#377, #425): its compressed + layout with banks and padding, and the written argument for mutable + sources; lydakis (#275): the bank patterns; DaysSky (#363): the + bridged layout, and the proposal, with a written proof in the + community's terms, to host helpers in idle arrays. + - Jeremy Kothe and DreamingOfClouds: the community's lemmas about where + a helper may end, on which the charge for exit frames rests. + - For package/u8x0hq: the community's U8 circuit, Jeremy Kothe's #869 on + his #799, with Chafik Boukhalfa's frame changes (#741) and + icekylinx's gauge44 construction (#645), as the community's notices + record it. + - The idea that Theorem 1G extends is Schönhage's τ-theorem for matrix + multiplication. + - OpenAI: the cost model, the statement and the step from the lengths + 2^k to every length. + i. research/. Written for this repository by the agents under my + direction. The sections cite papers by title or arXiv identifier and + the community's pull requests by number, commit and file hash; they + copy no paper text and no file of the community's repository, with + these exceptions: research/record-track/programs/a-fc/r2/tools/ + fallback_patch.py carries a modified copy (made tolerant, as its first + lines say) of the placement loop of construction/build_fill.py of + #3815 (Erwan Secousses / ChaloupeMusic; CrocSwap/integer-mult-bounds, + Apache License, Version 2.0); and research/record-track/programs/ + a-fc/r2/tools/fillmodel.py copies the window, family and width logic + of owner/verify.py of #3945 (Dugongue and the lineage its NOTICE.md + names, see (g); CrocSwap/integer-mult-bounds, Apache License, Version + 2.0), as its first lines say; research/why-the-savers-save/programs/ + helpers-narrower-ends-skeptic-rules/asm1042.json is an unmodified + copy of research/community-frontier1042-audit/ASSEMBLY.json of the + community's main branch (the community's #1042 audit, integrated by + its maintainer Douglas R. Colkitt on 13 October 2026); + research/record-track/data/a-fc/design/ + ORDER.json and DISPATCH.json hold #2979's allocation/ORDER.json and + DISPATCH.json (Mason Hale's #2917, as #2979's NOTICE says), with the + same content, re-serialised; and research/why-the-savers-save/ + programs/helpers-half-reads/bit/xcheck_base.json, the report of the + community's checker xcheck_p.py (DreamingOfClouds' line, on eumemic's + #555) on #589's word, has the same bytes as + research/community-frontier741-audit/bit/xcheck.json of the + community's main branch, integrated by Douglas R. Colkitt on 13 + October 2026 (these three: CrocSwap/integer-mult-bounds, + Apache License, Version 2.0). fallback_patch.py, and tools/ + place_patch.py, tools/hall_fc.py and tools/mkpatch_fill17.py of + programs/a-fc/, apply their changes to copies of community files that + they read by path (#3815's build_fill.py and hall_lp.py, #2979's + fill17.py); those files are not changed. The other programs of + programs/a-fc/ are mine. + j. What is mine: the mixed-target recursion (Theorem 1G and its Lean + proof in OpenAI's model); the lever (the head test, the half cut, the + full cut and its 24 + 24 split, the head lemma, the heads-first + certificate, and the tables with every head kept); the A-fc table on + #2979's words (its class solution and its tolerant placement); the Lean ports of + #3945 and #2979 (the conversion of their + words, the warp checker and its soundness, the networks of many copies + with hosting across copies, the dispatch, the generators); the Lean + engine of this project and its extensions for exit frames, mutable + sources, centre reading and hosting; the checks; the search and pricing + programs in tools/lever/ and tools/lever-hub/; and the research in + research/. tools/lever-hub/network/Gates.lean is an unmodified copy of + lean/complex588/Work/Bridge401/Gates.lean of Ryan Shea's repository + (commit 52394849c7ecf3412d39b925d75fc7fdb98994d8, Apache License, + Version 2.0; its header credits Dugongue, lydakis, jacklightChen and + DaysSky); the other files of tools/lever-hub/ are mine. + + The new files of package/, scripts/, tools/lever/, data/, results/ and + research/ were made between 2026-10-13 and 2026-10-20 the way section 4 + describes: by AI agents (Claude) under my direction. The credits, with + links, are in README.md. + +Not included in this repository and fetched by the build: Mathlib +(https://github.com/leanprover-community/mathlib4, Apache License 2.0) and its +dependencies. The comparator (https://github.com/leanprover/comparator) is a +separate tool and is not included. diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Flow.lean b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Flow.lean new file mode 100644 index 0000000000..ee319585d9 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Flow.lean @@ -0,0 +1,507 @@ +import Mathlib.Data.Fin.Tuple.NatAntidiagonal +import Mathlib.Algebra.BigOperators.Fin +import Mathlib.Algebra.Order.BigOperators.Group.Finset +import Mathlib.Algebra.BigOperators.Group.Multiset.Basic +import Mathlib.Data.Nat.Find +import Mathlib.Algebra.Order.Group.Multiset +import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset +import Mathlib.Tactic.Ring +import Mathlib.Tactic.Linarith + +/-! +# R1: the job-model flow of Theorem 1G (rl-prove), natural numbers only + +rb-r12, 14 Oct 2026. The dilated lockstep tree for a SEPARATED mixed rule (every paid rank below every +target rank), with `L = n`. + +* `Rule` : K target ranks `kr k` with `Wk k` arrays each; P paid rounds `rk j` in word order. +* `uh h a` : Pascal weights on the points of height `h` (`u 0 = 1`, `u b = ∑_{b k ≥ 1} Wk k * u (b - e k)`). +* `pts n h ρ` : the points of a class of height `h` and multiplier `ρ`: `∑ a = h`, `m * ρ * col a ≤ n`. +* `inMS / outMS / srcMS` : the jobs (as multisets of sizes) a class consumes, emits in round `j`, takes from the top. +* `exitsSub / srcBelow / unitsSub` : the same summed over a subtree, by recursion on the height. +* `srcT n / exT n / unitsT n / Nsrc n` : the whole tree (root of height `G n`, multiplier 1). + +Lemmas: `balance` (Lemma B: a class's inputs = its parent's round-`j` outputs + its own sources), +`src_size` (Lemma C: sources lie in `(n/m, n]`), `exit_size`, `exit_lt` (Lemma D: exits are products of +`G+1` paid ranks, smaller than `n`), `sum_uh` (`∑ u = W^h`), `Nsrc_pos`, `card_srcT_le` (polynomial count). +The real-valued Phi facts (Lemma D's conservation, Lemma E) are in `Work.Mixed.Phi`. +-/ + +namespace OAI.PowerSaving.MX + +open Finset + +/-- A mixed rule with grouped paid blocks (Theorem 1G, rl-prove section 3), with the separation hypothesis. -/ +structure Rule where + K : ℕ + kr : Fin K → ℕ + Wk : Fin K → ℕ + m : ℕ + P : ℕ + rk : Fin P → ℕ + kmin : ℕ + rmax : ℕ + kmin_le : ∀ k, kmin ≤ kr k + kr_le : ∀ k, kr k ≤ m + rk_pos : ∀ j, 1 ≤ rk j + rk_le : ∀ j, rk j ≤ rmax + /-- hypothesis (S): every paid rank is below every target rank -/ + sep : rmax < kmin + two_le_kmin : 2 ≤ kmin + /-- the smallest target rank is attained by a target type that occurs -/ + kmin_att : ∃ k, kr k = kmin ∧ 1 ≤ Wk k + +lemma mem_nsmul_single {c x s : ℕ} (hx : x ∈ c • ({s} : Multiset ℕ)) : x = s := by + rcases Multiset.mem_nsmul.1 hx with ⟨_, h⟩ + simpa using h + +namespace Rule + +variable (R : Rule) + +/-- points of the lattice `ℕ^K` -/ +abbrev Pt := Fin R.K → ℕ + +/-- column factor `c(a) = ∏ k, kr k ^ a k` -/ +def col (a : R.Pt) : ℕ := ∏ k, R.kr k ^ a k + +/-- unit vector -/ +def e (k : Fin R.K) : R.Pt := Pi.single k 1 + +/-- total number of arrays of a unit -/ +def W : ℕ := ∑ k, R.Wk k + +/-- Pascal weights by height: `uh 0 = [a = 0]`, `uh (h+1) a = ∑_{a k ≥ 1} Wk k * uh h (a - e k)`. -/ +def uh : ℕ → R.Pt → ℕ + | 0, a => if a = 0 then 1 else 0 + | h + 1, a => ∑ k, if 1 ≤ a k then R.Wk k * uh h (a - R.e k) else 0 + +/-- the tree depth `G n = max {g ≤ n : m * kmin^g ≤ n}` -/ +def G (n : ℕ) : ℕ := Nat.findGreatest (fun g => R.m * R.kmin ^ g ≤ n) n + +/-- points of height `h`, multiplier `ρ` (the set `R(ρ)` of rl-prove, restricted to `|a| = h`) -/ +def pts (n h ρ : ℕ) : Finset R.Pt := + (Finset.Nat.antidiagonalTuple R.K h).filter (fun b => R.m * ρ * R.col b ≤ n) + +/-- what a class of height `h`, multiplier `ρ` consumes: at each point `b`, `uh h b * Wk k` jobs of size +`ρ * kr k * col b` -/ +def inMS (n h ρ : ℕ) : Multiset ℕ := + ∑ b ∈ R.pts n h ρ, ∑ k, (R.uh h b * R.Wk k) • ({ρ * R.kr k * R.col b} : Multiset ℕ) + +/-- what round `j` of a class emits: at each point `b`, `uh h b` jobs of size `ρ * rk j * col b` -/ +def outMS (n h ρ : ℕ) (j : Fin R.P) : Multiset ℕ := + ∑ b ∈ R.pts n h ρ, R.uh h b • ({ρ * R.rk j * R.col b} : Multiset ℕ) + +/-- sources of a (non-root) class whose parent has multiplier `ρp`: inputs whose producer point `b + e k` +lies outside the parent's region -/ +def srcMS (n h ρ ρp : ℕ) : Multiset ℕ := + ∑ b ∈ R.pts n h ρ, ∑ k, + if n < R.m * ρp * R.col (b + R.e k) then (R.uh h b * R.Wk k) • ({ρ * R.kr k * R.col b} : Multiset ℕ) + else 0 + +/-- number of fluid units of a class -/ +def unitsC (n h ρ : ℕ) : ℕ := ∑ b ∈ R.pts n h ρ, R.uh h b + +/-- exits of a subtree (outputs of its leaf classes) -/ +def exitsSub (n : ℕ) : ℕ → ℕ → Multiset ℕ + | 0, ρ => ∑ j, R.outMS n 0 ρ j + | h + 1, ρ => ∑ j, exitsSub n h (ρ * R.rk j) + +/-- sources of the strict descendants of a class -/ +def srcBelow (n : ℕ) : ℕ → ℕ → Multiset ℕ + | 0, _ => 0 + | h + 1, ρ => ∑ j, (R.srcMS n h (ρ * R.rk j) ρ + srcBelow n h (ρ * R.rk j)) + +/-- fluid units of a subtree -/ +def unitsSub (n : ℕ) : ℕ → ℕ → ℕ + | 0, ρ => R.unitsC n 0 ρ + | h + 1, ρ => R.unitsC n (h + 1) ρ + ∑ j, unitsSub n h (ρ * R.rk j) + +/-- all sources of the tree for `n` (the root's inputs are all sources) -/ +def srcT (n : ℕ) : Multiset ℕ := R.inMS n (R.G n) 1 + R.srcBelow n (R.G n) 1 + +/-- all exits of the tree for `n` -/ +def exT (n : ℕ) : Multiset ℕ := R.exitsSub n (R.G n) 1 + +/-- all fluid units of the tree for `n` -/ +def unitsT (n : ℕ) : ℕ := R.unitsSub n (R.G n) 1 + +/-- number of source vectors of one tree copy -/ +def Nsrc (n : ℕ) : ℕ := Multiset.card (R.srcT n) + +/-! ### basic facts -/ + +lemma e_apply (k k' : Fin R.K) : R.e k k' = if k' = k then 1 else 0 := by + unfold e; rw [Pi.single_apply] + +lemma sum_add_e (b : R.Pt) (k : Fin R.K) : ∑ i, (b + R.e k) i = ∑ i, b i + 1 := by + simp only [Pi.add_apply, Finset.sum_add_distrib, e_apply, Finset.sum_ite_eq', Finset.mem_univ, ite_true] + +lemma col_add_e (b : R.Pt) (k : Fin R.K) : R.col (b + R.e k) = R.kr k * R.col b := by + unfold col + simp only [Pi.add_apply, pow_add, Finset.prod_mul_distrib, e_apply] + rw [mul_comm] + congr 1 + rw [Finset.prod_eq_single k] + · simp + · intro b _ hb; simp [hb] + · simp + +lemma sub_e_add_e (a : R.Pt) (k : Fin R.K) (h : 1 ≤ a k) : a - R.e k + R.e k = a := by + funext i + simp only [Pi.add_apply, Pi.sub_apply, e_apply] + split_ifs with hi + · subst hi; omega + · omega + +lemma add_e_sub_e (b : R.Pt) (k : Fin R.K) : b + R.e k - R.e k = b := by + funext i + simp only [Pi.add_apply, Pi.sub_apply, e_apply] + omega + +lemma add_e_apply_self (b : R.Pt) (k : Fin R.K) : 1 ≤ (b + R.e k) k := by + simp [e_apply] + +/-- The reindexing behind Lemma B: pairs `(b, k)` with `∑ b = h` correspond to pairs `(a, k)` with +`∑ a = h + 1`, `a k ≥ 1`, through `a = b + e k`. -/ +lemma sum_shift {M : Type*} [AddCommMonoid M] (h : ℕ) (F : R.Pt → Fin R.K → M) : + ∑ b ∈ Finset.Nat.antidiagonalTuple R.K h, ∑ k, F b k = + ∑ a ∈ Finset.Nat.antidiagonalTuple R.K (h + 1), ∑ k, if 1 ≤ a k then F (a - R.e k) k else 0 := by + rw [Finset.sum_comm, Finset.sum_comm (s := Finset.Nat.antidiagonalTuple R.K (h + 1))] + refine Finset.sum_congr rfl (fun k _ => ?_) + rw [← Finset.sum_filter] + refine Finset.sum_nbij' (fun b => b + R.e k) (fun a => a - R.e k) ?_ ?_ ?_ ?_ ?_ + · intro b hb + simp only [Finset.mem_coe, Finset.mem_filter, Finset.Nat.mem_antidiagonalTuple] at hb ⊢ + exact ⟨by rw [R.sum_add_e, hb], R.add_e_apply_self b k⟩ + · intro a ha + simp only [Finset.mem_coe, Finset.mem_filter, Finset.Nat.mem_antidiagonalTuple] at ha ⊢ + have h1 := R.sum_add_e (a - R.e k) k + rw [R.sub_e_add_e a k ha.2, ha.1] at h1 + omega + · intro b _; exact R.add_e_sub_e b k + · intro a ha + simp only [Finset.mem_filter] at ha + exact R.sub_e_add_e a k ha.2 + · intro b _; rw [R.add_e_sub_e] + +lemma rk_lt_kr (j : Fin R.P) (k : Fin R.K) : R.rk j < R.kr k := + lt_of_le_of_lt (R.rk_le j) (lt_of_lt_of_le R.sep (R.kmin_le k)) + +lemma two_le_m : 2 ≤ R.m := by + obtain ⟨k, hk, _⟩ := R.kmin_att + have := R.kr_le k; have := R.two_le_kmin; omega + +lemma kmin_pos : 0 < R.kmin := by have := R.two_le_kmin; omega + +lemma kr_pos (k : Fin R.K) : 0 < R.kr k := lt_of_lt_of_le R.kmin_pos (R.kmin_le k) + +lemma col_pos (b : R.Pt) : 0 < R.col b := by + unfold col; exact Finset.prod_pos (fun k _ => pow_pos (R.kr_pos k) _) + +lemma pow_le_col (b : R.Pt) : R.kmin ^ (∑ i, b i) ≤ R.col b := by + unfold col + rw [← Finset.prod_pow_eq_pow_sum] + exact Finset.prod_le_prod' (fun k _ => Nat.pow_le_pow_left (R.kmin_le k) _) + +lemma col_zero : R.col 0 = 1 := by simp [col] + +lemma mem_T_zero {b : R.Pt} (hb : b ∈ Finset.Nat.antidiagonalTuple R.K 0) : b = 0 := by + rw [Finset.Nat.antidiagonalTuple_zero_right, Finset.mem_singleton] at hb; exact hb + +/-- `G n` is the largest `g` with `m * kmin^g ≤ n` (when `m ≤ n`). -/ +lemma m_kmin_pow_G_le {n : ℕ} (hn : R.m ≤ n) : R.m * R.kmin ^ R.G n ≤ n := + Nat.findGreatest_spec (P := fun g => R.m * R.kmin ^ g ≤ n) (Nat.zero_le n) (by simpa using hn) + +lemma lt_m_kmin_pow_G_succ (n : ℕ) : n < R.m * R.kmin ^ (R.G n + 1) := by + by_cases hle : R.G n + 1 ≤ n + · have := Nat.findGreatest_is_greatest (P := fun g => R.m * R.kmin ^ g ≤ n) (Nat.lt_succ_self _) hle + exact Nat.lt_of_not_le this + · have h1 : n < 2 ^ (R.G n + 1) := lt_of_lt_of_le (by omega : n < R.G n + 1) (Nat.lt_two_pow_self).le + have h2 : 2 ^ (R.G n + 1) ≤ R.kmin ^ (R.G n + 1) := Nat.pow_le_pow_left R.two_le_kmin _ + have h3 : R.kmin ^ (R.G n + 1) ≤ R.m * R.kmin ^ (R.G n + 1) := + Nat.le_mul_of_pos_left _ (by have := R.two_le_m; omega) + omega + + +/-! ### Lemma B: exact balance -/ + +/-- **Lemma B** (exact balance, rl-prove section 3). The inputs of the child class `ρp * rk j` of height `h` +are exactly the round-`j` outputs of its parent (height `h+1`, multiplier `ρp`) plus the child's sources. -/ +theorem balance (n h ρp : ℕ) (j : Fin R.P) : + R.inMS n h (ρp * R.rk j) = R.outMS n (h + 1) ρp j + R.srcMS n h (ρp * R.rk j) ρp := by + set ρ := ρp * R.rk j with hρ + -- split every input by whether its producer lies in the parent's region + have hsplit : R.inMS n h ρ = + (∑ b ∈ R.pts n h ρ, ∑ k, if n < R.m * ρp * R.col (b + R.e k) then 0 + else (R.uh h b * R.Wk k) • ({ρ * R.kr k * R.col b} : Multiset ℕ)) + R.srcMS n h ρ ρp := by + unfold inMS srcMS + rw [← Finset.sum_add_distrib] + refine Finset.sum_congr rfl (fun b _ => ?_) + rw [← Finset.sum_add_distrib] + refine Finset.sum_congr rfl (fun k _ => ?_) + split_ifs <;> simp + rw [hsplit, add_left_inj] + -- move to sums over the whole antidiagonal and shift + unfold outMS pts + rw [Finset.sum_filter, Finset.sum_filter] + have hL : (∑ b ∈ Finset.Nat.antidiagonalTuple R.K h, if R.m * ρ * R.col b ≤ n then + ∑ k, (if n < R.m * ρp * R.col (b + R.e k) then 0 + else (R.uh h b * R.Wk k) • ({ρ * R.kr k * R.col b} : Multiset ℕ)) else 0) = + ∑ b ∈ Finset.Nat.antidiagonalTuple R.K h, ∑ k, + (if R.m * ρ * R.col b ≤ n ∧ ¬ n < R.m * ρp * R.col (b + R.e k) then + (R.uh h b * R.Wk k) • ({ρ * R.kr k * R.col b} : Multiset ℕ) else 0) := by + refine Finset.sum_congr rfl (fun b _ => ?_) + split_ifs with h1 + · refine Finset.sum_congr rfl (fun k _ => ?_) + split_ifs <;> simp_all + · symm; refine Finset.sum_eq_zero (fun k _ => ?_) + rw [if_neg]; tauto + rw [hL, R.sum_shift h] + refine Finset.sum_congr rfl (fun a ha => ?_) + -- the parent side: expand `uh (h+1) a` by Pascal's rule + have hR : (if R.m * ρp * R.col a ≤ n then R.uh (h + 1) a • ({ρp * R.rk j * R.col a} : Multiset ℕ) else 0) = + ∑ k, if 1 ≤ a k then (if R.m * ρp * R.col a ≤ n then + (R.Wk k * R.uh h (a - R.e k)) • ({ρp * R.rk j * R.col a} : Multiset ℕ) else 0) else 0 := by + split_ifs with h1 + · simp only [uh] + rw [← Finset.sum_nsmul_assoc] + refine Finset.sum_congr rfl (fun k _ => ?_) + split_ifs <;> simp + · simp + rw [hR] + refine Finset.sum_congr rfl (fun k _ => ?_) + by_cases hak : 1 ≤ a k + · rw [if_pos hak, if_pos hak] + have hcol : R.col a = R.kr k * R.col (a - R.e k) := by + conv_lhs => rw [← R.sub_e_add_e a k hak] + exact R.col_add_e _ _ + rw [R.sub_e_add_e a k hak] + have hiff : (R.m * ρ * R.col (a - R.e k) ≤ n ∧ ¬ n < R.m * ρp * R.col a) ↔ R.m * ρp * R.col a ≤ n := by + constructor + · rintro ⟨_, h2⟩; omega + · intro h2 + refine ⟨?_, by omega⟩ + rw [hρ] + have := R.rk_lt_kr j k + calc R.m * (ρp * R.rk j) * R.col (a - R.e k) = R.m * ρp * (R.rk j * R.col (a - R.e k)) := by ring + _ ≤ R.m * ρp * (R.kr k * R.col (a - R.e k)) := + Nat.mul_le_mul_left _ (Nat.mul_le_mul_right _ this.le) + _ = R.m * ρp * R.col a := by rw [hcol] + _ ≤ n := h2 + by_cases hc : R.m * ρp * R.col a ≤ n + · rw [if_pos (hiff.2 hc), if_pos hc, mul_comm (R.uh h _), hcol, hρ, + show ρp * R.rk j * R.kr k * R.col (a - R.e k) = ρp * R.rk j * (R.kr k * R.col (a - R.e k)) by ring] + · rw [if_neg (fun h' => hc (hiff.1 h')), if_neg hc] + · rw [if_neg hak, if_neg hak] + +/-! ### Lemma C: sources lie in `(n/m, n]` -/ + +lemma mem_srcMS {n h ρp : ℕ} {j : Fin R.P} {x : ℕ} (hx : x ∈ R.srcMS n h (ρp * R.rk j) ρp) : + n < R.m * x ∧ x ≤ n := by + unfold srcMS at hx + obtain ⟨b, hb, hx⟩ := Multiset.mem_sum.1 hx + obtain ⟨k, -, hx⟩ := Multiset.mem_sum.1 hx + simp only [pts, Finset.mem_filter] at hb + split_ifs at hx with hc + · have hx := mem_nsmul_single hx + subst hx + rw [R.col_add_e] at hc + have h1 := R.rk_pos j + have h2 := R.kr_le k + constructor + · calc n < R.m * ρp * (R.kr k * R.col b) := hc + _ = 1 * (R.m * (ρp * R.kr k * R.col b)) := by ring + _ ≤ R.rk j * (R.m * (ρp * R.kr k * R.col b)) := Nat.mul_le_mul_right _ h1 + _ = R.m * (ρp * R.rk j * R.kr k * R.col b) := by ring + · calc ρp * R.rk j * R.kr k * R.col b ≤ ρp * R.rk j * R.m * R.col b := + Nat.mul_le_mul_right _ (Nat.mul_le_mul_left _ h2) + _ = R.m * (ρp * R.rk j) * R.col b := by ring + _ ≤ n := hb.2 + · simp at hx + +lemma mem_srcBelow {n h ρ x : ℕ} (hx : x ∈ R.srcBelow n h ρ) : n < R.m * x ∧ x ≤ n := by + induction h generalizing ρ with + | zero => simp [srcBelow] at hx + | succ h ih => + simp only [srcBelow] at hx + obtain ⟨j, -, hx⟩ := Multiset.mem_sum.1 hx + rcases Multiset.mem_add.1 hx with hx | hx + · exact R.mem_srcMS hx + · exact ih hx + +lemma mem_inMS_root {n x : ℕ} (hx : x ∈ R.inMS n (R.G n) 1) : n < R.m * x ∧ x ≤ n := by + unfold inMS at hx + obtain ⟨b, hb, hx⟩ := Multiset.mem_sum.1 hx + obtain ⟨k, -, hx⟩ := Multiset.mem_sum.1 hx + have hx := mem_nsmul_single hx + subst hx + simp only [pts, Finset.mem_filter, Finset.Nat.mem_antidiagonalTuple] at hb + constructor + · have h1 := R.lt_m_kmin_pow_G_succ n + have h2 := R.pow_le_col (b + R.e k) + rw [R.sum_add_e, hb.1, R.col_add_e] at h2 + calc n < R.m * R.kmin ^ (R.G n + 1) := h1 + _ ≤ R.m * (R.kr k * R.col b) := Nat.mul_le_mul_left _ h2 + _ = R.m * (1 * R.kr k * R.col b) := by ring + · have h2 := R.kr_le k + calc 1 * R.kr k * R.col b ≤ R.m * R.col b := by rw [one_mul]; exact Nat.mul_le_mul_right _ h2 + _ = R.m * 1 * R.col b := by ring + _ ≤ n := hb.2 + +/-- **Lemma C**: every source of the tree for `n` has size in `(n/m, n]`. -/ +theorem src_size {n x : ℕ} (hx : x ∈ R.srcT n) : n < R.m * x ∧ x ≤ n := by + rcases Multiset.mem_add.1 hx with hx | hx + · exact R.mem_inMS_root hx + · exact R.mem_srcBelow hx + +lemma src_pos {n x : ℕ} (hx : x ∈ R.srcT n) : 1 ≤ x := by + have := (R.src_size hx).1 + rcases Nat.eq_zero_or_pos x with h | h + · subst h; simp at this + · exact h + +/-! ### Lemma D (sizes): exits are products of `G + 1` paid ranks, below `n` -/ + +lemma mem_exitsSub {n h ρ x : ℕ} (hx : x ∈ R.exitsSub n h ρ) : + ∃ ρ' : ℕ, ∃ j : Fin R.P, x = ρ' * R.rk j ∧ R.m * ρ' ≤ n ∧ ρ ≤ ρ' ∧ ρ' ≤ ρ * R.rmax ^ h := by + induction h generalizing ρ with + | zero => + simp only [exitsSub] at hx + obtain ⟨j, -, hx⟩ := Multiset.mem_sum.1 hx + unfold outMS at hx + obtain ⟨b, hb, hx⟩ := Multiset.mem_sum.1 hx + have hx := mem_nsmul_single hx + simp only [pts, Finset.mem_filter] at hb + have hb0 := R.mem_T_zero hb.1 + subst hb0 + rw [R.col_zero] at hx hb + exact ⟨ρ, j, by rw [hx]; ring, by simpa using hb.2, le_rfl, by simp⟩ + | succ h ih => + simp only [exitsSub] at hx + obtain ⟨j, -, hx⟩ := Multiset.mem_sum.1 hx + obtain ⟨ρ', j', h1, h2, h3, h4⟩ := ih hx + refine ⟨ρ', j', h1, h2, ?_, ?_⟩ + · exact le_trans (Nat.le_mul_of_pos_right _ (R.rk_pos j)) h3 + · calc ρ' ≤ ρ * R.rk j * R.rmax ^ h := h4 + _ ≤ ρ * R.rmax * R.rmax ^ h := Nat.mul_le_mul_right _ (Nat.mul_le_mul_left _ (R.rk_le j)) + _ = ρ * R.rmax ^ (h + 1) := by ring + +/-- **Lemma D (sizes)**: every exit of the tree for `n` is at least 1, at most `rmax^(G n + 1)`, and below `n`. -/ +theorem exit_size {n x : ℕ} (hx : x ∈ R.exT n) : 1 ≤ x ∧ x ≤ R.rmax ^ (R.G n + 1) ∧ x < n := by + obtain ⟨ρ', j, rfl, h2, h3, h4⟩ := R.mem_exitsSub hx + have hr := R.rk_pos j + refine ⟨Nat.one_le_iff_ne_zero.2 (Nat.mul_ne_zero (by omega) (by omega)), ?_, ?_⟩ + · calc ρ' * R.rk j ≤ (1 * R.rmax ^ R.G n) * R.rmax := Nat.mul_le_mul h4 (R.rk_le j) + _ = R.rmax ^ (R.G n + 1) := by ring + · have h5 : R.rk j < R.m := lt_of_lt_of_le (R.rk_lt_kr j (R.kmin_att.choose)) (R.kr_le _) + calc ρ' * R.rk j < ρ' * R.m := Nat.mul_lt_mul_of_pos_left h5 (by omega) + _ = R.m * ρ' := by ring + _ ≤ n := h2 + +/-! ### counts -/ + +lemma uh_single (k : Fin R.K) (h : ℕ) (hW : 1 ≤ R.Wk k) : 1 ≤ R.uh h (Pi.single k h) := by + induction h with + | zero => simp [uh] + | succ h ih => + simp only [uh] + have hk : 1 ≤ (Pi.single k (h + 1) : R.Pt) k := by simp + have hsub : (Pi.single k (h + 1) : R.Pt) - R.e k = Pi.single k h := by + funext i; simp only [Pi.sub_apply, e_apply, Pi.single_apply]; split_ifs <;> omega + calc 1 ≤ R.Wk k * R.uh h (Pi.single k h) := Nat.one_le_iff_ne_zero.2 (Nat.mul_ne_zero (by omega) (by omega)) + _ = (fun k' => if 1 ≤ (Pi.single k (h + 1) : R.Pt) k' then + R.Wk k' * R.uh h ((Pi.single k (h + 1) : R.Pt) - R.e k') else 0) k := by + simp only [if_pos hk, hsub] + _ ≤ ∑ k', (if 1 ≤ (Pi.single k (h + 1) : R.Pt) k' then + R.Wk k' * R.uh h ((Pi.single k (h + 1) : R.Pt) - R.e k') else 0) := + Finset.single_le_sum (f := fun k' => if 1 ≤ (Pi.single k (h + 1) : R.Pt) k' then + R.Wk k' * R.uh h ((Pi.single k (h + 1) : R.Pt) - R.e k') else 0) + (fun _ _ => Nat.zero_le _) (Finset.mem_univ k) + +lemma card_inMS (n h ρ : ℕ) : + Multiset.card (R.inMS n h ρ) = ∑ b ∈ R.pts n h ρ, ∑ k, R.uh h b * R.Wk k := by + simp [inMS, Multiset.card_sum, Multiset.card_nsmul] + +/-- the tree for `n ≥ m` has at least one source -/ +theorem Nsrc_pos {n : ℕ} (hn : R.m ≤ n) : 1 ≤ R.Nsrc n := by + obtain ⟨k, hk, hW⟩ := R.kmin_att + obtain ⟨b, hbdef⟩ : ∃ b : R.Pt, b = Pi.single k (R.G n) := ⟨_, rfl⟩ + have hub : 1 ≤ R.uh (R.G n) b := by rw [hbdef]; exact R.uh_single k (R.G n) hW + have hbT : b ∈ R.pts n (R.G n) 1 := by + simp only [pts, Finset.mem_filter, Finset.Nat.mem_antidiagonalTuple] + refine ⟨by simp [hbdef], ?_⟩ + have : R.col b = R.kmin ^ R.G n := by + simp only [col, hbdef, Pi.single_apply] + rw [Finset.prod_eq_single k] + · simp [hk] + · intro i _ hi; simp [hi] + · simp + rw [this, mul_one]; exact R.m_kmin_pow_G_le hn + have h1 : 1 ≤ R.uh (R.G n) b * R.Wk k := Nat.one_le_iff_ne_zero.2 + (Nat.mul_ne_zero (by omega) (by omega)) + have h2 : R.uh (R.G n) b * R.Wk k ≤ ∑ k', R.uh (R.G n) b * R.Wk k' := + Finset.single_le_sum (f := fun k' => R.uh (R.G n) b * R.Wk k') (fun _ _ => Nat.zero_le _) (Finset.mem_univ k) + have h3 : ∑ k', R.uh (R.G n) b * R.Wk k' ≤ ∑ b' ∈ R.pts n (R.G n) 1, ∑ k', R.uh (R.G n) b' * R.Wk k' := + Finset.single_le_sum (f := fun b' => ∑ k', R.uh (R.G n) b' * R.Wk k') (fun _ _ => Nat.zero_le _) hbT + unfold Nsrc srcT + rw [Multiset.card_add, card_inMS] + omega + +/-- the multinomial theorem in Pascal form: `∑_{|b| = h} u b = W^h` -/ +theorem sum_uh (h : ℕ) : ∑ b ∈ Finset.Nat.antidiagonalTuple R.K h, R.uh h b = R.W ^ h := by + induction h with + | zero => + rw [Finset.Nat.antidiagonalTuple_zero_right]; simp [uh] + | succ h ih => + have := R.sum_shift h (fun b k => R.Wk k * R.uh h b) + simp only [uh] + rw [← this] + simp_rw [← Finset.sum_mul] + rw [← Finset.mul_sum, ih]; unfold W; ring + +lemma card_inMS_le (n h ρ : ℕ) : Multiset.card (R.inMS n h ρ) ≤ R.W ^ (h + 1) := by + rw [card_inMS] + calc ∑ b ∈ R.pts n h ρ, ∑ k, R.uh h b * R.Wk k + = ∑ b ∈ R.pts n h ρ, R.uh h b * R.W := by simp_rw [← Finset.mul_sum]; rfl + _ ≤ ∑ b ∈ Finset.Nat.antidiagonalTuple R.K h, R.uh h b * R.W := + Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun _ _ _ => Nat.zero_le _) + _ = R.W ^ (h + 1) := by rw [← Finset.sum_mul, sum_uh, pow_succ] + +lemma card_srcMS_le (n h ρ ρp : ℕ) : Multiset.card (R.srcMS n h ρ ρp) ≤ Multiset.card (R.inMS n h ρ) := by + unfold srcMS inMS + simp only [Multiset.card_sum] + refine Finset.sum_le_sum (fun b _ => Finset.sum_le_sum (fun k _ => ?_)) + split_ifs <;> simp + +lemma card_sub_le (n h ρ : ℕ) : + Multiset.card (R.inMS n h ρ) + Multiset.card (R.srcBelow n h ρ) ≤ (R.P + R.W + 1) ^ (h + 1) := by + induction h generalizing ρ with + | zero => + simp only [srcBelow, Multiset.card_zero, add_zero, zero_add, pow_one] + have := R.card_inMS_le n 0 ρ + simp at this; omega + | succ h ih => + have h1 := R.card_inMS_le n (h + 1) ρ + have h2 : Multiset.card (R.srcBelow n (h + 1) ρ) ≤ R.P * (R.P + R.W + 1) ^ (h + 1) := by + simp only [srcBelow, Multiset.card_sum, Multiset.card_add] + calc ∑ j, (Multiset.card (R.srcMS n h (ρ * R.rk j) ρ) + Multiset.card (R.srcBelow n h (ρ * R.rk j))) + ≤ ∑ _j : Fin R.P, (R.P + R.W + 1) ^ (h + 1) := Finset.sum_le_sum (fun j _ => + le_trans (Nat.add_le_add_right (R.card_srcMS_le n h _ ρ) _) (ih _)) + _ = R.P * (R.P + R.W + 1) ^ (h + 1) := by simp + have h3 : R.W ^ (h + 1 + 1) ≤ R.W * (R.P + R.W + 1) ^ (h + 1) := by + rw [pow_succ', ] + exact Nat.mul_le_mul_left _ (Nat.pow_le_pow_left (by omega) _) + calc Multiset.card (R.inMS n (h + 1) ρ) + Multiset.card (R.srcBelow n (h + 1) ρ) + ≤ R.W * (R.P + R.W + 1) ^ (h + 1) + R.P * (R.P + R.W + 1) ^ (h + 1) := by omega + _ ≤ (R.P + R.W + 1) * (R.P + R.W + 1) ^ (h + 1) := by + rw [← add_mul]; exact Nat.mul_le_mul_right _ (by omega) + _ = (R.P + R.W + 1) ^ (h + 1 + 1) := by ring + +/-- the number of sources is polynomial in `n`: at most `(P + W + 1)^(G n + 1)` -/ +theorem Nsrc_le (n : ℕ) : R.Nsrc n ≤ (R.P + R.W + 1) ^ (R.G n + 1) := by + unfold Nsrc srcT; rw [Multiset.card_add]; exact R.card_sub_le n (R.G n) 1 + +end Rule + +end OAI.PowerSaving.MX diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/ClassBody.lean b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/ClassBody.lean new file mode 100644 index 0000000000..59859aee19 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/ClassBody.lean @@ -0,0 +1,286 @@ +import Work.Mixed.Prog.SweepLeaf + +/-! +# Mixed engine, R3 program part 17: the body of a class call (agent e-r3prog) + +From the port input `(n, I, h+1, μ, o, own jobs, supply)`: `2^n`, the class tables, load, the sweep of +the whole word (child calls or exits), unload, answer `(own answers, consumed answers, sizes)`. + +* `Eng.class_up` a class of height `h+1` (children); +* `Eng.class_leaf` a class of height `0` (exits). +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving.RAM +open Binary Matrix Ty Finset GF MXR +noncomputable section + +universe U + +namespace Eng +variable {α ρ σ : Type*} [Fintype α] [DecidableEq α] [Fintype ρ] [DecidableEq ρ] + [Fintype σ] [DecidableEq σ] +variable (E : Eng α ρ σ) +variable {ι : Type U} {m : Bool} {cl : Paint} {b : Bank (portM cl) ι} +variable {n h μ o BS : ι → ℕ} {J sup : ∀ i, ℕ → Bits (n i) → ℂ} {τ : ι → MDesc → ℕ} + +/-- The empty tapes are all equal. -/ +lemma jobsTape_zero (n : ℕ) (f g : ℕ → Bits n → ℂ) : jobsTape n 0 f = jobsTape n 0 g := by + unfold jobsTape; exact tape_std_ext rfl _ _ (fun k h1 _ => absurd h1 (Nat.not_lt_zero _)) + +/-- The own answers after the word are `hlow` of the own jobs. -/ +lemma own_answers (n h μ : ℕ) (J : ℕ → Bits n → ℂ) : + jobsTape n (E.NUc n h μ * E.W) (E.unl n h μ (E.runV (E.V0 n h μ J) E.wd)) = + jobsTape n (E.NUc n h μ * E.W) (fun t => hlow n (E.slotSize n h μ t) (J t)) := by + unfold jobsTape + apply tape_std_ext rfl + intro k h1 _ + show (bits n).tape (E.unl n h μ (E.runV (E.V0 n h μ J) E.wd) k) = (bits n).tape (hlow n (E.slotSize n h μ k) (J k)) + rw [E.unl_run n h μ J k h1] + +/-- The decoded port input of a class. -/ +lemma decode {x : ι → T (portIn cl)} + (hx : ∀ i, x i = mkIn cl (n i) (h i + 1) (μ i) (o i) (jobsTape (n i) (E.NUc (n i) (h i) (μ i) * E.W) (J i)) + (jobsTape (n i) (BS i) (sup i))) : + Knows m b.B (portIn cl) x n ∧ Has m b.B (portIn cl) sc x (fun _ => Complex.I) ∧ + Knows m b.B (portIn cl) x (fun i => h i + 1) ∧ Knows m b.B (portIn cl) x μ ∧ Knows m b.B (portIn cl) x o ∧ + Has m b.B (portIn cl) (jobsT cl) x (fun i => jobsTape (n i) (E.NUc (n i) (h i) (μ i) * E.W) (J i)) ∧ + Has m b.B (portIn cl) (jobsT cl) x (fun i => jobsTape (n i) (BS i) (sup i)) := by + refine ⟨Can.first.cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.fst Can.second).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.fst (Can.snd Can.second)).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.fst (Can.snd (Can.snd Can.second))).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.fst (Can.snd (Can.snd (Can.snd Can.second)))).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.fst (Can.snd (Can.snd (Can.snd (Can.snd Can.second))))).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl), + (Can.snd (Can.snd (Can.snd (Can.snd (Can.snd Can.second))))).cong (fun _ => rfl) (fun i => by rw [hx i]; rfl)⟩ + +/-- Local work of a class of height `h+1`. -/ +def locUp (n h μ : ℕ) : ℕ := + (n + 1) + E.idxCost n (h+1) μ + E.NUc n (h+1) μ * (4 * n + 4 * (h + 1) + 8) + + E.NUc n (h+1) μ * (2 * 2^n + 2) + E.NUc n (h+1) μ * 2^n + E.swCost n h μ 0 E.wd + + E.NUc n (h+1) μ * E.W * 2^n + +lemma szT_nil : szT [] = (Layout.std 0).tape (fun _ : Fin 0 => (0 : ℕ)) := by + unfold szT; exact tape_std_ext rfl _ _ (fun k h1 _ => absurd h1 (Nat.not_lt_zero _)) + +/-- **The body of a class of height `h+1`.** -/ +theorem class_up (hB2 : Small b.B (fun i => 2^(n i))) (hBS : Small b.B BS) + (hle : ∀ i, h i + 1 ≤ n i) (hμ : Small b.B μ) + (hfit : ∀ i, o i + E.cons (n i) (h i + 1) (μ i) ≤ BS i) + (hk : ∀ i c, c ∈ E.kidsC (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i)) → + (b.use i (E.enc cl c)).OK (E.out cl c) (τ i c) (b.ext i)) : + Able m b (portIn cl) (portOut cl) + (fun i => E.enc cl (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.out cl (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.locUp (n i) (h i) (μ i)) + (fun i => ((E.kidsC (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i))).map (τ i)).sum) := by + let x0 (i : ι) := E.enc cl (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i)) + obtain ⟨hn, hI, hkd, hμK, hoK, hown, hsup⟩ := E.decode (b:=b) (x:=x0) (n:=n) (h:=fun i => h i + 1) (μ:=μ) + (o:=o) (BS:=BS) (J:=J) (sup:=sup) (fun i => rfl) + -- 2^n + have hnS : Small b.B n := hB2.mono (fun i => Nat.lt_two_pow_self.le) + have H2n : Can m b.B (portIn cl) w x0 (fun i => 2^(n i)) (fun i => n i + 1) := + (hn.mono (fun _ => Nat.zero_le _)).wTwo hnS hB2 (Dom.of_le (fun i => by omega)) + let x1 (i : ι) : T (p (portIn cl) w) := (x0 i, 2^(n i)) + have c1 : CIn m b.B (p (portIn cl) w) x1 n (fun i => h i + 1) μ := + ⟨Can.first.then_do hn, Can.first.then_do ((hkd.wsub (Can.wc 1) (hnS.mono (fun i => by have := hle i; omega))).cong + (fun _ => rfl) (fun i => by simp)), Can.first.then_do hμK, Can.second, Can.first.then_do hI, hB2, + hle, hμ⟩ + have Hidx := E.can_index c1 + have Hun := E.can_units c1 + have Hge := E.can_gears c1 + have cc := E.cctx c1 + let idx (i : ι) := E.idxVal (n i) (h i + 1) (μ i) + let un (i : ι) := E.unitVal (n i) (h i + 1) (μ i) + let ge (i : ι) := E.gearVal (n i) (h i + 1) (μ i) + let s4 : Ty := ClsT (p (portIn cl) w) + let x4 (i : ι) : T s4 := (((x1 i, idx i), un i), ge i) + have F4 : ∀ {u : Ty} {z : ι → T u}, Has m b.B (portIn cl) u x0 z → Has m b.B s4 u x4 z := + fun g => Can.first.then_do (Can.first.then_do (Can.first.then_do (Can.first.then_do g))) + have hidx4 : Has m b.B s4 IdxT x4 idx := Can.first.then_do (Can.first.then_do Can.second) + have c4 : CIn m b.B s4 x4 n (fun i => h i + 1) μ := + ((c1.ext (t:=IdxT) idx).ext (t:=UnT) un).ext (t:=GearT) ge + -- load + have Hld := E.load cc (J:=J) (F4 hown) + let ct0 (i : ι) : T (a (a (a (c cl)))) := E.ctabE (n i) (h i + 1) (μ i) (E.V0 (n i) (h i + 1) (μ i) (J i)) + let s5 : Ty := p s4 (a (a (a (c cl)))) + let x5 (i : ι) : T s5 := (x4 i, ct0 i) + have F5 : ∀ {u : Ty} {z : ι → T u}, Has m b.B s4 u x4 z → Has m b.B s5 u x5 z := fun g => Can.first.then_do g + -- the sweep + have tacc : Has m b.B s5 (jobsT cl) x5 (fun i => accT (n i) (o i) (sup i) ([] : List ℕ)) := + (has_noJobs (cl:=cl) n).cong (fun _ => rfl) (fun i => jobsTape_zero _ _ _) + have tsz : Has m b.B s5 (a w) x5 (fun i => szT ([] : List ℕ)) := by + have := can_tabIx (t:=w) (N:=fun _ => 0) (x:=x5) (B:=b.B) (m:=m) (fun i (t : Fin 0) => (0 : ℕ)) (Can.wc 0) + (Small.const _ 0) (can_of_empty (fun u => Nat.not_lt_zero _ u.2.2)) + exact this.cong (fun _ => rfl) (fun i => szT_nil.symm) + have horun : Knows m b.B s5 x5 (fun i => o i + ([] : List ℕ).length) := F5 (F4 hoK) + have Hsw := E.sweepUp (b:=b) (n:=n) (h:=h) (μ:=μ) (o:=o) (BS:=BS) (sup:=sup) (τ:=τ) E.wd + (s:=s5) (x:=x5) (V:=fun i => E.V0 (n i) (h i + 1) (μ i) (J i)) (A:=fun _ => []) + (c4.ext (t:=a (a (a (c cl)))) ct0) (cc.ext (t:=a (a (a (c cl)))) ct0) + (F5 (Can.fst hidx4)) (F5 (Can.fst (Can.snd (Can.snd hidx4)))) (F5 (Can.snd (Can.snd (Can.snd hidx4)))) + (F5 (F4 hsup)) hBS Can.second tacc tsz horun E.hP E.sep + (fun i => by + have h1 := hfit i + have e : (E.accW (n i) (h i) (μ i) E.wd).length = E.cons (n i) (h i + 1) (μ i) := rfl + simp only [List.length_nil, Nat.add_zero]; omega) + (fun i c hc => hk i c hc) + -- unload and answer + let stF (i : ι) : T (StT cl) := (E.ctabE (n i) (h i + 1) (μ i) (E.runV (E.V0 (n i) (h i + 1) (μ i) (J i)) E.wd), + accT (n i) (o i) (sup i) ([] ++ E.accW (n i) (h i) (μ i) E.wd), + szT ([] ++ E.accW (n i) (h i) (μ i) E.wd), o i + ([] ++ E.accW (n i) (h i) (μ i) E.wd).length) + let s6 : Ty := p s5 (StT cl) + let x6 (i : ι) : T s6 := (x5 i, stF i) + have cc6 := (cc.ext (t:=a (a (a (c cl)))) ct0).ext (t:=StT cl) stF + have hst : Has m b.B s6 (StT cl) x6 stF := Can.second + have HU := E.unload cc6 (V:=fun i => E.runV (E.V0 (n i) (h i + 1) (μ i) (J i)) E.wd) (Can.fst hst) + have Hout : Can m b.B s6 (portOut cl) x6 + (fun i => E.out cl (MDesc.cls (n i) (h i + 1) (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.NUc (n i) (h i + 1) (μ i) * E.W * 2^(n i)) := by + refine ((HU.fork ((Can.fst (Can.snd hst)).fork (Can.fst (Can.snd (Can.snd hst))))).cong (fun _ => rfl) + (fun i => ?_)).mono (fun i => by simp) + show (_, _, _) = (_, _, _) + rw [E.own_answers] + rfl + have H := (Able.of_can (b:=b) H2n).save ((Able.of_can (b:=b) Hidx).save ((Able.of_can (b:=b) Hun).save + ((Able.of_can (b:=b) Hge).save ((Able.of_can (b:=b) Hld).save (Hsw.save (Able.of_can (b:=b) Hout)))))) + refine H.mono (fun i => le_of_eq ?_) (fun i => le_of_eq ?_) + · simp only [locUp, List.length_nil]; ring + · simp only [Nat.zero_add, Nat.add_zero, List.length_nil] + rfl + +/-- Local work of a leaf class. -/ +def locLeaf (n μ : ℕ) : ℕ := + (n + 1) + E.idxCost n 0 μ + E.NUc n 0 μ * (4 * n + 4 * 0 + 8) + + E.NUc n 0 μ * (2 * 2^n + 2) + E.NUc n 0 μ * 2^n + E.lfCost n μ E.wd + E.NUc n 0 μ * E.W * 2^n + +/-- **The body of a leaf class.** -/ +theorem class_leaf (hB2 : Small b.B (fun i => 2^(n i))) (hμ : Small b.B μ) (hfit : ∀ i, E.X.n * μ i ≤ n i) + (hk : ∀ i c, c ∈ E.kidsC (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i)) → + (b.use i (E.enc cl c)).OK (E.out cl c) (τ i c) (b.ext i)) : + Able m b (portIn cl) (portOut cl) + (fun i => E.enc cl (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.out cl (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.locLeaf (n i) (μ i)) + (fun i => ((E.kidsC (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i))).map (τ i)).sum) := by + let x0 (i : ι) := E.enc cl (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i)) + obtain ⟨hn, hI, hkd, hμK, hoK, hown, hsup⟩ := E.decode (b:=b) (x:=x0) (n:=n) (h:=fun _ => 0) (μ:=μ) + (o:=o) (BS:=BS) (J:=J) (sup:=sup) (fun i => rfl) + have hnS : Small b.B n := hB2.mono (fun i => Nat.lt_two_pow_self.le) + have H2n : Can m b.B (portIn cl) w x0 (fun i => 2^(n i)) (fun i => n i + 1) := + (hn.mono (fun _ => Nat.zero_le _)).wTwo hnS hB2 (Dom.of_le (fun i => by omega)) + let x1 (i : ι) : T (p (portIn cl) w) := (x0 i, 2^(n i)) + have c1 : CIn m b.B (p (portIn cl) w) x1 n (fun _ => 0) μ := + ⟨Can.first.then_do hn, Can.wc 0, Can.first.then_do hμK, Can.second, Can.first.then_do hI, hB2, + fun i => Nat.zero_le _, hμ⟩ + have Hidx := E.can_index c1 + have Hun := E.can_units c1 + have Hge := E.can_gears c1 + have cc := E.cctx c1 + let idx (i : ι) := E.idxVal (n i) 0 (μ i) + let un (i : ι) := E.unitVal (n i) 0 (μ i) + let ge (i : ι) := E.gearVal (n i) 0 (μ i) + let s4 : Ty := ClsT (p (portIn cl) w) + let x4 (i : ι) : T s4 := (((x1 i, idx i), un i), ge i) + have F4 : ∀ {u : Ty} {z : ι → T u}, Has m b.B (portIn cl) u x0 z → Has m b.B s4 u x4 z := + fun g => Can.first.then_do (Can.first.then_do (Can.first.then_do (Can.first.then_do g))) + have c4 : CIn m b.B s4 x4 n (fun _ => 0) μ := + ((c1.ext (t:=IdxT) idx).ext (t:=UnT) un).ext (t:=GearT) ge + have Hld := E.load cc (J:=J) (F4 hown) + let ct0 (i : ι) : T (a (a (a (c cl)))) := E.ctabE (n i) 0 (μ i) (E.V0 (n i) 0 (μ i) (J i)) + let s5 : Ty := p s4 (a (a (a (c cl)))) + let x5 (i : ι) : T s5 := (x4 i, ct0 i) + have Hsw := E.sweepLeaf (b:=b) (n:=n) (μ:=μ) (τ:=τ) E.wd (s:=s5) (x:=x5) + (V:=fun i => E.V0 (n i) 0 (μ i) (J i)) (c4.ext (t:=a (a (a (c cl)))) ct0) (cc.ext (t:=a (a (a (c cl)))) ct0) + Can.second E.hP hfit (fun i c hc => hk i c hc) + let ctF (i : ι) : T (a (a (a (c cl)))) := E.ctabE (n i) 0 (μ i) (E.runV (E.V0 (n i) 0 (μ i) (J i)) E.wd) + let s6 : Ty := p s5 (a (a (a (c cl)))) + let x6 (i : ι) : T s6 := (x5 i, ctF i) + have cc6 := (cc.ext (t:=a (a (a (c cl)))) ct0).ext (t:=a (a (a (c cl)))) ctF + have HU := E.unload cc6 (V:=fun i => E.runV (E.V0 (n i) 0 (μ i) (J i)) E.wd) Can.second + have hz : Has m b.B s6 (jobsT cl) x6 (fun i => jobsTape (n i) 0 (fun _ _ => 0)) := has_noJobs n + have hz2 : Has m b.B s6 (a w) x6 (fun i => (Layout.std 0).tape (fun _ : Fin 0 => (0 : ℕ))) := + can_tabIx (t:=w) (N:=fun _ => 0) (x:=x6) (B:=b.B) (m:=m) (fun i (t : Fin 0) => (0 : ℕ)) (Can.wc 0) + (Small.const _ 0) (can_of_empty (fun u => Nat.not_lt_zero _ u.2.2)) + have Hout : Can m b.B s6 (portOut cl) x6 + (fun i => E.out cl (MDesc.cls (n i) 0 (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.NUc (n i) 0 (μ i) * E.W * 2^(n i)) := by + refine ((HU.fork (hz.fork hz2)).cong (fun _ => rfl) (fun i => ?_)).mono (fun i => by simp) + show (_, _, _) = (_, _, _) + rw [E.own_answers] + refine Prod.ext rfl (Prod.ext ?_ ?_) + · dsimp only + unfold jobsTape + exact tape_std_ext (show 0 = E.cons (n i) 0 (μ i) from rfl) _ _ (fun k h1 _ => absurd h1 (Nat.not_lt_zero _)) + · dsimp only + exact tape_std_ext (show 0 = E.cons (n i) 0 (μ i) from rfl) _ _ (fun k h1 _ => absurd h1 (Nat.not_lt_zero _)) + have H := (Able.of_can (b:=b) H2n).save ((Able.of_can (b:=b) Hidx).save ((Able.of_can (b:=b) Hun).save + ((Able.of_can (b:=b) Hge).save ((Able.of_can (b:=b) Hld).save (Hsw.save (Able.of_can (b:=b) Hout)))))) + refine H.mono (fun i => le_of_eq ?_) (fun i => le_of_eq ?_) + · simp only [locLeaf]; ring + · simp only [Nat.zero_add, Nat.add_zero] + rfl + +/-- Local work of a class. -/ +def clsLoc (n h μ : ℕ) : ℕ := if h = 0 then E.locLeaf n μ else E.locUp n (h - 1) μ + +/-- **The body of a class call** (any height): a leaf class makes exits, a higher class makes child calls. -/ +theorem class_body (hB2 : Small b.B (fun i => 2^(n i))) (hBS : Small b.B BS) + (hv1 : ∀ i, h i = 0 → E.X.n * μ i ≤ n i) (hv2 : ∀ i, h i ≤ n i) (hv3 : ∀ i, μ i ≤ n i) + (hv4 : ∀ i, o i + E.cons (n i) (h i) (μ i) ≤ BS i) + (hk : ∀ i c, c ∈ E.kidsC (MDesc.cls (n i) (h i) (μ i) (o i) (BS i) (J i) (sup i)) → + (b.use i (E.enc cl c)).OK (E.out cl c) (τ i c) (b.ext i)) : + Able m b (portIn cl) (portOut cl) + (fun i => E.enc cl (MDesc.cls (n i) (h i) (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.out cl (MDesc.cls (n i) (h i) (μ i) (o i) (BS i) (J i) (sup i))) + (fun i => E.clsLoc (n i) (h i) (μ i)) + (fun i => ((E.kidsC (MDesc.cls (n i) (h i) (μ i) (o i) (BS i) (J i) (sup i))).map (τ i)).sum) := by + let x0 (i : ι) := E.enc cl (MDesc.cls (n i) (h i) (μ i) (o i) (BS i) (J i) (sup i)) + have hkd : Knows m b.B (portIn cl) x0 (fun i => h i + 1) := Can.fst (Can.snd Can.second) + have hnS : Small b.B n := hB2.mono (fun i => Nat.lt_two_pow_self.le) + have hh : Knows m b.B (portIn cl) x0 h := + (hkd.wsub (Can.wc 1) (hnS.mono (fun i => by have := hv2 i; omega))).cong (fun _ => rfl) (fun i => by simp) + have hμs : Small b.B μ := hnS.mono hv3 + apply Able.branch (hh.mono (W':=fun i => E.clsLoc (n i) (h i) (μ i)) (fun _ => Nat.zero_le _)) + (fun i => h i = 0) (fun i => Iff.rfl) + · -- leaf classes + let K := {i // h i = 0} + have H := E.class_leaf (m:=m) (b:=b.comap (fun i : K => i.1)) (n:=fun i : K => n i.1) (μ:=fun i : K => μ i.1) + (o:=fun i : K => o i.1) (BS:=fun i : K => BS i.1) (J:=fun i : K => J i.1) (sup:=fun i : K => sup i.1) + (τ:=fun i : K => τ i.1) (hB2.reindex _) (hμs.reindex _) (fun i => hv1 i.1 i.2) + (fun i c hc => hk i.1 c (by rw [i.2]; exact hc)) + refine (H.cong (fun i => ?_) (fun i => ?_)).mono (fun i => le_of_eq ?_) (fun i => le_of_eq ?_) + · show E.enc cl (MDesc.cls (n i.1) 0 _ _ _ _ _) = E.enc cl (MDesc.cls (n i.1) (h i.1) _ _ _ _ _) + rw [i.2] + · show E.out cl (MDesc.cls (n i.1) 0 _ _ _ _ _) = E.out cl (MDesc.cls (n i.1) (h i.1) _ _ _ _ _) + rw [i.2] + · show E.locLeaf (n i.1) (μ i.1) = E.clsLoc (n i.1) (h i.1) (μ i.1) + unfold clsLoc; rw [if_pos i.2] + · show _ = ((E.kidsC (MDesc.cls (n i.1) (h i.1) (μ i.1) (o i.1) (BS i.1) (J i.1) (sup i.1))).map (τ i.1)).sum + rw [i.2] + · -- classes of height h'+1 + let K := {i // ¬ h i = 0} + have hsucc : ∀ i : K, h i.1 - 1 + 1 = h i.1 := fun i => by have := i.2; omega + have H := E.class_up (m:=m) (b:=b.comap (fun i : K => i.1)) (n:=fun i : K => n i.1) (h:=fun i : K => h i.1 - 1) + (μ:=fun i : K => μ i.1) (o:=fun i : K => o i.1) (BS:=fun i : K => BS i.1) (J:=fun i : K => J i.1) + (sup:=fun i : K => sup i.1) (τ:=fun i : K => τ i.1) (hB2.reindex _) (hBS.reindex _) + (fun i => by rw [hsucc i]; exact hv2 i.1) (hμs.reindex _) + (fun i => by rw [hsucc i]; exact hv4 i.1) + (fun i c hc => hk i.1 c (by rw [← hsucc i]; exact hc)) + refine (H.cong (fun i => ?_) (fun i => ?_)).mono (fun i => le_of_eq ?_) (fun i => le_of_eq ?_) + · show E.enc cl (MDesc.cls (n i.1) (h i.1 - 1 + 1) _ _ _ _ _) = E.enc cl (MDesc.cls (n i.1) (h i.1) _ _ _ _ _) + rw [hsucc i] + · show E.out cl (MDesc.cls (n i.1) (h i.1 - 1 + 1) _ _ _ _ _) = E.out cl (MDesc.cls (n i.1) (h i.1) _ _ _ _ _) + rw [hsucc i] + · show E.locUp (n i.1) (h i.1 - 1) (μ i.1) = E.clsLoc (n i.1) (h i.1) (μ i.1) + unfold clsLoc; rw [if_neg i.2] + · show _ = ((E.kidsC (MDesc.cls (n i.1) (h i.1) (μ i.1) (o i.1) (BS i.1) (J i.1) (sup i.1))).map (τ i.1)).sum + rw [hsucc i] + +end Eng + +end +end PowerSaving.RAM +end OAI diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/Route.lean b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/Route.lean new file mode 100644 index 0000000000..c178d4876e --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/package/lever/overlay/Work/Mixed/Prog/Route.lean @@ -0,0 +1,392 @@ +import Work.Mixed.Prog.CTab + +/-! +# Mixed engine, R3 program part 3: routing a block letter to the child class (agent e-r3prog) + +A block letter of rank `k` on role `l` turns every unit of the class into ONE job of `2^n` +entries (`blockJob`: the role read through `placeS`, block coordinates low). The child class +of the lockstep tree receives, for each of its input slots, either such a job of a parent unit +(the slot is FED) or a fresh top vector (the slot is a SOURCE). This file builds that input +(`gather`) and reads the answers back into the class table (`scatter`). + +The routing is abstract here: tables `fl` (fed flag), `pp` (parent unit of a fed slot), `si` +(supply index of a source slot) and `slot` (child slot of a parent unit). Their values for the +lockstep tree are computed elsewhere. +-/ + +set_option linter.unusedSectionVars false +set_option linter.unusedVariables false + +namespace OAI +namespace PowerSaving.RAM +open Binary Matrix Ty Finset GF +noncomputable section + +universe U + +/-! ## 1. Arithmetic of `placeS` addresses -/ + +lemma split3 {F Rr Qq A Bb : ℕ} (hF : F < 2^A) (hR : Rr < 2^Bb) : + (F + 2^A * Rr + 2^(A+Bb) * Qq) % 2^A = F ∧ + ((F + 2^A * Rr + 2^(A+Bb) * Qq) / 2^A) % 2^Bb = Rr ∧ + (F + 2^A * Rr + 2^(A+Bb) * Qq) / 2^(A+Bb) = Qq := by + have hA : 0 < 2^A := Nat.two_pow_pos A + have hB : 0 < 2^Bb := Nat.two_pow_pos Bb + have e1 : F + 2^A * Rr + 2^(A+Bb) * Qq = F + 2^A * (Rr + 2^Bb * Qq) := by + rw [pow_add]; ring + have e2 : (F + 2^A * (Rr + 2^Bb * Qq)) / 2^A = Rr + 2^Bb * Qq := by + rw [Nat.add_mul_div_left _ _ hA, Nat.div_eq_of_lt hF, zero_add] + refine ⟨?_, ?_, ?_⟩ + · rw [e1, Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hF] + · rw [e1, e2, Nat.add_mul_mod_self_left, Nat.mod_eq_of_lt hR] + · rw [e1, pow_add, ← Nat.div_div_eq_div_mul, e2, Nat.add_mul_div_left _ _ hB, + Nat.div_eq_of_lt hR, zero_add] + +section Place +variable {α β ρ : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] + [Fintype ρ] [DecidableEq ρ] +variable {k : ℕ} + +/-- The three parts of the address of a job entry. -/ +lemma placeS_parts (S : Split α β (Fin k)) (X' : Layout β) (f r n : ℕ) + (h : k*f + X'.n*f + r = n) (e : Bits n) : + let qy := (placeS S X' f r n h).symm e + (bits n).loc e % 2^(k*f) = (bits (k*f)).loc (S.peel 0 f (0, qy.2)).2 ∧ + ((bits n).loc e / 2^(k*f)) % 2^(X'.n*f) = (book X' 0 f).loc (S.peel 0 f (0, qy.2)).1 ∧ + (bits n).loc e / 2^(k*f + X'.n*f) = (bits r).loc qy.1 := by + intro qy + have he : e = placeS S X' f r n h (qy.1, qy.2) := by simp [qy] + have hl := placeS_loc S X' f r n h qy.1 qy.2 + rw [← he] at hl + rw [hl] + have hF := (bits (k*f)).bd (S.peel 0 f (0, qy.2)).2 + have hR := (book X' 0 f).bd (S.peel 0 f (0, qy.2)).1 + rw [bits_len] at hF + rw [book_len, Nat.add_zero] at hR + exact split3 hF hR + +/-- The job of one unit for a block of rank `k` on role `l`: the role read through `placeS`. -/ +def blockJob (S : Split α β (Fin k)) (X' : Layout β) (f r n : ℕ) + (h : k*f + X'.n*f + r = n) (l : ρ) (v : Bits r → Grain α ρ 0 f) : Bits n → ℂ := + fun e => v ((placeS S X' f r n h).symm e).1 (l, (0, ((placeS S X' f r n h).symm e).2)) + +lemma blockJob_placeS (S : Split α β (Fin k)) (X' : Layout β) (f r n : ℕ) + (h : k*f + X'.n*f + r = n) (l : ρ) (v : Bits r → Grain α ρ 0 f) (q : Bits r) (y : Sky α f) : + blockJob S X' f r n h l v (placeS S X' f r n h (q, y)) = v q (l, (0, y)) := by + simp [blockJob] + +/-- **The child's answer for a fed slot is the block letter on the parent unit.** -/ +lemma blockJob_hlow (S : Split α β (Fin k)) (X' : Layout β) (f r n : ℕ) + (h : k*f + X'.n*f + r = n) (l : ρ) (v : Bits r → Grain α ρ 0 f) (q : Bits r) (b0 : Bits 0) + (y : Sky α f) : + hlow n (k*f) (blockJob S X' f r n h l v) (placeS S X' f r n h (q, y)) = + ferryM 0 f l (S.lift (kernel (Fin k))) (v q) (l, (b0, y)) := by + rw [hlow_placeS] + have hb : b0 = 0 := Subsingleton.elim _ _ + subst hb + unfold ferryM roleAct + simp only [if_true] + congr 1 + funext y' + rw [blockJob_placeS] + +lemma fitk {X : Layout α} (X' : Layout β) (hX' : X'.n + k = X.n) {f r n : ℕ} + (hf : X.n * f + r = n) : k*f + X'.n*f + r = n := by + rw [← hf, ← hX', Nat.add_mul]; ring + +end Place + +/-! ## 2. The child's input -/ + +section Gather +variable {α β ρ : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] + [Fintype ρ] [DecidableEq ρ] +variable {k : ℕ} + +/-- Input job of child slot `t`: a parent unit's block job if fed, else a supply vector. -/ +def cjob (X : Layout α) (S : Split α β (Fin k)) (X' : Layout β) (hX' : X'.n + k = X.n) (l : ρ) + {NU n : ℕ} {col ib : ℕ → ℕ} (hfit : ∀ p : Fin NU, X.n * col p + ib p = n) + (V : ∀ p : Fin NU, Bits (ib p) → Grain α ρ 0 (col p)) + (fl pp si : ℕ → ℕ) (hpp : ∀ t, fl t ≠ 0 → pp t < NU) (sup : ℕ → Bits n → ℂ) (t : ℕ) : + Bits n → ℂ := + if hf : fl t ≠ 0 then + blockJob S X' (col (pp t)) (ib (pp t)) n (fitk X' hX' (hfit ⟨pp t, hpp t hf⟩)) l (V ⟨pp t, hpp t hf⟩) + else sup (si t) + +end Gather + +section GatherRAM +variable {ι : Type U} {m : Bool} {B : ι → ℕ} {s : Ty} {x : ι → T s} + {n NU : ι → ℕ} {col ib : ι → ℕ → ℕ} +variable {α β ρ : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] + [Fintype ρ] [DecidableEq ρ] +variable {cl : Paint} {X : Layout α} {R : Layout ρ} {k : ℕ} + +/-- Entries of the child's input: slot `t`, entry `e`. -/ +abbrev EntF (NJ : ι → ℕ) (n : ι → ℕ) := Σ u : UnitF NJ, Bits (n u.1) + +/-- **The child's input** (fed slots from the parent's units, source slots from the supply): +linear work. -/ +theorem CCtx.gather (h : CCtx m B s x X.n n NU col ib) + {V : ∀ i (p : Fin (NU i)), Bits (ib i p) → Grain α ρ 0 (col i p)} + (ct : Has m B s (a (a (a (c cl)))) x (fun i => ctab X R (NU i) (col i) (ib i) (V i))) + (S : Split α β (Fin k)) (X' : Layout β) (hX' : X'.n + k = X.n) (l : ρ) + {NJ BS : ι → ℕ} {fl pp si : ι → ℕ → ℕ} {sup : ∀ i, ℕ → Bits (n i) → ℂ} + (hpp : ∀ i t, fl i t ≠ 0 → pp i t < NU i) + (hsi : ∀ i t, t < NJ i → fl i t = 0 → si i t < BS i) + (hNJ : Knows m B s x NJ) (hNJs : Small B NJ) + (tfl : Has m B s (a w) x (fun i => (Layout.std (NJ i)).tape (fun t => fl i t))) + (tpp : Has m B s (a w) x (fun i => (Layout.std (NJ i)).tape (fun t => pp i t))) + (tsi : Has m B s (a w) x (fun i => (Layout.std (NJ i)).tape (fun t => si i t))) + (tsup : Has m B s (a (a (c cl))) x + (fun i => (Layout.std (BS i)).tape (fun t => (bits (n i)).tape (sup i t)))) : + Can m B s (a (a (c cl))) x + (fun i => (Layout.std (NJ i)).tape (fun t => (bits (n i)).tape + (cjob X S X' hX' l (h.fit i) (V i) (fl i) (pp i) (si i) (hpp i) (sup i) t))) + (fun i => NJ i * 2^(n i)) := by + let J (i : ι) (t : ℕ) : Bits (n i) → ℂ := + cjob X S X' hX' l (h.fit i) (V i) (fl i) (pp i) (si i) (hpp i) (sup i) t + let yE (en : EntF NJ n) : T (p (p s w) w) := ((x en.1.1, en.1.2.val), (bits (n en.1.1)).loc en.2) + let BE (en : EntF NJ n) := B en.1.1 + -- the environment of an entry contains the slot environment and the class environment + have hslot : ∀ {t : Ty} {y : UnitF NJ → T t}, + Has m (fun u : UnitF NJ => B u.1) (p s w) t (fun u => (x u.1, u.2.val)) y → + Has m BE (p (p s w) w) t yE (fun en => y en.1) := + fun g => Can.first.then_do (g.reindex (fun en : EntF NJ n => en.1)) + have hcls : ∀ {t : Ty} {y : ι → T t}, Has m B s t x y → Has m BE (p (p s w) w) t yE (fun en => y en.1.1) := + fun g => (Can.first.then_do Can.first).then_do (g.reindex (fun en : EntF NJ n => en.1.1)) + have hE : Knows m BE (p (p s w) w) yE (fun en => (bits (n en.1.1)).loc en.2) := Can.second + have hbig : Small BE (fun en => 2^(n en.1.1)) := h.big.reindex (fun en : EntF NJ n => en.1.1) + have hEs : Small BE (fun en => (bits (n en.1.1)).loc en.2) := + hbig.mono (fun en => ((bits (n en.1.1)).bd en.2).le) + have hfl : Knows m BE (p (p s w) w) yE (fun en => fl en.1.1 en.1.2) := hslot (unit_fetch _ tfl) + -- one entry + have hent : Has m BE (p (p s w) w) (c cl) yE (fun en => J en.1.1 en.1.2 en.2) := by + apply Can.testz hfl + · -- source slot + let K0 := {en : EntF NJ n // fl en.1.1 en.1.2 = 0} + have hsiK : Knows m (fun κ : K0 => BE κ.1) (p (p s w) w) (fun κ => yE κ.1) + (fun κ => si κ.1.1.1 κ.1.1.2) := + (hslot (unit_fetch _ tsi)).reindex (fun κ : K0 => κ.1) + let ix (κ : K0) : Fin (BS κ.1.1.1) := ⟨si κ.1.1.1 κ.1.1.2, hsi _ _ κ.1.1.2.2 κ.2⟩ + have hv := Can.fetch (t:=a (c cl)) (L:=fun κ : K0 => Layout.std (BS κ.1.1.1)) + (v:=fun κ j => (bits (n κ.1.1.1)).tape (sup κ.1.1.1 j)) (i:=ix) + ((hcls tsup).reindex (fun κ : K0 => κ.1)) hsiK + have hw := Can.fetch (t:=c cl) (L:=fun κ : K0 => bits (n κ.1.1.1)) + (v:=fun κ => sup κ.1.1.1 (ix κ).val) (i:=fun κ => κ.1.2) hv (hE.reindex (fun κ : K0 => κ.1)) + refine hw.cong (fun _ => rfl) (fun κ => ?_) + change sup κ.1.1.1 (si κ.1.1.1 κ.1.1.2) κ.1.2 = J κ.1.1.1 κ.1.1.2 κ.1.2 + simp only [J, cjob, κ.2, ne_eq, not_true_eq_false, dif_neg, not_false_eq_true] + · -- fed slot: read the parent unit's role `l` through `placeS` + let K1 := {en : EntF NJ n // fl en.1.1 en.1.2 ≠ 0} + let BK (κ : K1) := BE κ.1 + let yK (κ : K1) := yE κ.1 + let i_ (κ : K1) := κ.1.1.1 + let p_ (κ : K1) : Fin (NU (i_ κ)) := ⟨pp (i_ κ) κ.1.1.2, hpp _ _ κ.2⟩ + let fit_ (κ : K1) := fitk X' hX' (h.fit (i_ κ) (p_ κ)) + let PL (κ : K1) := placeS S X' (col (i_ κ) (p_ κ)) (ib (i_ κ) (p_ κ)) (n (i_ κ)) (fit_ κ) + let qy (κ : K1) := (PL κ).symm κ.1.2 + let j (κ : K1) : ElemF NU ib := ⟨⟨i_ κ, p_ κ⟩, (qy κ).1⟩ + have hppK : Knows m BK (p (p s w) w) yK (fun κ => (p_ κ).val) := + (hslot (unit_fetch _ tpp)).reindex (fun κ : K1 => κ.1) + have hpcK : Knows m BK (p (p s w) w) yK (fun κ => 2^(col (i_ κ) (p_ κ))) := + Can.fetch (t:=w) (L:=fun κ : K1 => Layout.std (NU (i_ κ))) + (v:=fun κ q => 2^(col (i_ κ) q)) (i:=p_) + ((hcls h.tpc).reindex (fun κ : K1 => κ.1)) hppK + have hcbig : Small BK (fun κ => 2^(col (i_ κ) (p_ κ))) := + (hbig.reindex (fun κ : K1 => κ.1)).mono (fun κ => h.two_c_le (i_ κ) (p_ κ)) + have hXc : Knows m BK (p (p s w) w) yK (fun κ => 2^(k * col (i_ κ) (p_ κ) + + X'.n * col (i_ κ) (p_ κ))) := + (hpcK.wpow_fixed X.n hcbig).cong (fun _ => rfl) (fun κ => by + rw [← pow_mul, ← hX']; ring_nf) + have hq : Knows m BK (p (p s w) w) yK (fun κ => (bits (ib (i_ κ) (p_ κ))).loc (qy κ).1) := + ((hE.reindex (fun κ : K1 => κ.1)).dvdW hXc (hEs.reindex (fun κ : K1 => κ.1))).cong + (fun _ => rfl) (fun κ => (placeS_parts S X' _ _ _ (fit_ κ) κ.1.2).2.2) + have hx0 : Has m BE (p (p s w) w) s yE (fun en => x en.1.1) := + Can.first.then_do Can.first + have hxK : Has m BK (p (p s w) w) s yK (fun κ => x (i_ κ)) := + hx0.reindex (fun κ : K1 => κ.1) + have g : Has m (fun κ => B (j κ).1.1) (p (p s w) w) (p (p s w) w) yK + (fun κ => ((x (j κ).1.1, (j κ).1.2.val), (bits (ib (j κ).1.1 (j κ).1.2)).loc (j κ).2)) := + (hxK.pair hppK).pair hq + have G1 := (h.erig ct).gather j yK g + have hk1 : Knows m BK (p (p s w) w) yK (fun κ => 2^(k * col (i_ κ) (p_ κ))) := + G1.rig.gear.powerB k + have hk2 : Knows m BK (p (p s w) w) yK (fun κ => 2^(X'.n * col (i_ κ) (p_ κ))) := + G1.rig.gear.powerB X'.n + have hfib : Knows m BK (p (p s w) w) yK + (fun κ => (bits (k * col (i_ κ) (p_ κ))).loc (S.peel 0 (col (i_ κ) (p_ κ)) (0, (qy κ).2)).2) := + ((hE.reindex (fun κ : K1 => κ.1)).modW hk1 (hEs.reindex (fun κ : K1 => κ.1))).cong + (fun _ => rfl) (fun κ => (placeS_parts S X' _ _ _ (fit_ κ) κ.1.2).1) + have hrest : Knows m BK (p (p s w) w) yK + (fun κ => (book X' 0 (col (i_ κ) (p_ κ))).loc (S.peel 0 (col (i_ κ) (p_ κ)) (0, (qy κ).2)).1) := + (((hE.reindex (fun κ : K1 => κ.1)).dvdW hk1 (hEs.reindex (fun κ : K1 => κ.1))).modW hk2 + ((hEs.reindex (fun κ : K1 => κ.1)).div)).cong + (fun _ => rfl) (fun κ => (placeS_parts S X' _ _ _ (fit_ κ) κ.1.2).2.1) + have hgrid := G1.rig.gear.to_gridB X 0 S X' + (fun κ => S.peel 0 (col (i_ κ) (p_ κ)) (0, (qy κ).2)) hrest hfib + have hb : Knows m BK (p (p s w) w) yK + (fun κ => (book X 0 (col (i_ κ) (p_ κ))).loc ((0 : Bits 0), (qy κ).2)) := + hgrid.cong (fun _ => rfl) (fun κ => by rw [Equiv.symm_apply_apply]) + have hr := G1.rig.read (fun κ => (l, ((0 : Bits 0), (qy κ).2))) (Can.wc (R.loc l)) hb + refine hr.cong (fun _ => rfl) (fun κ => ?_) + change V (i_ κ) (p_ κ) (qy κ).1 (l, (0, (qy κ).2)) = J κ.1.1.1 κ.1.1.2 κ.1.2 + simp only [J, cjob, dif_pos κ.2] + rfl + -- inner layout over the entries of one slot, then outer layout over the slots + have hlen : Knows m (fun u : UnitF NJ => B u.1) (p s w) (fun u => (x u.1, u.2.val)) + (fun u => (bits (n u.1)).n) := + Can.first.then_do (h.p2n.reindex (fun u : UnitF NJ => u.1)) + have HI := Can.layout (t:=c cl) (L:=fun u : UnitF NJ => bits (n u.1)) + (v:=fun u e => J u.1 u.2 e) (P:=fun _ => 0) (W:=fun _ => 0) hlen + (hent.mono (fun _ => Nat.zero_le _)) (h.big.reindex (fun u : UnitF NJ => u.1)) + have HO := Can.layout (t:=a (c cl)) (L:=fun i => Layout.std (NJ i)) + (v:=fun i t => (bits (n i)).tape (J i t)) (P:=fun i => 2^(n i)) (W:=fun _ => 0) hNJ + (HI.mono (fun u => by simp)) hNJs + refine HO.weaken ⟨3, fun i => ?_⟩ + change 0 + NJ i + NJ i * 2^(n i) + 1 ≤ 3 * (NJ i * 2^(n i) + 1) + have h1 : 1 ≤ 2^(n i) := Nat.one_le_two_pow + have h2 : NJ i ≤ NJ i * 2^(n i) := Nat.le_mul_of_pos_right _ h1 + omega + +end GatherRAM + +section ScatterRAM +variable {ι : Type U} {m : Bool} {B : ι → ℕ} {s : Ty} {x : ι → T s} + {n NU : ι → ℕ} {col ib : ι → ℕ → ℕ} +variable {α β ρ : Type*} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] + [Fintype ρ] [DecidableEq ρ] +variable {cl : Paint} {X : Layout α} {R : Layout ρ} {k : ℕ} + +/-- **Reading the child's answers back** into the class table: role `l` of unit `p` comes from +child slot `slot p` (read through `placeS`), the other roles are kept. Linear work. -/ +theorem CCtx.scatter (h : CCtx m B s x X.n n NU col ib) + {V : ∀ i (p : Fin (NU i)), Bits (ib i p) → Grain α ρ 0 (col i p)} + (ct : Has m B s (a (a (a (c cl)))) x (fun i => ctab X R (NU i) (col i) (ib i) (V i))) + (S : Split α β (Fin k)) (X' : Layout β) (hX' : X'.n + k = X.n) (l : ρ) + {NJ : ι → ℕ} {slot : ι → ℕ → ℕ} {Y : ∀ i, ℕ → Bits (n i) → ℂ} + (hslot : ∀ i (p : Fin (NU i)), slot i p < NJ i) + (tslot : Has m B s (a w) x (fun i => (Layout.std (NU i)).tape (fun p => slot i p))) + (tout : Has m B s (a (a (c cl))) x + (fun i => (Layout.std (NJ i)).tape (fun t => (bits (n i)).tape (Y i t)))) + (hY : ∀ i (p : Fin (NU i)) (q : Bits (ib i p)) (y : Sky α (col i p)), + Y i (slot i p) (placeS S X' (col i p) (ib i p) (n i) (fitk X' hX' (h.fit i p)) (q, y)) = + ferryM 0 (col i p) l (S.lift (kernel (Fin k))) (V i p q) (l, (0, y))) : + Can m B s (a (a (a (c cl)))) x + (fun i => ctab X R (NU i) (col i) (ib i) + (fun p q => ferryM 0 (col i p) l (S.lift (kernel (Fin k))) (V i p q))) + (fun i => NU i * 2^(n i)) := by + have G := h.erig ct + apply h.ctab_build _ rfl + let yE (e : ElemF NU ib) : T (p (p s w) w) := + ((x e.1.1, e.1.2.val), (bits (ib e.1.1 e.1.2)).loc e.2) + let BE (e : ElemF NU ib) := B e.1.1 + have hel : Can m BE (p (p s w) w) (a (c cl)) yE + (fun e => (fleet X R 0 (col e.1.1 e.1.2)).tape + (ferryM 0 (col e.1.1 e.1.2) l (S.lift (kernel (Fin k))) (V e.1.1 e.1.2 e.2))) + (fun e => (fleet X R 0 (col e.1.1 e.1.2)).n) := by + refine G.rig.chart _ ?_ + -- one beach of one element + let D := Σ e : ElemF NU ib, Beach α ρ 0 (col e.1.1 e.1.2) + let yD (d : D) : T (p (p (p s w) w) w) := (yE d.1, (fleet X R 0 (col d.1.1.1 d.1.1.2)).loc d.2) + let BD (d : D) := BE d.1 + have GD := G.rig.gather (fun d : D => d.1) yD (t:=p (p (p s w) w) w) Can.first + obtain ⟨hl, hk⟩ := GD.addrs (fun d => d.2) Can.second + have hbD : Has m BD (p (p (p s w) w) w) (c cl) yD (fun d => V d.1.1.1 d.1.1.2 d.1.2 d.2) := + GD.read (fun d => d.2) hl hk + apply Can.cases_code (q:=fun d : D => d.2.1) R hl + intro r' + by_cases hr : r' = l + · -- role `l`: from the child's answer + let K := {d : D // d.2.1 = r'} + let BK (κ : K) := BD κ.1 + have GK := GD.gather (fun κ : K => κ.1) (fun κ => yD κ.1) (t:=p (p (p s w) w) w) Can.id + obtain ⟨hrest, hfib⟩ := GK.gear.from_gridB X 0 S X' (fun κ => κ.1.2.2) + (hk.reindex (fun κ : K => κ.1)) + have hsU : Knows m (fun u : UnitF NU => B u.1) (p s w) (fun u => (x u.1, u.2.val)) + (fun u => slot u.1 u.2) := unit_fetch _ tslot + have hsE : Knows m BE (p (p s w) w) yE (fun e => slot e.1.1 e.1.2) := + Can.first.then_do (hsU.reindex (fun e : ElemF NU ib => e.1)) + have hsD : Knows m BD (p (p (p s w) w) w) yD (fun d => slot d.1.1.1 d.1.1.2) := + Can.first.then_do (hsE.reindex (fun d : D => d.1)) + have hslotK : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => slot κ.1.1.1.1 κ.1.1.1.2) := hsD.reindex (fun κ : K => κ.1) + have hqD : Knows m BD (p (p (p s w) w) w) yD (fun d => (bits (ib d.1.1.1 d.1.1.2)).loc d.1.2) := + Can.first.then_do Can.second + have hq : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => (bits (ib κ.1.1.1.1 κ.1.1.1.2)).loc κ.1.1.2) := hqD.reindex (fun κ : K => κ.1) + have hk1 : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => 2^(k * col κ.1.1.1.1 κ.1.1.1.2)) := GK.gear.powerB k + have hk3 : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => 2^(k * col κ.1.1.1.1 κ.1.1.1.2 + X'.n * col κ.1.1.1.1 κ.1.1.1.2)) := + (GK.gear.powerB X.n).cong (fun _ => rfl) (fun κ => by rw [← hX']; ring_nf) + have hbig : Small BK (fun κ => 2^(n κ.1.1.1.1)) := + h.big.reindex (fun κ : K => κ.1.1.1.1) + have hcb : Small BK (fun κ => 2^(col κ.1.1.1.1 κ.1.1.1.2)) := + hbig.mono (fun κ => h.two_c_le _ _) + have hs1 : Small BK (fun κ => 2^(k * col κ.1.1.1.1 κ.1.1.1.2)) := + GK.gear.boundB k + have hs3 : Small BK (fun κ => 2^(k * col κ.1.1.1.1 κ.1.1.1.2 + X'.n * col κ.1.1.1.1 κ.1.1.1.2)) := + (GK.gear.boundB X.n).mono (fun κ => le_of_eq (by rw [← hX']; ring_nf)) + have hsr : Small BK (fun κ => (book X' 0 (col κ.1.1.1.1 κ.1.1.1.2)).loc + (S.peel 0 (col κ.1.1.1.1 κ.1.1.1.2) κ.1.2.2).1) := + (GK.gear.grid_small X' 0).mono (fun κ => ((book X' 0 _).bd _).le) + have hsq : Small BK (fun κ => (bits (ib κ.1.1.1.1 κ.1.1.1.2)).loc κ.1.1.2) := + hbig.mono (fun κ => le_trans ((bits _).bd _).le (h.two_ib_le _ _)) + have hsf : Small BK (fun κ => (bits (k * col κ.1.1.1.1 κ.1.1.1.2)).loc + (S.peel 0 (col κ.1.1.1.1 κ.1.1.1.2) κ.1.2.2).2) := + hs1.mono (fun κ => ((bits _).bd _).le) + -- the address inside the child's job + have haddr : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => (bits (k * col κ.1.1.1.1 κ.1.1.1.2)).loc + (S.peel 0 (col κ.1.1.1.1 κ.1.1.1.2) κ.1.2.2).2 + + 2^(k * col κ.1.1.1.1 κ.1.1.1.2) * (book X' 0 (col κ.1.1.1.1 κ.1.1.1.2)).loc + (S.peel 0 (col κ.1.1.1.1 κ.1.1.1.2) κ.1.2.2).1 + + 2^(k * col κ.1.1.1.1 κ.1.1.1.2 + X'.n * col κ.1.1.1.1 κ.1.1.1.2) * + (bits (ib κ.1.1.1.1 κ.1.1.1.2)).loc κ.1.1.2) := + (hfib.plus (hk1.times hrest hs1 hsr) hsf (hs1.mul hsr)).plus (hk3.times hq hs3 hsq) + (hsf.add (hs1.mul hsr)) (hs3.mul hsq) + let jq (κ : K) := κ.1.1.2 + let jy (κ : K) := κ.1.2.2.2 + let PL (κ : K) := placeS S X' (col κ.1.1.1.1 κ.1.1.1.2) (ib κ.1.1.1.1 κ.1.1.1.2) + (n κ.1.1.1.1) (fitk X' hX' (h.fit κ.1.1.1.1 κ.1.1.1.2)) + have haddr' : Knows m BK (p (p (p s w) w) w) (fun κ => yD κ.1) + (fun κ => (bits (n κ.1.1.1.1)).loc (PL κ (jq κ, jy κ))) := by + refine haddr.cong (fun _ => rfl) (fun κ => ?_) + rw [placeS_loc] + have hb0 : κ.1.2.2 = ((0 : Bits 0), jy κ) := Prod.ext (Subsingleton.elim _ _) rfl + rw [hb0] + have hoU : Has m (fun u : UnitF NU => B u.1) (p s w) (a (a (c cl))) (fun u => (x u.1, u.2.val)) + (fun u => (Layout.std (NJ u.1)).tape (fun t => (bits (n u.1)).tape (Y u.1 t))) := + Can.first.then_do (tout.reindex (fun u : UnitF NU => u.1)) + have hoE : Has m BE (p (p s w) w) (a (a (c cl))) yE + (fun e => (Layout.std (NJ e.1.1)).tape (fun t => (bits (n e.1.1)).tape (Y e.1.1 t))) := + Can.first.then_do (hoU.reindex (fun e : ElemF NU ib => e.1)) + have hoD : Has m BD (p (p (p s w) w) w) (a (a (c cl))) yD + (fun d => (Layout.std (NJ d.1.1.1)).tape (fun t => (bits (n d.1.1.1)).tape (Y d.1.1.1 t))) := + Can.first.then_do (hoE.reindex (fun d : D => d.1)) + have hI := hoD.reindex (fun κ : K => κ.1) + have hJ := Can.fetch (t:=a (c cl)) (L:=fun κ : K => Layout.std (NJ κ.1.1.1.1)) + (v:=fun κ t => (bits (n κ.1.1.1.1)).tape (Y κ.1.1.1.1 t)) + (i:=fun κ => (⟨slot κ.1.1.1.1 κ.1.1.1.2, hslot _ _⟩ : Fin (NJ κ.1.1.1.1))) hI hslotK + have hv := Can.fetch (t:=c cl) (L:=fun κ : K => bits (n κ.1.1.1.1)) + (v:=fun κ => Y κ.1.1.1.1 (slot κ.1.1.1.1 κ.1.1.1.2)) (i:=fun κ => PL κ (jq κ, jy κ)) hJ haddr' + refine hv.cong (fun _ => rfl) (fun κ => ?_) + change Y κ.1.1.1.1 (slot κ.1.1.1.1 κ.1.1.1.2) (PL κ (jq κ, jy κ)) = + ferryM 0 _ l (S.lift (kernel (Fin k))) (V κ.1.1.1.1 κ.1.1.1.2 κ.1.1.2) κ.1.2 + rw [hY] + have hd : κ.1.2 = (l, ((0 : Bits 0), jy κ)) := + Prod.ext (κ.2.trans hr) (Prod.ext (Subsingleton.elim _ _) rfl) + rw [hd] + · -- other roles: unchanged + refine (hbD.reindex (fun κ : {d : D // d.2.1 = r'} => κ.1)).cong (fun _ => rfl) (fun κ => ?_) + have H : κ.1.2.1 ≠ l := fun hh => hr (κ.2.symm.trans hh) + simp only [ferryM, roleAct, H, ite_false, Prod.mk.eta] + refine hel.mono (fun e => le_of_eq ?_) + simp [fleet] + +end ScatterRAM + +end +end PowerSaving.RAM +end OAI diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/README.md b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/README.md new file mode 100644 index 0000000000..d853d7090d --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/README.md @@ -0,0 +1,29 @@ +# Programs: Theorem 1G, the dilated lockstep tree, and its two skeptics + +These programs check the construction of [../../proofs/theorem-1g.md](../../proofs/theorem-1g.md) in the job model. +They are Python 3.9 or newer, standard library only. Run each with `python3 -I -B` from this folder; each one took +under 40 seconds and under 300 MB on an Apple M4 Mac mini. All were re-run on 19 October 2026 for this revision, and +every output was identical to the one saved on 14 October (`../../data/theorem-1g/`). + +The first four were written with the proof; the other five were written by two AI skeptics from the proof's text +alone, without reusing its code. + +| program | what it does | command | output | +|---|---|---|---| +| `tree_flow.py` | builds the dilated lockstep tree for three test rules with grouped paid blocks, literally (explicit round sequences, L = 40 and 150) and merged (L up to 10^7); checks exact balance, sources in (L/m, L], Φ(exits) = Φ(sources), the exit bound and the unit bound | `python3 -I -B tree_flow.py` | `out_tree_flow.txt` | +| `recur.py` | the cost of the whole schedule (top split, tree, exits by fresh calls) on the test rules and the rank-1 control: T(n)/n^τ up to n | `python3 -I -B recur.py 800` and `python3 -I -B recur.py 6400` | `out_recur_800.txt`, `out_recur_6400.txt` | +| `recheck_t1.py` | re-runs the rank-1 τ-theorem's lockstep simulation and exact balance; imports `lockstep_sim.py` and `frontier_flow.py` from [../../../existence-routes/programs/tau-theorem-and-laser/](../../../existence-routes/programs/tau-theorem-and-laser/) | `python3 -I -B recheck_t1.py` | `out_recheck_t1.txt` | +| `hub_whatif.py` | the mixed price of the community's T74 and #1042 if every idle block were a head (the assumption later refuted by the head test), with the constants θ, η, κ of the theorem; control: the community's T74 figure | `python3 -I -B hub_whatif.py` | `out_hub_whatif.txt` | +| `indep_tree.py` | skeptic 1: an independent merged tree (toy rule and the T74 mixed rule, L up to 10^8) | `python3 -I -B indep_tree.py` | `out_indep_tree.txt` | +| `partial.py` | skeptic 1: the T74 price when only a fraction of the idle blocks can be cut and the rest are split 24 + 24 to stay separated; the smallest κ | `python3 -I -B partial.py` | `out_partial.txt` | +| `explicit_sim.py` | skeptic 2: explicit unit and job objects, random matching by size inside the prescribed classes, a generic topological sort of start, issue and finish events as a deadlock test; a "careless" matching across classes must deadlock | `python3 -I -B explicit_sim.py` | `out_explicit_sim.txt` | +| `cost_fit.py` | skeptic 2: its own cost recursion for the schedule; the third argument is the per-unit overhead as a multiple of W | `for r in D A B; do python3 -I -B cost_fit.py $r 12800 0,3; done` and `for r in C H J X; do python3 -I -B cost_fit.py $r 12800 0; done` | `out_cost_fit_12800_DAB.txt`, `out_cost_fit_12800_CHJX.txt` | +| `check_E.py` | skeptic 2: hypothesis (E) of Lemma A3 on the actual flows at every n (the last run, on the T74 rule up to 3,000,000, at every 997th n), the unit bound of Lemma E, the exit bound, and the ψ = 1 version (Lemma A4 predicts it fails at every n) | `for r in A B C H J; do python3 -I -B check_E.py $r 12800 1; done; python3 -I -B check_E.py T 20000 1; python3 -I -B check_E.py T 3000000 997` | `out_check_E.txt` | + +What to expect: 0 unbalanced points and Φ ratio 1 everywhere; every lockstep run deadlock-free and every careless +run deadlocked; T(n)/n^τ flat for the rank-1 control and slowly growing, far below (ln n)^κ, for the grouped rules; +(E) never violated and its ψ = 1 version violated at every n. Rule X of `cost_fit.py` is not separated; it is +included as a first look at the open case (README of this section, section 6, item 2). + +Status: these runs are *checked by computer* for the cases they cover; they do not prove the theorem, whose proof +is *argued by hand, not checked*. diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/explicit_sim.py b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/explicit_sim.py new file mode 100644 index 0000000000..0f6fb31f69 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/explicit_sim.py @@ -0,0 +1,163 @@ +# explicit_sim.py (bound-check/rl-vprove-compute, 14 Oct 2026). Skeptic's independent check of the dilated lockstep +# tree (rl-prove PROOF.md section 3, Theorem 1G; with all ranks 1 it is ex-laser's Theorem 1 / Lemma 2). +# Written from the PROOF.md text only (no code reused from rl-prove or ex-laser). +# +# Every unit is an explicit object; every job is an explicit object with a size. Matching is done BY SIZE ONLY (the +# physical requirement), in a RANDOM order, within the class pairs the proof prescribes (round j of class s feeds class +# s+(j)). Unmatched demand is a source and must have size in (L/m, L]; unmatched supply is an error. Then a generic +# event graph (start/issue/finish of every unit) is topologically sorted (Kahn) to test deadlock-freedom independently +# of Lemma F's procedure. Phi(sources) vs Phi(exits) is compared at tau_c in 50-digit Decimal arithmetic. +# Mode "careless": the same tree, but each child's inputs are drawn by size from the pooled outputs of ALL rounds of +# its parent's whole depth (crossing class boundaries); the deadlock test must then usually find a cycle (teeth check). +import sys, random +from decimal import Decimal as D, getcontext +from math import factorial, prod +from collections import defaultdict, deque +getcontext().prec = 50 + +def tau_root(Wt, rounds): + def f(t): + return sum(D(r).ln().__mul__(t).exp() for r in rounds) - sum(w * (D(k).ln() * t).exp() for k, w in Wt.items()) + lo, hi = D(0), D(1) + for _ in range(170): + mid = (lo + hi) / 2 + if f(mid) > 0: lo = mid # paid side bigger at small tau + else: hi = mid + return (lo + hi) / 2 + +def build(Wt, rounds, L, mode="lockstep", seed=1): + rng = random.Random(seed) + K = sorted(Wt); m = max(K); P = len(rounds) + def c(a): return prod(k ** x for k, x in zip(K, a)) + def u(a): return factorial(sum(a)) // prod(factorial(x) for x in a) * prod(Wt[k] ** x for k, x in zip(K, a)) + def pts(rho, size): # a in R(rho) with |a| = size + out = [] + def rec(i, a, cc, left): + if i == len(K) - 1: + a2 = a + [left]; c2 = cc * K[i] ** left + if m * rho * c2 <= L: out.append(tuple(a2)) + return + for e in range(left + 1): + c2 = cc * K[i] ** e + if m * rho * c2 > L: break + rec(i + 1, a + [e], c2, left - e) + rec(0, [], 1, size); return out + G = 0 + while pts(1, G + 1): G += 1 + # units: id -> dict(cls, col, slots=[(k,size)], inputs=[job], jobs=[job per round]) + units = []; jobs = [] # job: dict(size, producer=(uid,j) or None, consumer=uid or None, exit=bool) + classes = {} # path tuple -> list of unit ids + def mk_class(path, rho): + d = len(path); ids = [] + for a in pts(rho, G - d): + for _ in range(u(a)): + col = rho * c(a) + units.append(dict(cls=path, col=col, slots=[(k, k * col) for k in K for _ in range(Wt[k])], inputs=[None] * sum(Wt.values()), jobs=[None] * P)) + ids.append(len(units) - 1) + classes[path] = (ids, rho) + return ids + def rec_tree(path, rho): + mk_class(path, rho) + if len(path) < G: + for j, r in enumerate(rounds): rec_tree(path + (j,), rho * r) + rec_tree((), 1) + # issue jobs + for uid, U in enumerate(units): + d = len(U['cls']) + for j, r in enumerate(rounds): + jobs.append(dict(size=r * U['col'], producer=(uid, j), consumer=None, exit=(d == G))) + U['jobs'][j] = len(jobs) - 1 + errors = []; sources = [] + def fill(child_ids, supply_jobs): + # match by size, random order + pool = defaultdict(list) + for jid in supply_jobs: pool[jobs[jid]['size']].append(jid) + for lst in pool.values(): rng.shuffle(lst) + slots = [(uid, i) for uid in child_ids for i in range(len(units[uid]['slots']))] + rng.shuffle(slots) + for uid, i in slots: + k, size = units[uid]['slots'][i] + if pool.get(size): + jid = pool[size].pop(); jobs[jid]['consumer'] = uid; units[uid]['inputs'][i] = jid + else: + if not (L / m < size <= L): errors.append(('source size out of band', size)) + jobs.append(dict(size=size, producer=None, consumer=uid, exit=False)); units[uid]['inputs'][i] = len(jobs) - 1 + sources.append(size) + for lst in pool.values(): + for jid in lst: errors.append(('unconsumed job', jobs[jid]['size'])) + if mode == "lockstep": + for path, (ids, rho) in classes.items(): + if len(path) == 0: fill(ids, []) + else: + parent_ids, _ = classes[path[:-1]]; j = path[-1] + fill(ids, [units[p]['jobs'][j] for p in parent_ids]) + else: # careless: pool by depth + bydepth = defaultdict(list) + for path, (ids, rho) in classes.items(): bydepth[len(path)].extend(ids) + for d in range(G + 1): + if d == 0: fill(bydepth[0], []) + else: fill(bydepth[d], [units[p]['jobs'][j] for p in bydepth[d - 1] for j in range(P)]) + return dict(G=G, units=units, jobs=jobs, sources=sources, errors=errors, K=K, m=m, P=P, nclasses=len(classes)) + +def deadlock_free(T): + units, jobs, P = T['units'], T['jobs'], T['P'] + # events: ('S',u), ('I',u,j), ('F',u). Edges per the data dependencies. + succ = defaultdict(list); indeg = defaultdict(int); nodes = set() + def edge(a, b): succ[a].append(b); indeg[b] += 1; nodes.add(a); nodes.add(b) + for uid, U in enumerate(units): + nodes.add(('S', uid)); nodes.add(('F', uid)) + for jid in U['inputs']: + pr = jobs[jid]['producer'] + if pr is not None: edge(('I',) + pr, ('S', uid)) # input must have been issued + edge(('S', uid), ('I', uid, 0)) + for j in range(P): + jid = U['jobs'][j]; nxt = ('I', uid, j + 1) if j + 1 < P else ('F', uid) + edge(('I', uid, j), nxt) # program order inside the word + if not jobs[jid]['exit']: + edge(('F', jobs[jid]['consumer']), nxt) # round j's job must be COMPLETE before going on + q = deque(n for n in nodes if indeg[n] == 0); seen = 0 + while q: + n = q.popleft(); seen += 1 + for s in succ[n]: + indeg[s] -= 1 + if indeg[s] == 0: q.append(s) + return seen == len(nodes), seen, len(nodes) + +def report(name, Wt, rounds, L, mode="lockstep", seed=1): + T = build(Wt, rounds, L, mode, seed) + tau = tau_root(Wt, rounds) + units, jobs = T['units'], T['jobs'] + exits = defaultdict(int) + for jb in jobs: + if jb['exit']: exits[jb['size']] += 1 + unconsumed_nonexit = sum(1 for jb in jobs if not jb['exit'] and jb['consumer'] is None) + phis = sum((D(s).ln() * tau).exp() for s in T['sources']) + phie = sum(n * (D(e).ln() * tau).exp() for e, n in exits.items()) + ok, seen, tot = deadlock_free(T) + paid = len(units) * T['P'] + print(f"{name} L={L} mode={mode} seed={seed}: tau_c={float(tau):.6f} G={T['G']} classes={T['nclasses']} units={len(units)} " + f"paid moves issued={paid} sources={len(T['sources'])} (sizes {min(T['sources'])}..{max(T['sources'])}, L/m={L/T['m']:.2f}) " + f"exits={sum(exits.values())} {dict(sorted(exits.items()))} errors={len(T['errors'])} {T['errors'][:2]} unconsumed={unconsumed_nonexit} " + f"Phi_src={float(phis):.12f} Phi_exit={float(phie):.12f} |diff|={abs(phis-phie):.2E} deadlock_free={ok} ({seen}/{tot} events)", flush=True) + return T, ok + +if __name__ == "__main__": + rules = [ + ("D rank-1 control 1+4 in 10 (Thm 1)", {3: 1, 2: 4}, [1] * 10), + ("E rank-1 7+38 in 88 (Thm 1)", {3: 7, 2: 38}, [1] * 88), + ("A {4:1,3:2} r=[1,1,1,1,2,2]", {4: 1, 3: 2}, [1, 1, 1, 1, 2, 2]), + ("B {7:1,4:1} r=[1,1,1,1,2,2,2]", {7: 1, 4: 1}, [1, 1, 1, 1, 2, 2, 2]), + ("C {5:1,4:3} r=[1]*6+[2]*2+[3]*2", {5: 1, 4: 3}, [1] * 6 + [2] * 2 + [3] * 2), + ("H {3:2} r=[2,1,2] (word order mixed)", {3: 2}, [2, 1, 2]), + ("J {5:2} r=[4,1,4] theta=.861", {5: 2}, [4, 1, 4]), + ("X non-separated {3:1,2:4} r=[1]*6+[2]*2", {3: 1, 2: 4}, [1, 2, 1, 1, 2, 1, 1, 1]), + ] + Ls = {"D": (12, 30, 54), "E": (12,), "A": (40, 150, 400), "B": (150, 600), "C": (150, 400), "H": (300, 2000), "J": (600, 15000), "X": (30, 60)} + for name, Wt, rl in rules: + for L in Ls[name[0]]: + for seed in (1, 2): + report(name, Wt, rl, L, "lockstep", seed) + print("--- careless (cross-class) matching: deadlock expected ---", flush=True) + for name, Wt, rl in rules[:5]: + L = {"D": 30, "E": 12, "A": 150, "B": 150, "C": 150}[name[0]] + report(name, Wt, rl, L, "careless", 3) diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/tree_flow.py b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/tree_flow.py new file mode 100644 index 0000000000..31a9c1ae4d --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/programs/theorem-1g/tree_flow.py @@ -0,0 +1,167 @@ +# tree_flow.py (bound-check/rl-prove, 14 Oct 2026). The DILATED LOCKSTEP TREE of PROOF.md section 3 (Theorem 1G), +# for a mixed rule with GROUPED paid blocks: targets {k: W_k} (k in K), rounds r_1..r_P (P_r rounds of rank r), and +# the separation hypothesis r_max < k_min. Exact integers / fractions. +# u(a) = multinomial(|a|; a) prod_k W_k^a_k (a in N^K), c(a) = prod_k k^a_k, m = max K, +# R(rho) = {a : m rho c(a) <= L}, G = max{|a| : a in R(1)}. +# Class s (a round sequence of length d <= G, rho_s = product of its ranks): u(a) units at column rho_s c(a) for +# every a in R(rho_s) with |a| = G - d. Round j of class s feeds class s+(j) only. +# Checks (merged by (d, rho) with path multiplicities N(d, rho)): +# (1) exact balance: at every point b in R(rho') the child's demand equals the parent's round output; +# (2) every other demand is a SOURCE of size in (r_j L/m, L] (root: (L/m, L]); +# (3) exits only at generation 0: P jobs of sizes rho*r_j per class, all <= r_max * (L/m)^theta, theta = ln r_max/ln k_min; +# (4) Phi conservation at tau_c: Phi(sources) = Phi(exits); (5) unit count <= Phi(sources)/(W k_min^tau (1-k_min^-tau)). +# Literal lockstep simulation (lockstep_literal): explicit recursion over round sequences, a pool of issued jobs per +# class, every class must find all its inputs issued (parent round j + sources) and consume every job it is given. +import sys +from fractions import Fraction as Fr +from math import factorial, log, prod +from itertools import product as iprod + +def tau_c(Wt, P): + lo, hi = 0.0, 1.0 + for _ in range(200): + mid = (lo + hi) / 2 + if sum(w * k ** mid for k, w in Wt.items()) > sum(p * r ** mid for r, p in P.items()): hi = mid + else: lo = mid + return (lo + hi) / 2 + +def lattice(Wt, L, rho): + """points a in R(rho), as tuples over sorted K, grouped by |a|""" + K = sorted(Wt); m = max(K) + pts = {} + def rec(i, a, c): + if i == len(K): + pts.setdefault(sum(a), []).append(tuple(a)); return + k = K[i]; e = 0; cc = c + while m * rho * cc <= L: + rec(i + 1, a + [e], cc); e += 1; cc *= k + rec(0, [], 1) + return pts + +def mult(a): + return factorial(sum(a)) // prod(factorial(x) for x in a) + +class Tree: + def __init__(self, Wt, P, L): + self.Wt, self.P, self.L = dict(Wt), dict(P), L + self.K = sorted(Wt); self.m = max(self.K) + assert max(P) < min(self.K), "separation r_max < k_min fails" + self.Wv = [Wt[k] for k in self.K] + self.lat = {} + self.G = max(lattice(Wt, L, 1)) + def R(self, rho): + if rho not in self.lat: self.lat[rho] = lattice(self.Wt, self.L, rho) + return self.lat[rho] + def u(self, a): return mult(a) * prod(w ** x for w, x in zip(self.Wv, a)) + def c(self, a): return prod(k ** x for k, x in zip(self.K, a)) + def inR(self, a, rho): return self.m * rho * self.c(a) <= self.L + def units(self, d, rho): + return {a: self.u(a) for a in self.R(rho).get(self.G - d, [])} + def demand(self, d, rho): + need = {} + for a, ua in self.units(d, rho).items(): + for i, k in enumerate(self.K): + b = list(a); b[i] += 1; b = tuple(b) + need[b] = need.get(b, 0) + self.Wt[k] * ua + return need + +def check(Wt, P, L, tau=None): + T = Tree(Wt, P, L); tau = tau_c(Wt, P) if tau is None else tau + G, m, kmin, rmax = T.G, T.m, min(T.K), max(P) + theta = log(rmax) / log(kmin) if rmax > 1 else 0.0 + N = {(0, 1): 1} + src_w, src_phi, ex_w, ex_phi, units, bad = 0, 0.0, 0, 0.0, 0, [] + src_sizes, ex_sizes = set(), {} + # root + for b, x in T.demand(0, 1).items(): + s = T.c(b) + if T.inR(b, 1): bad.append(('root demand inside R(1)', b)) + if not (L / m < s <= L): bad.append(('root source size', s)) + src_w += x; src_phi += x * s ** tau; src_sizes.add(s) + for d in range(0, G + 1): + layer = {k: v for k, v in N.items() if k[0] == d} + for (_, rho), n in layer.items(): + us = T.units(d, rho) + units += n * sum(us.values()) + if d == G: # generation 0: exits + for a, ua in us.items(): + assert a == tuple(0 for _ in T.K) + for r, pr in P.items(): + e = rho * r + ex_w += n * pr * ua; ex_phi += n * pr * ua * e ** tau + ex_sizes[e] = ex_sizes.get(e, 0) + n * pr * ua + continue + for r, pr in P.items(): # children via a round of rank r + rc = rho * r + N[(d + 1, rc)] = N.get((d + 1, rc), 0) + n * pr + need = T.demand(d + 1, rc) + supply = {a: ua for a, ua in us.items()} # parent round output at point a, size rc*c(a) + for b in set(need) | set(supply): + sup = supply.get(b, 0); dem = need.get(b, 0) + if T.inR(b, rho): + if sup != dem: bad.append(('unbalanced', d, rho, r, b, sup, dem)) + else: + if sup: bad.append(('supply outside R', b)) + s = rc * T.c(b) + if dem and not (r * L / m < s <= L): bad.append(('source size', s, r)) + src_w += n * pr * dem; src_phi += n * pr * dem * s ** tau + if dem: src_sizes.add(s) + emax = max(ex_sizes) + bound = rmax * (L / m) ** theta if rmax > 1 else 1 + ubound = src_phi / (sum(Wt.values()) * kmin ** tau * (1 - kmin ** -tau)) + return dict(L=L, G=G, tau=tau, unbalanced_or_bad=len(bad), bad=bad[:3], sources=src_w, Phi_src=src_phi, + exits=ex_w, Phi_exit=ex_phi, Phi_ratio=ex_phi / src_phi, src_size_range=(min(src_sizes), max(src_sizes)), + exit_max=emax, exit_bound=bound, units=units, unit_bound=ubound, classes=sum(N.values()), + ex_sizes=ex_sizes, src_sizes=sorted(src_sizes)) + +def lockstep_literal(Wt, P_list, L): + """Explicit recursion over round sequences (P_list = ranks of rounds 1..P in word order).""" + Pd = {} + for r in P_list: Pd[r] = Pd.get(r, 0) + 1 + T = Tree(Wt, Pd, L); G = T.G + st = dict(classes=0, src=0, exits={}, issued_before_start=True) + def process(path, rho, fluid): + d = len(path); st['classes'] += 1 + us = T.units(d, rho) + need = T.demand(d, rho) + # fluid = parent's round output (points in R(parent rho)); top it up with sources, then it must equal need + for b, x in need.items(): + have = fluid.get(b, 0) + if have > x: raise AssertionError(('surplus', path, b, have, x)) + if have < x: st['src'] += x - have # source from the top split (always available) + for b in fluid: + if b not in need and fluid[b]: raise AssertionError(('unconsumed', path, b)) + for j, r in enumerate(P_list): # rounds in word order; round j+1 only after round j completes + out = dict(us) # one job of rank r per unit, at point a, size rho*r*c(a) + if d == G: + for a, ua in out.items(): + e = rho * r; st['exits'][e] = st['exits'].get(e, 0) + ua # handled by a new top call Do(e) + else: + process(path + (j,), rho * r, out) + process((), 1, {}) + return st, T + +if __name__ == "__main__": + rules = [("A: targets {4:1,3:2}, rounds [1,1,1,1,2,2]", {4: 1, 3: 2}, [1, 1, 1, 1, 2, 2]), + ("B: pure {7:2}+idle 3 -> targets {7:1,4:1}, rounds [1,1,1,1,2,2,2]", {7: 1, 4: 1}, [1, 1, 1, 1, 2, 2, 2]), + ("C: targets {5:1,4:3}, rounds [1]*6+[2]*2+[3]*2", {5: 1, 4: 3}, [1] * 6 + [2] * 2 + [3] * 2)] + print("(A) literal lockstep (explicit round sequences):", flush=True) + for name, Wt, rl in rules: + for L in (40, 150): + Pd = {} + for r in rl: Pd[r] = Pd.get(r, 0) + 1 + ref = check(Wt, Pd, L) + st, T = lockstep_literal(Wt, rl, L) + ok = (st['src'] == ref['sources'] and st['exits'] == ref['ex_sizes']) + print(f" {name}, L={L}: G={T.G}, classes {st['classes']}, every class exactly fed, sources {st['src']} " + f"(= merged count: {ok}), exits by size {dict(sorted(st['exits'].items()))}", flush=True) + print("(B) merged checks at larger L:", flush=True) + for name, Wt, rl in rules: + Pd = {} + for r in rl: Pd[r] = Pd.get(r, 0) + 1 + for L in (1000, 10 ** 5, 10 ** 7): + ch = check(Wt, Pd, L) + print(f" {name}, L={L}: tau_c={ch['tau']:.6f} G={ch['G']} classes={ch['classes']} bad={ch['unbalanced_or_bad']} " + f"sources in {ch['src_size_range']} (L/m={L/max(Wt):.1f}); Phi_exit/Phi_src={ch['Phi_ratio']:.12f}; " + f"max exit {ch['exit_max']} <= bound {ch['exit_bound']:.1f}; units {float(ch['units']):.4g} <= " + f"bound {ch['unit_bound']:.4g}", flush=True) diff --git a/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/proofs/theorem-1g.md b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/proofs/theorem-1g.md new file mode 100644 index 0000000000..36d2399f35 --- /dev/null +++ b/research/kappa-rebuild/architecture/sources/Sussman-563b4e6/research/record-track/proofs/theorem-1g.md @@ -0,0 +1,264 @@ +# Theorem 1G: mixed jobs with grouped paid blocks + +This note holds the proof behind the recursion lever of the record track: an extension of the τ-theorem (in the +form argued in [../../existence-routes/](../../existence-routes/), "Theorem 1") from paid moves of rank 1 to paid +blocks of any rank. It was written on 14 October 2026 by an AI agent working for me, in a simple model of jobs, and +it is reproduced here with its notation, cleaned of internal file names. + +**Status.** *Argued by hand, not checked.* Two independent AI skeptics tried to refute it on 14 October: one +re-derived the logic and simulated the construction with its own code, the other built the construction explicitly +(units, random matching inside the prescribed classes, a generic deadlock test) and checked the hypotheses of the +induction on the actual flows. Neither found an error. No person and no proof checker has read this text. A Lean +form of the theorem inside OpenAI's model was built on 19 October as part of the lever record (section 6); its +status is the record's: *proved in Lean*, as part of each of the five lever packages (in `package/leverF` and `package/leverB`, the modules under `Work/Mixed/` and `Work/MixedCost/`), with the official comparator accepting each result and three skeptics upholding each. + +Labels used inside the proof: [argued] (an argument given here in full, by hand; no proof checker has read it), +[computed] (a program, named in section 4), [what-if] (a price that rests on an unchecked step), [open]. + +## 0. The job model + +This is the model of the τ-theorem's proof, recalled so that this note can be read alone. + +- A **job of size s** is: apply the Walsh-Hadamard transform H on s chosen address bits of a register holding + 2^ν entries (ν ≥ s), that is, a batch of 2^(ν-s) independent transforms of size 2^s. Its cost is counted per entry: + T(s) is the number of full passes over the register (one per single-bit Hadamard leaf, one per free pass). A pass + over a fraction w of the entries costs w. +- A **rule** (a mixed unit) at block m: a word, correct for every number f of digit columns, with W_k registers + carrying a **rank-k target** (k of the m rows transformed, the other m - k untouched), paid blocks, and F free + register-passes. Applied at column c it turns jobs of the target sizes kc into jobs of the paid sizes r_j c. +- **Split**: a job of size a + b is the job of size a followed by the job of size b (Hadamards on disjoint bits + commute), at cost T(a) + T(b). A **plain leaf** of size s costs s passes. +- **Φ_τ** of a set of jobs (weighted by fractions of entries) is the sum of weight times size^τ. +- **τ** of a rule is the root in (0, 1) of Σ_j r_j^τ = Σ_k W_k k^τ (the *balance*). With every r_j = 1 this is the + τ-theorem's M = Σ_k W_k k^τ. + +The community's recursion is this model with one target size: a paid block is a recursive call on a whole register, +and the W arrays of the next level are W slices of that one call's batch. A rule with targets of two sizes needs jobs +of two sizes at once, which no single call's batch provides; that is what the construction below supplies. + +## 1. Re-check of the rank-1 case (the τ-theorem) + +- **Step 1 (multinomial flow).** With p_k = W_k/M and u(a) = multinomial(|a|; a) Π p_k^(a_k), Pascal's rule gives + u(b) = Σ over k with b_k ≥ 1 of p_k u(b - e_k) for b ≠ 0, so M u(b) = Σ_k W_k u(b - e_k). The region + R = {a : m c(a) ≤ L} is down-closed. A frontier point b (not in R, with some b - e_k in R) has size + c(b) = k c(b - e_k) ≤ kL/m ≤ L and c(b) > L/m. Confirmed [argued]. An exact re-run found 0 unbalanced sizes, + Φ(sources) = M to 10 digits, and sources in (L/3, L] [computed]. +- **Step 2 (Φ conservation).** Each unit is Φ-neutral at τ, so Φ(sources) = Φ(leaves) = M. Confirmed. +- **Step 3 (free passes).** The sum of u over generation j is (W/M)^j, so the sum of u is at most M/(M - W), and the + free passes come to F/(M - W) per unit of leaf Φ. The passes spent re-grouping portions (one per class level) cost + O(1) per register used by a unit, so they fold into F. +- **Lemma 2 (lockstep).** Process(g, λ) issues move k of all units in the class. The outputs plus (λ/M) Source at + generation g equal (λ/M) J at generation g, which is exactly what the child consumes. At level g there are + M^(g_max - g) calls with λ = M^-(g_max - g), so the sources used add up to Source exactly [argued]. The literal + simulation was re-run on 1+4 in 10 at L = 12, 30, 54, on 1+1 in 4 at L = 30, and on 7+38 in 88 at L = 12 + (7,833 classes): every class was fed exactly, the sources used equal Source, and the leaves equal M u(0) + [computed]. +- **Step 5 (top split and constant).** Every rest has size s - c(a) < (1 - 1/m)s. The induction closes with + C' = (A + c0)/(1 - (1 - 1/m)^τ). A sharper constant is C' = (A + c0)/η_m with + η_m = min over t in [1/m, 1] of (1 - (1-t)^τ)/t^τ, which is at least 1 - (1-1/m)^τ (Lemma A below). +- **Rounding.** The number of fluid portions is the number of lockstep classes, at most M^(g_max+1) = n^O(ln M/ln k_min). + That is polynomial in n, not the number of lattice points, and still far below 2^(n/2) batch entries, so dyadic + rounding costs a relative 2^-Ω(n). The claim stands; only the count in the earlier text was wrong. +- **A hypothesis that both theorems need and the earlier text did not state.** Each unit must be a correct word at + every column count f that occurs: products of target ranks in the rank-1 case, and products of target and paid ranks + in Theorem 1G. The community's engine also asserts m^2 > 2 max(r)^2; what that halving constraint asks of a new + recursion was not examined here. + +## 2. Lemma A: the top-split induction with leftovers (the exact condition) + +**Setting.** T(n) is the per-entry cost of the best schedule for a job of size n (with at least n/2 batch bits). A +*flow for n* consists of: +- source weights w(s) ≥ 0 on sizes 1 ≤ s ≤ n, with Σ w = 1; +- exit weights x(e) ≥ 0 on sizes e < n: jobs the flow leaves over, each completed by a separate call of the whole + procedure; +- an internal cost I ≤ A Φ̂ (free passes, re-grouping, plain size-1 leaves), where Φ̂ = Σ_s w(s) s^τ; +- a deadlock-free order, given that each exit call returns a completed job. + +The schedule splits the top job into portions w(s), each portion being s + (n - s). The s-parts feed the flow, and +the (n - s)-parts and the exits are done recursively. So + + T(n) ≤ c0 + I + Σ_e x(e) T(e) + Σ_s w(s) T(n - s). + +**Lemma A** [argued]. Let ψ ≥ 1 be nondecreasing, and let β > 0 and n0 be such that for every n ≥ n0 there is a flow +with + + (E) Σ_e x(e) e^τ ψ(e) + ψ(n) Σ_s w(s) (n - s)^τ ≤ ψ(n) (n^τ - β Φ̂). + +Then T(n) ≤ C n^τ ψ(n) for all n, with C = max{ max_{n L gives ρ_s c(b) > r_j L/m. Some b - e_k lies in R(ρ_s), so ρ_s c(b) ≤ kL/m ≤ L. ∎ + +**Lemma D (exits)** [argued]. Outputs leave the tree only at generation 0 (d = G). There each class has at most one +unit (at column ρ_s, present when m ρ_s ≤ L), and its P outputs have sizes ρ_s r_j. Φ(exits) = Φ(sources) exactly. +Every exit has size e ≤ r_max · r_max^G ≤ r_max (L/m)^θ with θ = ln r_max / ln k_min < 1, so ln e ≤ θ ln L. Exits of +size 1 are plain leaves. +*Proof.* By Lemma B, outputs at generation ≥ 1 are all consumed. Every unit is Φ-neutral at τ, and every consumed job +was produced or sourced, so Φ(exits) = Φ(sources). G ≤ log_(k_min)(L/m) because c(a) ≥ k_min^|a|, which gives +ρ_s ≤ r_max^G ≤ (L/m)^θ. Finally r_max ≤ m^θ (since r_max = k_min^θ ≤ m^θ), which gives ln e ≤ θ ln L. ∎ + +**Lemma E (number of units)** [argued]. units ≤ Φ(sources) / (W k_min^τ (1 - k_min^-τ)), with no log factor. +*Proof.* A unit at depth d sits at column ≥ c(a) ≥ k_min^(G-d), and its input Φ is +Σ_k W_k (k · col)^τ ≥ W k_min^τ k_min^(τ(G-d)). The total input Φ at each depth is at most Φ(sources), since Φ only +moves down through the tree or leaves it. Summing the geometric series over G - d gives the bound. ∎ + +**Lemma F (order: no deadlock)** [argued]. Procedure Process(s), called with the parent's round-j outputs pending: +1. the class's inputs are exactly those outputs plus top sources (Lemmas B and C); +2. for j = 1..P: issue round j of every unit of s, running the word's free steps between moves as the word says; if + d(s) = G, complete each exit job by a top call Do(e); otherwise call Process(s.j); +3. return. +A round j+1 is issued only after Process(s.j) has returned, which means every unit that consumed a round-j job has +finished. A unit starts only after all its inputs exist. So this is a straight-line program. Its recursion depth is +G + 1, and every exit call is a smaller instance (e < n). ∎ + +A careless order does deadlock: if a unit took one input from round 1 of unit A and another from round 2 of A, while +A's round 2 waits for that very unit, nothing can start. Lockstep classes, in which each child class takes only the +outputs of one round, rule this out. + +**Theorem 1G** [argued, job model]. Under (S): T(n) ≤ C n^τ (ln n)^κ for all n ≥ 3, where +κ = ⌈ln(2/η_m) / ln(1/θ)⌉ (κ = 0 when r_max = 1) and C depends only on the rule. So T(n) = O(n^(τ+δ)) for every +δ > 0. +*Proof.* Take L = n. The flow is the tree normalised to source weight 1. The sources lie in (n/m, n] (Lemma C), so +K = m. The internal cost I ≤ A Φ̂ with A = (F + O(W))/(W k_min^τ (1 - k_min^-τ)) (Lemma E, plus one size-1 leaf per +unit of leaf Φ). The exits satisfy Φ(exits) = Φ̂ and ln e ≤ θ ln n (Lemma D). The order is valid (Lemma F). Lemma A3 +applies with ψ = max(1, ln)^κ and β = η_m/2. The batch-bit and dyadic-portion arrangements are as in section 1: the +number of classes is at most Σ_d P^d = n^O(ln P / ln k_min). ∎ + +*Remarks.* +- With every r_j = 1 the tree is the rank-1 construction (ρ = 1, exits = leaves), and κ = 0 recovers O(n^τ). +- Exact O(n^τ) for grouped rules, with no log factor, would need a flow with Φ(exits) ≤ (η - β) Φ̂ (Lemma A2). This + tree has Φ(exits) = Φ̂, so by A4 it cannot give that [open]. +- The log factor costs nothing in the exponent: every saving below 1 - τ is reached with a fixed constant. OpenAI's + goal is a big-O statement, so an existential constant is allowed. +- Without (S): Lemma B needs r_j ≤ k for the pairs that meet, and Lemma D needs r_max < k_min. The case r_max ≥ k_min + is open. An example (targets {3:1, 2:4}, paid {1:6, 2:2}) has a fluid linear programme whose optimum was still rising + at n = 480. + +## 4. Numerical checks [computed] + +Every program named here is in [../programs/theorem-1g/](../programs/theorem-1g/) and its output in +[../data/theorem-1g/](../data/theorem-1g/). All were re-run on 19 October 2026 for this revision, on an Apple +M4 Mac mini with Python 3.9, in under a minute each and under 300 MB; every output was identical to the one made on +14 October. + +Three test rules, plus a control: +- A: targets {4:1, 3:2}, rounds 1,1,1,1,2,2 (τ 0.746375; θ 0.631, κ 3); +- B: targets {7:1, 4:1}, rounds 1,1,1,1,2,2,2 (τ 0.928234). This is the truncation of the pure rule {7:2} with paid + {1:4, 2:3} plus a rank-3 idle block; +- C: targets {5:1, 4:3}, rounds 1×6, 2×2, 3×2 (τ 0.930659; θ 0.792, κ 4); +- D: the rank-1 control 1+4 in 10. + +1. **Literal lockstep** (`tree_flow.py`): explicit round sequences, each round in word order. A, B and C were run at + L = 40 and 150 (up to 259 classes). Every class was exactly fed (no surplus, no deficit), the sources used equal the + merged count, and the exits match the merged count by size. +2. **Merged exact checks** at L = 10^3, 10^5, 10^7 (up to 1.6e10 classes, counted with multiplicity): 0 unbalanced + points; all sources in (L/m, L]; Φ(exits)/Φ(sources) = 1.000000000000; the largest exit at or below the bound + r_max (L/m)^θ (16,384 ≤ 21,761 for A at L = 10^7, 2,048 ≤ 2,391 for B, 177,147 ≤ 295,508 for C); the unit count + within the bound of Lemma E. +3. **Cost of the whole schedule** (`recur.py`; F = 0, L = n, exits by recursive calls), T(n)/n^τ: + + | rule | n = 50 | n = 400 | n = 3,200 | n = 6,400 | + |---|---|---|---|---| + | A | 1.612 | 1.896 | 2.034 | 2.052 | + | B | 1.144 | 1.197 | 1.218 | 1.229 | + | C | 1.177 | 1.192 | 1.249 | 1.246 | + | D (rank-1 control) | 1.162 | 1.180 | 1.183 | 1.183 | + + The grouped rules grow slowly, far below (ln n)^κ; A's local slope over the last doubling is 0.759 against + τ = 0.746 and falling. The control is flat, as the rank-1 theorem says. +4. **Re-check of the rank-1 theorem** (`recheck_t1.py`, which imports the τ-theorem's own programs from + [../../existence-routes/programs/tau-theorem-and-laser/](../../existence-routes/programs/tau-theorem-and-laser/)). +5. **The community's circuit T74, as a what-if** (`hub_whatif.py`). Control: the balance with one job size reproduces + the community's figure for T74, 1.6797550e-3 (1.6797549561e-3). The mixed price, assuming every idle block of T74 + is a head or a tail on its own array, is 1.8169468e-3 (+8.17 %), separated (8 < 25). **This assumption turned out + to be false**: only 11,010 of the 22,748 idle blocks are heads and none is a tail (the README of this section, + section 4.5), so this figure is **refuted** as a price of T74. It stays here as the test it was. + +The skeptics' own programs, written from this text without reusing its code: +- `indep_tree.py` rebuilds the merged tree independently (toy rule A and the T74 mixed rule up to L = 10^8): + 0 unbalanced points, Φ ratio 1, exits within the bound. `partial.py` prices partial truncation, which showed that + the result depends on how many idle blocks really are heads. +- `explicit_sim.py` builds explicit unit objects, matches outputs to inputs at random inside the prescribed classes, + and runs a generic topological sort of start, issue and finish events as a deadlock test independent of Lemma F; + Φ is tracked to 50 digits. Every lockstep run is deadlock-free with Φ(exits) = Φ(sources); a deliberately careless + matching across classes deadlocks, as it should. +- `cost_fit.py` and `check_E.py` compute the actual flows up to n = 12,800 and check hypothesis (E) of Lemma A3 at + every n there, the unit bound of Lemma E and the exit bound; for the T74 rule `check_E.py` checks every n up to + 20,000 and every 997th n up to 3,000,000. No violation anywhere. They also + confirm Lemma A4: (E) with ψ = 1 fails at every n, as predicted. + +## 5. What the new recursion needs that the community's uniform recursion does not have + +The community's recursion is one call per job: a call of size s slices its own batch into W rank-m arrays at column +s/m, and each paid block is an independent recursive call; no call ever needs a job of another size. Theorem 1G needs: +1. **Several sizes at once.** A unit at column c takes rank-73 arrays of size 73c and smaller ones (rank 49 for the + half cut, rank 25 for the full cut) in a fixed ratio. These come from different producers' outputs, never from one + call's batch. +2. **Fluid portions.** Classes are fractional portions of batches, which needs a portion and relabel step in the cost + model and dyadic rounding with at least n/2 batch bits (the class count is polynomial in n). +3. **A class tree in lockstep**, ordered as in Lemma F. +4. **A top split** of the top job into frontier portions of sizes in (n/m, n] plus rests, done recursively. +5. **Exit calls.** Jobs of size ρ_s r_j are completed by fresh top calls. +6. **Mixed-target words.** The word with the head blocks removed or shortened must be proved correct for the reduced + targets at every column count f that occurs. +7. **Hypothesis (S).** On the circuits of the record track it holds with room: for the half cut every paid rank is at + most 48 and the smaller job is 49; for the full cut every paid rank is at most 24 and the smaller job is 25. +8. **The structure of the circuit.** Each block that is cut must be the first operation on its array, acting on that + array alone (the truncation lemma of [../../existence-routes/](../../existence-routes/)). This is the head test + of the README, section 4.5. + +## 6. The Lean form + +On 19 October the recursion was written in Lean inside OpenAI's RAM model as part of the lever record. Two things +changed against the plan in section 5: + +- **No log factor.** The Lean cost bound does not use Lemma A3. It applies Lemma D at a rate point z1 and argues that + for every exponent z above z1 the exits' share of the sources' Φ shrinks by a factor at most 2 (E m/n)^(z - z1), + where E = r_max^(G+1) bounds the exits; since E m/n tends to 0, Lemma A closes with ψ = 1 for all large n, and a + finite maximum covers the small ones. The statement is: if the balance holds at z1, then for every z in (z1, 1] there + is a constant C with cost ≤ C · 2^n · (n + 1)^z. The constants exist but are astronomically large; only their + existence is used, which OpenAI's big-O goal allows. Lemma A4 is not contradicted: it concerns the exponent τ itself. +- **The program.** Units are coded as sequences of roles (the Pascal bijection of Lemma B as arithmetic on codes); + each paid block of a class becomes one child call whose answers are read back ("Hadamards on disjoint bits + commute"); the leaf class hands its exits to one fresh top call; the order is the lockstep order of Lemma F, by + induction over the word. + +Status of this Lean form: *proved in Lean*, as part of each of the five lever packages (in `package/leverF` and `package/leverB`, the modules under `Work/Mixed/` and `Work/MixedCost/`), with the official comparator accepting each result and three skeptics upholding each. Its module and theorem names are listed in the lever package +(package/leverB/). diff --git a/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json b/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json new file mode 100644 index 0000000000..fbe6da7241 --- /dev/null +++ b/research/kappa-rebuild/bit/BIT-ALLSIZE-PARAMETERS.json @@ -0,0 +1,69 @@ +{ + "C0_before_cover": "69/25000", + "C0_minus_C1": "1/100000", + "C1_after_cover": "11/4000", + "atom_power": "3/1000", + "finite_constants_to_replace": { + "ambient_dimension": "N, not50 or64", + "bad_fallback_children": "At most32*N^2 per bad final leaf", + "coefficient_data": "Every fixed rational entry, inverse, determinant, prefix descriptor and height", + "fixed_row_routes": "R, original port charts and every internal matcher/inverse", + "generic_affine_work": "At most4*N^2-2*N basic shears per rank1 normalizer together with its inverse", + "owned_storage": "C, bounded by one owned bank per allocated finite unit instance", + "paid_leaf_count": "P, including copied-center descendants", + "pointwise_and_copy_count": "L, literal compiled scalar/copy/erase records", + "productive_roles": "B, not3739 or661" + }, + "finite_template_hypothesis": "Fixed integers N>=2,B>=1,P>=1 and a whole-cover native schematic uniform word, with P/B < N^(1-C0), final paid leaves of rank1, arbitrary productive inputs and separately owned storage.", + "mixed_margin_at_C0": { + "display": 0.7944031286900184, + "lower": "67580344230009584908122845692691165477/85070591730234615865843651857942052864", + "strict_negative": false, + "strict_positive": true, + "upper": "270321376920038339632491382770764742407/340282366920938463463374607431768211456" + }, + "new_kappa": false, + "old_ordinary_saving": "384599/10000000000", + "ordinary_saving_admitted": "27/10000", + "ordinary_saving_before_final_relaxation": "27418653797/10000000000000", + "ordinary_saving_display": 0.0027418653797, + "prime_rule": { + "N_to_d_lower": "200001/200000", + "also_required": "Odd prime avoiding every fixed rational denominator, matcher determinant and chosen rank witness of the entire compiled word.", + "bound": "q > 25600000 * N^5", + "complete_moment_upper": "400001/400002", + "fallback_multiplier_upper": "400001/400000", + "good_moment_upper": "N^(-d), d=C0-C1", + "strict_moment_slack_lower": "1/400002" + }, + "root_cost_terms": [ + "e^(1-delta)", + "e^((1-delta)*(1-C1)+delta*(1-a_old))", + "e^delta * log(e)" + ], + "schema": "conditional-delayed-cover-BIT-allsize-parameters/1", + "source_sha256": { + "notes/batched-bit-rows.tex": "ef72c0ead19373cbc19c8a3aacb44dd0f75353ee5f4d055f775a986a4878022c", + "notes/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511", + "notes/three-stage-cover-rows.tex": "6ad97a5cd7d6aecf978a81226e1c2763d37840caca130c61f3bf5c4ac111c5c8", + "research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json": "9ece46fdfda6182e757e9a3b1f513fb3d029d730f3bafdcd6ed59180a14175b0", + "research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.json": "1e1a91871ad8d1098b6851c95d257e8e3fb130222933009b4edfb5e2c2d5c471", + "research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md": "9731b93026717f5fa15851e3633cbb4bdc76eba2bffb9d6b4a3b69f7dd502443", + "research/kappa-rebuild/bit/MIXED-PRODUCTIVE-BIT.json": "2f69a19f417c0e0d141c180f9b2fae3d358391b9a230f5848ee07c45450e70bf", + "research/kappa-rebuild/bit/bit_allsize_parameters.py": "59f92cf44019c27bde3db2f4482f2b5fcf25f0c3dcf071319d46b041f7b6a821" + }, + "status": "PASS_EXACT_PARAMETERS_CONDITIONAL_ON_FINITE_UNIFORM_PRECOMPILATION", + "stock": { + "asymptotic": "Fixed compiled N,B,P: k=O_(N,B,q)(w*log e)=o(e), w=ceil(e^(3/1000)); pad high rows once by factor<2.", + "borrowed_digits": "k <= D_N(ceil(e/w))*(N^2*(w-1)+ceil(log_q(B*|GL_N(F_q)|)))+ceil(log_q S_old)+1", + "depth_bound": "D_N(n)=floor(log_N n) for rank<=floor(n/N) children; any larger explicit integer bound is safe.", + "descendant_weights": "Only final leaves are lowered. This outer cover row and role type become ancestor data; every next ordinary invocation has fresh disjoint role stock.", + "reserve": "S(e,w)=W(w)^D_N(ceil(e/w))*S_old(e,w)", + "role_divisor": "W(w)=B*|GL_N(F_q)|*q^(N^2*(w-1))" + }, + "strict_power_gaps": { + "borrowed_stock_log_absorption": "9942581346203/10000000000000", + "final_relaxation": "418653797/10000000000000", + "linear_adapter": "2581346203/10000000000000" + } +} diff --git a/research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json b/research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json new file mode 100644 index 0000000000..e8c5e4f077 --- /dev/null +++ b/research/kappa-rebuild/bit/BIT-P3-NATIVE-BINDING.json @@ -0,0 +1,64074 @@ +{ + "all_f_lift": "Every head projector and chart is tensor-multiplied by I_f; the reversed tail block uses C_(Nf). Polynomial identities remain exact, and routes change only cover coordinates.", + "ambient_rank": 50, + "formal_cover_contract": "All local projectors, per-port right routes and unit matchers embed into one future ambient GL_N. 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"Linux-6.18.55-x86_64-with-glibc2.39" + }, + "elapsed_seconds": 23.195403568000074 +} diff --git a/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.json b/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.json new file mode 100644 index 0000000000..1c0c4bfb15 --- /dev/null +++ b/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.json @@ -0,0 +1,177 @@ +{ + "affine_high_row_checks": 7776, + "ambient_rank": 2, + "child_projectors": [ + [ + [ + 1, + 0 + ], + [ + 0, + 0 + ] + ], + [ + [ + 1, + 0 + ], + [ + 1, + 0 + ] + ] + ], + "common_external_weight": [ + 2, + 4 + ], + "controls": { + "illegal_fresh_row_weight_counterexample": { + "A_g": [ + 1, + 4 + ], + "A_h": [ + 2, + 4 + ], + "g": [ + [ + 0, + 1 + ], + [ + 1, + 0 + ] + ], + "h": [ + [ + 1, + 0 + ], + [ + 1, + 1 + ] + ], + "role": 0 + }, + "omitted_matcher_counterexample": { + "expected": [ + [ + 1, + 0 + ], + [ + 1, + 0 + ] + ], + "g": [ + [ + 0, + 1 + ], + [ + 1, + 0 + ] + ], + "role": 0, + "unrouted": [ + [ + 0, + 0 + ], + [ + 0, + 1 + ] + ] + } + }, + "cover_rows": 3888, + "exact_matches": 7776, + "fixed_matchers": [ + [ + [ + 1, + 1 + ], + [ + 1, + 0 + ] + ], + [ + [ + 0, + 1 + ], + [ + 1, + -1 + ] + ] + ], + "independent_target_roles": 2, + "last_step_leaf_normalization": { + "actual_weight_histogram": { + "1": 432, + "2": 1512, + "4": 432, + "5": 864, + "7": 432, + "8": 864 + }, + "dependency": "A_child is evaluated from the outer ancestor A and this outer cover row g; neither is a fresh field of the leaf invocation.", + "fallback_rows": 3240, + "good_rows": 4536 + }, + "new_kappa": false, + "not_checked": [ + "The complete mixed-flow precompiler", + "Materialization of a winning enormous uniform template", + "Final multiplication transfer" + ], + "parent_projectors": [ + [ + [ + 0, + 1 + ], + [ + 0, + 1 + ] + ], + [ + [ + 1, + 1 + ], + [ + 0, + 0 + ] + ] + ], + "proved_in_companion_note": "General rational projector matching, one global cover, common-gauge commutation, delayed weighted lowering and fixed routing costs.", + "ring": { + "atom_digits": 2, + "modulus": 9, + "prime": 3 + }, + "row_routing_bijections": [ + 3888, + 3888 + ], + "runtime_seconds": 0.6266116549995786, + "scalar_gauge_address_checks": 6561, + "schema": "delayed-bit-cover-weight-bridge/1", + "source_sha256": "96ea0f49a6fc07687e2f932859e5ce18e2b3f42d6e826fdc555c48069560a2ec", + "status": "PASS_EXACT_LOCAL_RING_AUDIT" +} diff --git a/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md b/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md new file mode 100644 index 0000000000..a58774e237 --- /dev/null +++ b/research/kappa-rebuild/bit/DELAYED-COVER-BRIDGE.md @@ -0,0 +1,401 @@ +# Delaying the BIT cover until the whole finite program is compiled + +Status: a constructive native-interface argument, with an exact local-ring +audit. This closes the particular common-weight mismatch identified in the +first mixed BIT proposal, conditional on the finite mixed-flow compiler +producing the stated finite projector program. It is not an independent +proof of that compiler's asymptotic termination or of integer multiplication. + +Substantial OpenAI assistance. Apache-2.0. This argument uses the arbitrary- +projector GL cover and weighted-child normal form in +[`notes/three-stage-cover-bit.tex`](../../../notes/three-stage-cover-bit.tex), +the retained PR589 local word and its predecessors, and the finite +precompilation proposal in `../architecture/FINITE-PRECOMPILATION.md`. +The decisive order of operations was proposed jointly with the architecture +research task: compile all internal jobs first; apply one common cover and +one common external weight only afterwards. + +## 1. The earlier obstruction was real + +A productive rank-eight helper is a proper projector job inside a larger +ambient address space. After an ambient GL cover, its endpoint is + +\[ +D_{gPg^{-1}}, +\] + +not the same fixed eight-coordinate interchange on every cover row. +Lowering this job immediately to a canonical recursive query introduces a +diagonal weight determined by the pivot coefficients of `g P g^-1`. +Jobs from different parents can have different such weights. The inherited +scalar word accepts one common external weight across every fresh role; +simply placing differently weighted jobs in its roles is invalid. + +The solution below does not identify these weights. It avoids creating +weighted internal jobs at all. Only leaves of the entire precompiled word +are lowered to canonical weighted calls. + +## 2. Native projector jobs before canonical lowering + +Fix an ambient rational head space of dimension `N` and the atom-reversal +matrix `C_N`. For any rational idempotent `P` put + +\[ +D_P=\begin{pmatrix} +I-P & PC_N\\ +C_NP & I-C_NPC_N +\end{pmatrix}. +\] + +For `G` invertible let + +\[ +Q_G=\operatorname{diag}(G,C_NGC_N). +\] + +Direct block multiplication gives + +\[ +Q_GD_PQ_G^{-1}=D_{GPG^{-1}},\qquad +D_PD_{I-P}=D_I. +\] + +No symmetry of `P` is required. The second identity says that any source +job and its complementary projector complete the full interchange on that +array. If a unit of rank `m` is used at column multiplicity `c`, first embed +its rational projectors into the chosen `mc`-dimensional coordinate block +of the common `N`-dimensional space. Complete unused coordinates are +spectators. All these dimensions and embeddings are constants of the +finite precompiled program. + +The local PR589 primitive, the new global controller, its pointwise boundary +adapters, and owned COPY storage can all be treated as a finite chronological +word of pointwise scalar operations, proper projector requests, restores, +and fixed right-multiplication symbols on one future cover coordinate. +Their exact finite operator identities are established independently of a +generic canonical child normalizer. An internal request may therefore be +replaced by a finite native word computing the same `D_P` on arbitrary data. + +This schematic qualification matters. The p3 construction already uses +per-port chart routes to share one local helper bank across 111 data ports. +It is not a 3,739-field word with those routes silently deleted. Before +choosing the final cover, embed every local projector and every port chart +into the common ambient `N`. Preserve all fixed right-route symbols during +substitution. Only then instantiate that single `GL_N` cover. A helper's +endpoint is the rank-eight local-space projector at its local invocation +vertex, while main data endpoints use their port-specific line and stage +projectors. `BIT-P3-NATIVE-BINDING.json` checks every fixed rational port +chart, this ownership distinction, and the complete recursive tariff. + +## 3. Equal rank gives a fixed rational matcher + +For rational idempotents `P,Q` of the same rank, choose a rational basis of +their images and kernels. Let `B_P` and `B_Q` be the corresponding full basis +matrices, with image columns first. Then + +\[ +S=B_PB_Q^{-1},\qquad SQS^{-1}=P. +\] + +This is a finite rational matrix construction. A physical execution of the +dense address map `Q_S` is **not** assumed. The matcher is used only as a +right multiplication of the full cover row, as follows. + +Choose a fixed odd prime `q` avoiding the denominators and nonzero +determinants of every native matrix and matcher in the finite program. +For atom width `w`, use one global cover + +\[ +\mathcal G=\operatorname{GL}_N(\mathbb Z/q^w\mathbb Z). +\] + +All matchers reduce to invertible matrices over this local ring. +An array role is accompanied by the full cover coordinate `g in G`. +Its entries at different `g` are independent data. + +Suppose a parent role at cover row `g` requests `D_(g P g^-1)` and a child +target role at its cover row `h` computes `D_(h Q h^-1)`. Before running +the child, route that complete target-role fiber by + +\[ +x_g\longmapsto y_{gS}=x_g. +\] + +Execute the child word on these routed inputs, then undo the role-specific +route. The endpoint on the parent's original `x_g` is exactly + +\[ +D_{(gS)Q(gS)^{-1}}=D_{gPg^{-1}}. +\] + +Every child target role can use its own fixed matcher `S`. Each mapping +`g -> gS` is a bijection of the *whole* cover fiber, so no input is copied, +discarded, or assumed zero. The child word can mix its roles at a common +child row `h`, because those routed inputs are arbitrary independent data. +After the completed word returns its independent endpoints, undoing the +routes restores the parent ordering. The finite flow's ownership and +chronology are still required; cover routing does not repair an aliased +or cyclic flow. + +## 4. One common external weight survives every internal routing + +At column multiplicity `f`, let the outer invocation's diagonal weight +`A` be any permitted generated descriptor on its `Nf` head atoms. It can +depend on exposed ancestor rows and atom indices. It must be independent +of all fresh role and cover fields of this entire compiled invocation. +Write `Lambda_A=diag(A,I)`. + +Conjugate the entire completed covered word by `Lambda_A`. Pointwise scalar +ADDs commute with this common address permutation across roles. Every +internal role route also commutes with it: a route changes only a fresh +role or the current cover row, while `A` is independent of both. Thus the +matching identity becomes + +\[ +\Lambda_A D_{(gS)Q(gS)^{-1}}\Lambda_A^{-1} +=\Lambda_A D_{gPg^{-1}}\Lambda_A^{-1}. +\] + +There is no child-specific weight to reconcile inside the precompiled +word. The source/complement product gives + +\[ +\Lambda_A D_I\Lambda_A^{-1}=F_A +\] + +on every productive source array. This is exactly the inherited weighted +full-interchange endpoint. Owned zero storage is unaffected by address +permutations and is excluded from productive output credit; its allocation, +initialization and final erasure still belong to the finite word. + +The condition on `A` is essential. The companion audit exhibits a concrete +failure when `A` is allowed to depend on the current fresh cover row. The +argument does not quietly weaken the old ancestor-only contract. + +## 5. Only the final leaves become weighted children + +After the mixed finite compiler has replaced every internal request, suppose +the remaining paid projector requests have rank one. At multiplicity `f`, +each is a rank-`f` residual on the outer atom space. Apply the inherited +generic ordered normal form to this *finished* leaf list. + +For a generic residue class, that proof produces unit lower intrabank and +cross-bank adapters around one contiguous weighted reversed interchange. +For column `j`, the resulting weight is + +\[ +A_{\rm child}[j]=A[i_0f+j]D_P[0], +\] + +with the inherited reversed tail pairing. The pivot coefficient is evaluated +from the current outer cover row and the fixed leaf projector. Those are +ancestor data when the recursive rank-`f` child is invoked. Its fresh roles +come from unused row stock, so the child's weight is independent of them. + +Different leaf calls can have different weights; the ordinary recursion +already permits this. Each such child itself receives one common weight +across all of *its* fresh roles and cover rows. This is the critical +distinction from calling a mixed unit on several already-normalized, +differently weighted parent jobs. + +Bad residue classes use the inherited genuine local-ring fallback, including +its swaps, paired scalings and reversals. This is charged explicitly rather +than declaring a nonunit pivot invertible. + +## 6. Routing is priced by the retained native lemma + +The cover row has the unique chart `g=g0+qG1`, with `g0` in `GL_N(F_q)`. +For each fixed matcher `S`, right multiplication gives a permutation of +the finite low type and a fixed affine map on the high matrix, with carry +determined by the low type. The retained proof of finite cover routing +implements this with a constant number of fixed scalings, ordered shifts +and old atom interchanges. Its cost is + +\[ +O(Vw^{1-a_{\rm old}}) +\] + +per fixed route. It is not a free address-gather operation. A precompiled +word contains finitely many matchers, so their total remains a finite +constant times this term. High matrices are streamed by fixed low type; +the number of tapes and program types does not grow with `w`. + +The same statement holds for nonintegral rational matchers: choose `q` +avoiding their fixed denominators, use their local-ring representatives, +and apply the retained fixed-rational routing lemma. The exact small audit +uses integral unimodular matchers to exercise the carry formula directly. + +## 7. Moment and row-stock consequences + +Let the finite uniform compiler have `B` productive source arrays, ambient +rank `N`, and `P` rank-one leaves, with a certified strict inequality + +\[ +\frac PB N^{-\tau}<1. +\] + +The covered, weighted word has the same normalized good-class leaf count. +Choose `q` large enough that the full fallback surcharge fits the displayed +slack. For example, the conservative inherited per-leaf bounds yield + +\[ +\mu_\tau\le +\frac PB N^{-\tau} +\left(1+\frac{2N^3}{q}\,32N^2\right)<1. +\] + +No old fixed choice of `q` or old numerical fallback envelope is reused for +the potentially enormous new `N`. Prime avoidance and a sufficiently large +prime exist because the whole precompiled program is finite. + +For the present candidate, use the exact mixed saving +`C0=69/25000=0.00276` for the finite compiler and retain +`C1=11/4000=0.00275` after covering. Put `d=C0-C1=1/100000`. +The former gives `P/B\frac{256N^5}{d}=25,600,000N^5. +\] + +Then the fallback multiplier is less than `1+d/4`, while +`N^d >= 1+d/2` for `N>=2`, using `log 2>1/2` and `exp x>=1+x`. +Thus the complete moment is strictly below +`(1+d/4)/(1+d/2)<1`. This uses an explicit exponent gap and does not +require knowing the eventual winning template's numerical moment slack. + +The cover consumes `N^2(w-1)` high-row digits plus a fixed low-type and role +reserve. Its finite owned-storage and scalar costs may be enormous. They +replace the old constants. For fixed `N` and a fixed finite word, the +ancestor reserve remains `O(w log e)`, and the previous eventual-threshold +arguments still apply after substituting those constants. They cannot be +reused numerically without that substitution. + +The existing weighted recursion can now be applied to this one uniform +supplier: every paid query has width at most `f`, the endpoint is `F_A`, and +all routing and scalar terms have the inherited permitted form. This is a +native-interface bridge. The complete final exponent still depends on the +mixed compiler's termination theorem, its exact leaf-count saving, and the +outer multiplication transfer. + +## 8. The ordinary BIT saving and explicit template constants + +The retained stopped recurrence gives + +\[ +T(n,w)=O(n+n^{1-C_1}w^{1-a_{\rm old}}),\qquad +a_{\rm old}=384599/10^{10}. +\] + +The old atom exponent `1/1000` cannot be reused here: its linear `n` term +would limit the ordinary saving to `0.001`. Instead choose +`delta=3/1000` and `w=ceil(e^delta)` from the original chunk width `e`. +For the whole compiled program set + +\[ +W(w)=B|\operatorname{GL}_N(\mathbb F_q)|q^{N^2(w-1)}. +\] + +A safe reserve is `W(w)^D * S_old`, with +`D >= floor(log_N ceil(e/w))`; all final ordinary children have rank at +most `floor(n/N)`. Therefore the borrowed high-digit count satisfies + +\[ +k\le D\left(N^2(w-1)+ +\left\lceil\log_q(B|\operatorname{GL}_N(\mathbb F_q)|)\right\rceil\right) ++\left\lceil\log_qS_{\rm old}\right\rceil+1 +=O_{N,B,q}(w\log e). +\] + +This is `o(e)`. For sufficiently large `e`, borrow `ka_BIT` and `delta<1-a_BIT` hold strictly. The linear adapter +term and the borrowed-stock logarithm are lower order. One may relax the +ordinary saving to the simpler rational `27/10000=0.0027`. +`BIT-ALLSIZE-PARAMETERS.json` records all strict rational gaps and the +exact mixed margin at `C0`. This remains conditional on the finite uniform +compiler premise; the enormous word is not materialized in this package. + +The final ambient dimension is `N`, the productive role count is `B`, and +the rank-one leaf count is `P`. They are not the old `64`, `661`, `3739`, +or the base unit's literal counts. Let `C` be the total allocated owned +array fields, `L` the literal scalar/copy/erase count, and `R` all fixed +right routes including inverses. Keep the full finite matrix and +coefficient-height list. The new fixed tape inventory can use one stream +per productive or owned role, with the inherited parking stack; it is +finite in `B+C`. Every route is charged, and every final leaf has its own +normalization and possible fallback. A rank-one generic normalizer and +its inverse use at most `4N^2-2N` elementary lower/cross-bank shears; the inherited bad +normalizer uses at most `32N^2` recursive calls of width at most `f`. + +These constants can be enormous and affect the eventual threshold. Their +dependence on the fixed compiled template does not introduce a new power +of the later input length. The mixed-flow argument is used to bound the +normalized number of paid rank-one leaves; it must not assume that dense +matrix routing costs stay uniformly bounded while the candidate template +dimension `N` is increased in the existence search. + +## 9. Exact finite audit + +Run: + +```sh +python3 -B research/kappa-rebuild/bit/delayed_cover_bridge.py +``` + +The audit enumerates every one of the 3,888 rows of `GL_2(Z/9Z)` for each +of two independently matched target roles. It checks 7,776 exact physical +projector identities, both full cover bijections, the affine high-row carry +formula, and every one of the 6,561 addresses for common-gauge scalar +commutation. Its external weight is `(2,4)`, so it is not restricted to an +unweighted or constant-across-axis input. The emitted leaf-weight histogram +is nonconstant. Two explicit negative controls reject omitting a matcher +and allowing the external weight to depend on the fresh cover row. + +These are independent checks of concrete algebra and a genuine local ring. +The general argument is the conjugacy calculation above, not an inference +from the finite sample. + +## Open next questions + +1. Independently audit that the finite lockstep compiler only substitutes + whole arbitrary-input jobs and gives each simultaneously live child role + a distinct physical array. The row matching here assumes that ownership. +2. Bind the new BIT p3 native physical projector list and its exact local + primitive into the abstract finite compiler, including all COPY centers. +3. Write the parameter substitutions for the eventual row reserve and the + final BIT/complex outer exponent. A finite numerical mixed moment alone + is still insufficient for a new multiplication claim. diff --git a/research/kappa-rebuild/bit/LITERATURE.md b/research/kappa-rebuild/bit/LITERATURE.md new file mode 100644 index 0000000000..b3ed942c29 --- /dev/null +++ b/research/kappa-rebuild/bit/LITERATURE.md @@ -0,0 +1,153 @@ +# Literature mapped to the actual accounting problem + +Research pass: 21 October 2026. These are primary papers and institutional +author copies. A paper's abstract notion of rank, space or circuit size does +not automatically price native recursive calls. The application statements +below are the current research team's deductions, with their finite tests +identified separately. + +## 1. Reversible linear circuits and shared producer state + +**K. N. Patel, I. L. Markov and J. P. Hayes, “Optimal synthesis of linear +reversible circuits” (2003 preprint; later journal version).** +[Primary preprint](https://arxiv.org/abs/quant-ph/0302002). +Their binary linear-circuit synthesis gives a systematic alternative to naive +elimination, with asymptotic `O(n^2/log n)` gate counts. The relevant import is +explicit reversible ADD/SWAP programs and shared intermediate linear forms. +It does not remove the native frame in which an addition must be delivered. + +**A. Visconti, C. V. Schiavo and R. Peralta, “Improved Upper Bounds for the +Expected Circuit Complexity of Dense Systems of Linear Equations over GF(2).”** +[NIST author record](https://www.nist.gov/publications/improved-upper-bounds-expected-circuit-complexity-dense-systems-linear-equations-over), +[NIST paper](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=925478). +This is relevant to dense shared sums followed by a smaller number of row +edits. Our deferred-original construction keeps the sum itself alive until +all its readers and dirty pools finish, rather than recomputing it for every +consumer. The finite exact result is `DEFERRED-RECONSTRUCTION.json`. + +**A. Parent, M. Roetteler and K. M. Svore, “Reversible circuit compilation +with space constraints” (2015).** +[Primary preprint](https://arxiv.org/abs/1510.00377). +The useful viewpoint is a joint time/space compilation problem, with explicit +uncomputation rather than a free cleanup assumption. Our version must also +respect a monotone subspace frame. The actual PR589 helper-pair census shows +why ordinary lifetime reuse is insufficient here: all five geometrically +eligible coalescings fail a subsequent exact primitive or response check. + +## 2. Rectifier networks, coding width and consumer interfaces + +**D. Chistikov, S. Iván, A. Lubiw and J. Shallit, “Fractional Coverings, +Greedy Coverings, and Rectifier Networks.”** +[Primary preprint](https://arxiv.org/abs/1509.07588), +[STACS 2017 publication](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23). +This connects shared intermediate computation with combinatorial covering +objectives. It motivated comparing the entire consumer map with individual +reads and minimizing a source-neighborhood path cover. Rectifier size alone +does not pay arbitrary dirty baselines, inverse operations or frame motion. + +**G. D. Forney Jr., “Minimal realizations of linear systems: The ‘shortest +basis’ approach” (2009; final revision 2010).** +[Primary shortest-basis paper](https://arxiv.org/html/0910.4336v3). +**N. Kashyap, “On Minimal Tree Realizations of Linear Codes” (2007).** +[Primary paper](https://arxiv.org/html/0711.1383v1). +The relevant question is the smallest state space needed across a specified +cut or decomposition, not just the rank of one arbitrary factorization. +Our actual nine-center consumer matrix has rank nine, as does the producer +response. This rejects a fixed-cut eight-signal carrier before its native +transport and cleanup bill is assessed. Moving the cut or changing the +consumer frame remains a different, open design choice. + +## 3. FIR delay factorization and exact projector controllers + +**Z. Doğanata, P. P. Vaidyanathan and T. Q. Nguyen, “General synthesis +procedures for FIR lossless transfer matrices, for perfect-reconstruction +multirate filter bank applications” (1988).** +[Caltech author copy](https://authors.library.caltech.edu/records/mv71c-vqj75), +[DOI](https://doi.org/10.1109/29.7544). +The paper constructs lossless FIR transfer matrices through a state-space +and cascaded-lattice approach. This suggested treating recursive stages as +formal transfer factors with a boundary basis, rather than preserving every +old intermediate frame alignment. + +**H.-J. Park, “A Computational Theory of Laurent Polynomial Rings and +Multidimensional FIR Systems” (Berkeley thesis, 1995).** +[Institutional PDF](https://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/ERL-95-39.pdf). +Section 7.4 records the one-variable projector/delay factorization +`I-P+zP`; Theorem 7.4.8 gives the lossless factorization in its original +setting. The multivariable discussion also warns that factorization does +not become automatic merely by adding variables. + +Our use is an exact algebraic analogue: transport each stage projector +through the constant scalar prefix, demand the appropriate commutation, +and check the complete coefficient identity. We import no SVD, unitary +rotation, irrational coefficient or automatically legal frame change. +This led to the successful p3 endpoint basis and the exact complex Laurent +lift, followed by the broader 4,096-mask and 123,904-choice negative rounds. +The same idea later proves the H controller factorization with genuinely +noncommuting stage symbols; the extra endpoint bill is then visible rather +than hidden by a commutative simplification. + +## 4. Catalytic computing as an operator identity + +**A. Vinciguerra, “An Operator Approach to Register Programs for Catalytic +Computing” (2026).** +[Primary paper](https://arxiv.org/abs/2609.18692), +[HTML](https://arxiv.org/html/2609.18692v1). +The paper represents register programs through derivation operators and +reduces catalytic restoration to an operator identity. Its polynomial +construction has an explicit characteristic restriction; it is not a +ready-made construction over the BIT word's `F2` scalar field. + +The useful methodological import is exact restoration of every arbitrary +input, including the supposedly incidental work fields. Our controller +audits initialize all productive fields independently. The owned stream +used in the nine-row and three-row completions is separately declared and +never receives productive credit. A future embedded-query logger would +alter the helper interface; its baseline must be restored before that +helper is again counted as a clean independent transform output. + +## 5. Matrix completion and low-rank feedback + +**I. Baragaña and A. Roca, “Fixed rank perturbations of regular matrix +pencils” (2019).** +[Primary paper](https://arxiv.org/html/1907.10657v2). +The paper distinguishes fixed-rank from bounded-rank perturbation problems +and treats prescribed determinant/structure under field-size hypotheses. +It also points back to Thompson's rank-one perturbation work (1980), +Silva's prescribed-similarity work (1988), and Zaballa's pole assignment +with fixed-rank perturbations (1991). Those hypotheses must be checked +before using a theorem in the two-element scalar field. + +**A. Amparan, I. Baragaña, S. Marcaida and A. Roca, “Row or column completion +of polynomial matrices of given degree” (2023; revised 2024).** +[Primary paper](https://arxiv.org/abs/2306.06104). +It studies completion with prescribed rows/columns and eigenstructure over +an arbitrary field. This is a better conceptual fit for our local problem +than unconstrained numerical low-rank approximation: only designated +output holes have the cheap rank-one delivery interface. + +The actual BIT completion is proved directly, rather than inferred from +these general theorems. For `J=I+UV`, selecting an invertible `V_A` and +changing `U` to `U+E_A V_A^-1 VU` makes the defect square-zero. This gives +a nine-row involution. Dropping the unnecessary involution requirement +reveals `rank(J)=108`, so only three row changes are necessary and +sufficient for invertibility. The native three-gather word and both +orientation-dependent wrappers are replayed exactly. This is the latest +concrete literature-to-probe refinement, but the global head normalization +still consumes more rank than its complete target credit provides. + +## Next research gate + +The highest-value next probe is an **actual owner-aware completion and +controller designed together**. Its objective must be the total recursive +moment after all three interval head corrections, not a local correction +rank or an elegant scalar identity. Candidate routes include a shared +query state with a proven final erase, a changed center family with +compatible physical monodromies, or a mixed controller that removes some +endpoint products without losing independent output credit. + +The first disproof gate should be exact noncommuting endpoint algebra; +the second should be actual rational port charts and arbitrary dirty +inputs; the third should be the full paid profile and strict child/job +separation. Only a survivor reaches the delayed-cover and outer-supplier +compiler. The stable p3 construction remains separate throughout. diff --git a/research/kappa-rebuild/bit/MIXED-CONTROLLER-INDEPENDENT.json b/research/kappa-rebuild/bit/MIXED-CONTROLLER-INDEPENDENT.json new file mode 100644 index 0000000000..38d831f1d3 --- /dev/null +++ b/research/kappa-rebuild/bit/MIXED-CONTROLLER-INDEPENDENT.json @@ -0,0 +1,100 @@ +{ + "arbitrary_function_input_columns": 1536, + "checker_sha256": "759c1814d6d0e24b739389107b1fabcd1c09a30425f3ee9efa84767d339c2816", + "controls": { + "omit_input_adapter_0": { + "wrong_coefficients": 256, + "wrong_rows": 256 + }, + "omit_input_adapter_1": { + "wrong_coefficients": 1536, + "wrong_rows": 768 + }, + "omit_input_adapter_2": { + "wrong_coefficients": 1024, + "wrong_rows": 512 + }, + "omit_input_adapter_3": { + "wrong_coefficients": 1536, + "wrong_rows": 768 + }, + "omit_input_adapter_4": { + "wrong_coefficients": 256, + "wrong_rows": 256 + }, + "omit_remaining_idle_13": { + "wrong_coefficients": 3072, + "wrong_rows": 1024 + }, + 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"all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 99 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 100 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 101 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 102 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 103 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 104 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 105 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 106 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 107 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 108 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 109 + }, + { + "all_other_fresh_blocks_fixed": true, + "old_hole_to_fresh": true, + "own_active_to_old_full": true, + "port": 110 + } + ], + "new_kappa": false, + "noncommuting_factor": { + "boundary_unit_ADDs_per_port": 22, + "coefficient_projectors": { + "(2, 1, 0)": [ + 5, + 2, + 0, + 2, + 0, + 0 + ], + "(6, 5, 4)": [ + 0, + 0, + 0, + 0, + 16, + 32 + ], + "(6, 5, 4, 3, 2, 1, 0)": [ + 4, + 0, + 4, + 10, + 0, + 0 + ] + }, + "diagonal_monomials": [ + [ + 2, + 1, + 0 + ], + [ + 2, + 1, + 0 + ], + [ + 6, + 5, + 4 + ], + [ + 6, + 5, + 4 + ], + [ + 6, + 5, + 4, + 3, + 2, + 1, + 0 + ], + [ + 6, + 5, + 4, + 3, + 2, + 1, + 0 + ] + ], + "full_coefficient_endpoint_sha256": "8bef855225bc0948c0a18557200543f0bc0cd16958b31595a2979ab5cb83589b", + "left_program": [ + [ + "SWAP", + 4, + 5 + ], + [ + "SWAP", + 3, + 4 + ], + [ + "ADD", + 0, + 2 + ], + [ + "SWAP", + 2, + 5 + ], + [ + "ADD", + 4, + 1 + ] + ], + "left_rows": [ + 5, + 2, + 16, + 32, + 10, + 4 + ], + "right_program": [ + [ + "ADD", + 0, + 5 + ], + [ + "SWAP", + 4, + 5 + ], + [ + "SWAP", + 3, + 5 + ], + [ + "SWAP", + 2, + 4 + ], + [ + "ADD", + 3, + 1 + ] + ], + "right_rows": [ + 33, + 2, + 32, + 18, + 4, + 8 + ] + }, + "owned_COPY_episodes": 63, + "physical_local_contract": "F_s/B_s = diag(T_s,Q T_s Q) times the Q-valued elementary shear; T_s=H_s b_s, H_s^2=I, [H_s,Q b_s]=0.", + "raw_margins": { + "0": { + "display": 603.0, + "lower": "603/1", + "strict_negative": false, + "strict_positive": true, + "upper": "603/1" + }, + "1/250": { + "display": 275.3993792858817, + "lower": "93713552631957099015026374420061506967857/340282366920938463463374607431768211456", + "strict_negative": false, + "strict_positive": true, + "upper": "93713552631957099015026374420061507047513/340282366920938463463374607431768211456" + }, + "3/1000": { + "display": 356.84094561657395, + "lower": "60713340794356833142404149776241551677751/170141183460469231731687303715884105728", + "strict_negative": false, + "strict_positive": true, + "upper": "60713340794356833142404149776241551717743/170141183460469231731687303715884105728" + }, + "7/2000": { + "display": 316.08210168281954, + "lower": "53778582850987287497314218191607309822857/170141183460469231731687303715884105728", + "strict_negative": false, + "strict_positive": true, + "upper": "107557165701974574994628436383214619725533/340282366920938463463374607431768211456" + } + }, + "raw_paid": { + "1": 9324, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1402 + }, + "raw_rank_margin": 603, + "runtime_seconds": 0.2392450520001148, + "scalar_costs": { + "common_frame_H_ADDs": 1452, + "controller_boundary_ADDs_per_port": 22, + "endpoint_inverse_monodromy_H_uses": 26, + "extra_fixed_cover_routes": "Every H use and inverse has its literal per-port right routes; no physical H use is free of linear/routing tariff.", + "global_unit_ADDs_with_direct_normalization": 163962, + "idle_common_frame_H_uses": 12, + "local_H_unit_ADDs": 11955, + "local_common_frame_wrappers": 14, + "owned_clear_or_erase_records": 126 + }, + "schema": "actual-BIT-H-noncommuting-p3-global-probe/1", + "source_sha256": { + "BIT-P3-NATIVE-BINDING.json": "9ece46fdfda6182e757e9a3b1f513fb3d029d730f3bafdcd6ed59180a14175b0", + "PR589-H-COMPLETION.json": "48006f2f9eba64a469cc087dd79209137335fb0d903d89ab35d7daad6cedceb8", + "PR589-H-NATIVE.json.gz": "4408d350db69102220a5eaef0448ef46207e832f27974ab7cec0f3818713f434", + "pr589_h_global_probe.py": "f0b9a6ac10b99c1a204fc3d9a0d26f20c6c413c436054b076ab2d9398864d391" + }, + "stage_commutation": "[H_s,b_t]=0 for s!=t, by the exact native chart identities. Hence each interval product M=K B, and [K,Q B]=0.", + "status": "PASS_EXACT_ENDPOINT_FACTOR_WITH_UNFUNDED_DIRECT_HEAD_REPAIR", + "targets": { + "8": 3073, + "22": 444, + "50": 222 + } +} diff --git a/research/kappa-rebuild/bit/PR589-H-NATIVE.json.gz b/research/kappa-rebuild/bit/PR589-H-NATIVE.json.gz new file mode 100644 index 0000000000..0b92077a56 Binary files /dev/null and b/research/kappa-rebuild/bit/PR589-H-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION-NATIVE.json.gz b/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION-NATIVE.json.gz new file mode 100644 index 0000000000..8c68c28b2d Binary files /dev/null and b/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION.json b/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION.json new file mode 100644 index 0000000000..d07d86b96b --- /dev/null +++ b/research/kappa-rebuild/bit/PR589-MINIMUM-COMPLETION.json @@ -0,0 +1,195 @@ +{ + "H_source_artifact_sha256": "4408d350db69102220a5eaef0448ef46207e832f27974ab7cec0f3818713f434", + "J_rank": 108, + "artifact_sha256": "3109f76749eab76da280826fd468a608c3ce05616e7ea66090a91ef0ffa36d5d", + "canonical_artifact_sha256": "492168c9f917f165a24d2f29367246e48f78738a5f300634b3cad900dc17b214", + "common_frame_G_ADDs": 6166, + "common_frame_G_inverse_ADDs": 6222, + "completed_rank": 111, + "correction_source_row_weights": [ + 59, + 55, + 57 + ], + "direct_head_repair_rank1_calls": 666, + "direct_repaired_paid": { + "1": 9948, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1402 + }, + "direct_repaired_rank_margin": -21, + "exact_raw_margin_at_004": { + "display": 317.3993792858817, + "lower": "108005412042636514480488107932195771849009/340282366920938463463374607431768211456", + "strict_negative": false, + "strict_positive": true, + "upper": "108005412042636514480488107932195771928665/340282366920938463463374607431768211456" + }, + "exact_repaired_margin_at_004": { + "display": -348.6006207141183, + "lower": "-118622644326708502186119380617361856980687/340282366920938463463374607431768211456", + "strict_negative": true, + "strict_positive": false, + "upper": "-118622644326708502186119380617361856901031/340282366920938463463374607431768211456" + }, + "full_arbitrary_input_checks": [ + { + "arbitrary_columns": 661, + "direction": "forward", + "owned_clear_erase_episodes": 3, + "owned_fields": 1, + "paid": { + "1": 1326, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 10 + }, + "paid_rank_mass": 5069, + "prefix_positive_majorant": "231552", + "prefix_positive_majorant_bits": 18, + "primitive_counts": { + "ADD": 11625, + "MOVE": 2914, + "OWNED_CLEAR": 3, + "OWNED_ERASE": 3 + }, + "restored_dirty_helpers": 439, + "wrapped": false + }, + { + "arbitrary_columns": 661, + "direction": "reverse", + "owned_clear_erase_episodes": 3, + "owned_fields": 1, + "paid": { + "1": 1326, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 10 + }, + "paid_rank_mass": 5069, + "prefix_positive_majorant": "395700", + "prefix_positive_majorant_bits": 19, + "primitive_counts": { + "ADD": 11625, + "MOVE": 2914, + "OWNED_CLEAR": 3, + "OWNED_ERASE": 3 + }, + "restored_dirty_helpers": 439, + "wrapped": false + }, + { + "arbitrary_columns": 661, + "direction": "forward", + "owned_clear_erase_episodes": 3, + "owned_fields": 1, + "paid": { + "1": 1326, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 10 + }, + "paid_rank_mass": 5069, + "prefix_positive_majorant": "553102610835552016316", + "prefix_positive_majorant_bits": 69, + "primitive_counts": { + "ADD": 23957, + "MOVE": 2914, + "OWNED_CLEAR": 3, + "OWNED_ERASE": 3 + }, + "restored_dirty_helpers": 439, + "wrapped": true + }, + { + "arbitrary_columns": 661, + "direction": "reverse", + "owned_clear_erase_episodes": 3, + "owned_fields": 1, + "paid": { + "1": 1326, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 10 + }, + "paid_rank_mass": 5069, + "prefix_positive_majorant": "515530264302074203076486786641586423723871034", + "prefix_positive_majorant_bits": 149, + "primitive_counts": { + "ADD": 24069, + "MOVE": 2914, + "OWNED_CLEAR": 3, + "OWNED_ERASE": 3 + }, + "restored_dirty_helpers": 439, + "wrapped": true + } + ], + "generator_sha256": "a3782f3aba5bdacdcf9d88a7663f19d6006543e936279288ffa6d1f137ba0f20", + "global_contract": "In the pair gauge S=diag(I,Q), forward wrapped multiplier is G_s b_s and reverse wrapped multiplier is G_s^-1 b_s. Both multiply the two fields equally, so the same noncommuting interval factorization applies.", + "global_raw_paid": { + "1": 9282, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1402 + }, + "minimum_proof": "rank(J+E)-rank(J)<=rank(E)<=number of modified rows; the three displayed row replacements attain the bound.", + "minimum_row_supported_correction_rank": 3, + "new_kappa": false, + "no_new_global_closure": true, + "omitted_row_negative_controls": [ + { + "omitted_selected_port": 106, + "resulting_rank": 110 + }, + { + "omitted_selected_port": 109, + "resulting_rank": 110 + }, + { + "omitted_selected_port": 110, + "resulting_rank": 110 + } + ], + "physical_completion": "Each selected target is its native q-perp; the completed source bank is fullI8. Their exact complement is native rank-one Pq, as in the independently audited H completion.", + "raw_rank_margin": 645, + "replacement_source_columns": [ + 0, + 1, + 2 + ], + "runtime_seconds": 0.44443022499945073, + "schema": "actual-PR589-minimum-invertible-completion-receipt/1", + "selected": [ + 106, + 109, + 110 + ], + "source_word_sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f", + "status": "PASS_MINIMUM_THREE_ROW_NATIVE_COMPLETION_GLOBAL_DEBT_UNFUNDED", + "targets": { + "8": 3073, + "22": 444, + "50": 222 + } +} diff --git a/research/kappa-rebuild/bit/PR589-STATE-SHARING-PROBE.json b/research/kappa-rebuild/bit/PR589-STATE-SHARING-PROBE.json new file mode 100644 index 0000000000..cb4a5da1cd --- /dev/null +++ b/research/kappa-rebuild/bit/PR589-STATE-SHARING-PROBE.json @@ -0,0 +1,290 @@ +{ + "accepted": [], + "baseline": { + "all_original_helpers_restored": true, + "arbitrary_columns": 661, + "dirty_helpers": 439, + "owned_COPY_episodes": 9, + "paid": { + "1": 1323, + "2": 1342, + "3": 79, + "4": 44, + "5": 102, + "6": 11, + "7": 19 + }, + "unit_ADDs": 24305 + }, + "baseline_global_p3": { + "exact_margin_at_004": { + "display": -99.18135962389455, + "lower": "-4218708475906954655215013037424231185379/42535295865117307932921825928971026432", + "strict_negative": true, + "strict_positive": false, + "upper": "-16874833903627818620860052149696924701471/170141183460469231731687303715884105728" + }, + "floating_root_for_screen": 0.0027698302509953643, + "paid": { + "1": 9261, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1465 + }, + "rank_margin": 225, + "targets": { + "8": 3073, + "22": 444, + "50": 222 + } + }, + "coverage": { + "disjoint_forward_lifetimes": 0, + "monotone_frame_candidates": 5, + "self_gate_reject": 2, + "source_response_reject": 3, + "total_helper_pairs": 96141 + }, + "not_claimed": [ + "No unrestricted scheduling lower bound.", + "No decoder-COPY charge removed without a new operator identity.", + "No exponent gain from a finite helper count alone." + ], + "owned_center_independence": { + "center_full_arbitrary_response_rank": 9, + "center_source_response_rank": 9, + "conclusion": "The nine actual center signals are independent, and their consumer map is injective. A same-cut linear compression to eight signals cannot preserve all responses even before its frame-conversion and cleanup cost is charged.", + "consumer_matrix_rank": 9, + "current_center_rank_bill": 63, + "owned_center_count": 9 + }, + "runtime_seconds": 0.12710095799957344, + "sample_response_rejections": [ + { + "pair": [ + 435, + 622 + ], + "wrong_source_coefficients": 66, + "wrong_targets": [ + 0, + 1, + 4, + 5, + 8, + 9, + 11, + 12, + 14, + 16, + 17, + 20, + 21, + 26, + 27, + 28, + 29, + 33, + 35, + 36, + 40, + 44, + 45, + 46, + 47, + 49, + 52, + 53, + 54, + 60, + 62, + 63, + 64, + 65, + 66, + 67, + 68, + 71, + 72, + 73, + 74, + 77, + 78, + 80, + 84, + 88, + 91, + 92, + 95, + 96, + 97, + 100, + 101, + 105, + 106, + 109, + 110 + ] + }, + { + "pair": [ + 472, + 484 + ], + "wrong_source_coefficients": 513, + "wrong_targets": [ + 2, + 3, + 4, + 5, + 10, + 11, + 12, + 14, + 15, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 27, + 28, + 29, + 30, + 32, + 33, + 34, + 35, + 37, + 38, + 40, + 41, + 42, + 43, + 45, + 46, + 47, + 50, + 51, + 55, + 56, + 57, + 58, + 59, + 61, + 62, + 65, + 67, + 68, + 69, + 70, + 73, + 74, + 79, + 80, + 84, + 85, + 86, + 87, + 89, + 90, + 92, + 95, + 98, + 99, + 100, + 101, + 103, + 104, + 109, + 110 + ] + }, + { + "pair": [ + 476, + 585 + ], + "wrong_source_coefficients": 576, + "wrong_targets": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 10, + 12, + 13, + 14, + 17, + 19, + 20, + 23, + 24, + 25, + 26, + 29, + 30, + 33, + 35, + 36, + 39, + 40, + 41, + 42, + 45, + 47, + 48, + 49, + 51, + 54, + 56, + 57, + 59, + 60, + 61, + 63, + 64, + 66, + 67, + 69, + 72, + 74, + 75, + 77, + 80, + 81, + 83, + 84, + 85, + 86, + 89, + 90, + 92, + 95, + 96, + 99, + 100, + 101, + 102, + 103, + 106, + 108, + 109 + ] + } + ], + "schema": "actual-PR589-monotone-helper-state-sharing/1", + "source_generator_sha256": "c75f14cb5de3609abca2c38b1b171b32d70b227bf4cb17c797cc2f4ebafececc", + "source_word_sha256": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f", + "status": "PASS_EXHAUSTIVE_FINITE_PROBE" +} diff --git a/research/kappa-rebuild/bit/README.md b/research/kappa-rebuild/bit/README.md new file mode 100644 index 0000000000..10132fe535 --- /dev/null +++ b/research/kappa-rebuild/bit/README.md @@ -0,0 +1,215 @@ +# Rebuilt BIT supplier, delayed-cover bridge, and larger-gain probes + +Prepared with substantial OpenAI assistance. Apache-2.0. This directory contains +new source files after the workspace loss. The retained predecessors are read +only. Fresh finite fixtures are not represented as recovered historical bytes. + +## Main result and scope + +The strongest closed BIT path in this directory changes the **global controller +and the accounting of independent helper outputs**, while retaining the actual +PR589 local word. Its exact mixed moment root lies in + +\[ +[0.002769830250,\;0.002769830251]. +\] + +The target inventory is `3073 H8 + 444 H22 + 222 H50`; the complete paid +inventory is + +\[ +9261H_1+9394H_2+553H_3+308H_4+714H_5+77H_6+1465H_7. +\] + +All 3,739 productive fields have independent arbitrary input arrays. In +particular, all 3,073 helper inputs are arbitrary, and their physical completed +rank-eight transforms are proved before receiving output credit. The 63 owned +COPY episodes remain charged. No owned zero stream receives independent-output +credit. Every paid rank is at most seven; every target rank is at least eight. + +The finite mixed-to-uniform compiler is documented in the sibling +[architecture proof](../architecture/FINITE-PRECOMPILATION.md). The BIT-specific +[delayed-cover bridge](DELAYED-COVER-BRIDGE.md) keeps one common external weight +until that entire finite program has been compiled. Only the final leaves are +lowered into ordinary weighted calls. This resolves the earlier mismatch +between weights inherited from different parent jobs; it is not a claim that +those mismatched weights can be normalized for free at each internal boundary. + +The parameter certificate uses + +\[ +C_0=0.00276,\quad C_1=0.00275,\quad \delta=0.003, +\] + +with an allowed odd prime `q > 25,600,000 N^5` for the fixed compiled dimension +`N`, also excluding the finite list of rational denominators and rank +witnesses. The ordinary BIT saving before relaxation is exactly + +\[ +(1-\delta)C_1+\delta a_{\rm old} +=0.0027418653797. +\] + +The admitted simple BIT saving is `0.0027`. The complete multiplication +parameter selection and its complex supplier belong to the parent package; +this directory does not independently declare a multiplication kappa. The +enormous finite uniform program is specified constructively, not materialized. +The exact finite audits and the all-size proof are separate evidence layers. + +## Evidence map + +| Question | Generator / proof | Receipt | +|---|---|---| +| What survived from the lost deferred mechanism? | `reconstruct_deferred.py` | `DEFERRED-RECONSTRUCTION.json` | +| Does the actual PR589 word replay on arbitrary helper inputs? | `mixed_productive_bit.py`, frozen independent `xcheck_p.py` | `MIXED-PRODUCTIVE-BIT.json`, `sources/pr589-regenerated/XCHECK.json` | +| Is the new p3 controller an exact operator identity? | `mixed_productive_bit.py`, independent `check_mixed_controller.py` | `MIXED-CONTROLLER-INDEPENDENT.json` | +| Are the endpoint ranks native, with actual owners? | `bit_native_binding.py` | `BIT-P3-NATIVE-BINDING.json` | +| Can internal mixed jobs keep a common weight? | `DELAYED-COVER-BRIDGE.md`, `delayed_cover_bridge.py` | `DELAYED-COVER-BRIDGE.json` | +| Are the prime, atom, row-stock and adapter powers funded? | `bit_allsize_parameters.py` | `BIT-ALLSIZE-PARAMETERS.json` | +| Does the controller lift to exact complex Laurent algebra? | `complex_controller_factor.py` | `COMPLEX-CONTROLLER-FACTOR.json`, `COMPLEX-CONTROLLER-FACTOR-P6.json` | +| Do wider controller searches improve the bill? | `single_field_controller_probe.py`, `stage_projector_probe.py` | `SINGLE-FIELD-CONTROLLER-P3.json`, `STAGE-PROJECTOR-P3.json` | +| Can actual PR589 helpers simply share state? | `pr589_state_sharing_probe.py` | `PR589-STATE-SHARING-PROBE.json` | +| Can the nine expensive centers be replaced locally? | `pr589_h_completion.py`, `pr589_minimum_completion.py` | `PR589-H-COMPLETION.json`, `PR589-MINIMUM-COMPLETION.json` | +| Does the cheaper local completion close globally? | `pr589_h_global_probe.py` | `PR589-H-GLOBAL-PROBE.json` | + +## Reconstruction and actual-source provenance + +The lost 15-output and 16-output fixtures were not recoverable byte for byte. +The fresh reconstruction keeps aggregate functions in original registers until +their consumers and pooled helpers have finished. Minimum path covers in a +reverse-inclusion DAG share nested functions on dirty helpers. The originals +are restored last. The fresh 15-output fixture changes 263 helpers / 34,386 ADDs +to 205 / 17,779; the 16-output fixture changes 527 / 39,362 to 265 / 18,903. +All arbitrary inputs, both orientations, intermediate cuts and deliberately +wrong cleanup orders are checked. These finite examples do not supply an +exponent by themselves. + +The actual local word used in the closed supplier is regenerated from the +retained PR589/PR555 sources into `sources/pr589-regenerated/`. Its compressed +SHA256 is +`e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f`; +its canonical decompressed JSON SHA256 is +`5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b`. +It has 111 data ports, 439 helpers and 661 independent persistent inputs. +Both the source-side and copied-center obligations are included in its bill. + +The immutable local `NOTICE.md` credits preparation to **eumemic with +substantial Anthropic Claude assistance**. The tuned source records the PR555 +R439 baseline, PR543 R420 skeleton, DreamingOfClouds's PR545 puncture and PR510 +nine-stage chain, and the cited mechanisms by Dugongue and rohanarun. The tuned +transport pass records Claude Opus 5.5 assistance. The E8 construction and +earlier global controller retain their original repository attribution, +including Jacob Sussman's construction. Current work changes the global +factorization, endpoint accounting, bridge proof and the explicitly named new +probes; it does not claim authorship of those predecessors. + +## What the additional controller rounds established + +The first exact search lets the old global controller omit selected idle +windows and then changes its constant boundary bases. It tests formal +coefficient matrices, not sampled numerical operators. For two through six +pairs, the complete paired-window survivor counts are 3, 8, 20, 48 and 112. +All 192 survivors across one through six pairs lift to exact signed Laurent +identities for the complex construction; no reduction modulo an operator order +is used to justify that lift. The best BIT member is the three-pair case with +keep mask 43. Simply extending the path to more pairs does not improve it. + +A broader three-pair probe separates the two fields of each idle window: +4,096 choices give 64 exact factorizations, including 56 asymmetric ones. A +second probe allows independent stage projectors and tests all 123,904 complete +compatible choices in its stated family. It recovers the old best all-fresh +three-pair bill, rather than a new saving. These are scoped exhaustive results; +they are not lower bounds for all possible global controllers. + +The actual PR589 state-sharing probe examines all 96,141 helper pairs. Only +five pairs have compatible full monotone frame timelines. Two cause a +noninvertible self-gate; the other three change the actual source response. +No pair has disjoint forward lifetimes. Separately, both the nine center +producer signals and their consumer matrix have rank nine. A same-cut +eight-signal compression therefore fails before transport and cleanup costs +are considered. + +## A real cheaper local primitive, and its remaining global debt + +Removing the nine old copied centers and rebuilding dirty compensation gives +an actual clean response + +\[ +J=I+UV +\] + +over `F2`, with `V` the nine center producer rows and `U` their actual scatter +map. Select nine source coordinates `A` for which `V_A` is invertible and set + +\[ +M=V_A^{-1}VU,\qquad U'=U+E_A M,\qquad H=I+U'V. +\] + +Then `VU'=0`, hence `H^2=I`. Only nine output rows are corrected. Each query is +gathered from unchanged completed source arrays, moved through its selected +rank-one target hole, read, and erased under the inherited owned-stream +contract. The complete local bill becomes + +\[ +1332H_1+1342H_2+79H_3+44H_4+102H_5+11H_6+10H_7, +\] + +with rank mass 5,075, saving 54 from the old 5,129. The native word has 11,955 +ADDs and one owned stream reused nine times. Both orientations restore all +439 arbitrary helpers. Its common-frame involution has a three-shear +implementation of only `474 + 504 + 474 = 1452` ADDs. A separate independent +audit in `../transfer-audit/h-branch/` reconstructs the matrices, every frame +rank, all 661 input columns and the owned-erasure model. + +The further minimum-completion probe observes that `rank(J)=108`. It changes +only rows `106,109,110`, replacing them by source coordinates `0,1,2`, and +obtains an invertible `G`. Three changed rows are optimal by the elementary +rank bound. This costs only three new rank-one gathers, giving rank mass +5,069 and 11,625 local ADDs. Forward common-frame wrappers use `G`; reverse +wrappers use `G^-1`. Four literal native replays check the plain and wrapped +orientations on all 661 inputs. Removing any one correction leaves rank 110. +The separate `BIT-MINIMUM-COMPLETION-AUDIT.json` in +`../transfer-audit/h-branch/` independently reconstructs the completion, +both scalar inverse programs, all native transitions and the three actual +line-to-hyperplane contacts; all four native variants pass. + +Neither local improvement is advertised as a new global supplier. The exact +noncommuting three-pair controller produces three interval products, for +stages `(0,1,2)`, `(4,5,6)` and `(0,...,6)`. One bank of each pair can erase its +remaining product by costed scalar/cover operations. The other has a +`Q K Q` endpoint. Direct normalization costs **666 rank-one calls**, not 222. +With the involution the raw rank surplus is 603 and the completed direct bill +has surplus -63. With the minimum invertible completion these are 645 and -21. +Both direct constructions fail even at exponent zero. These receipts prevent +a cheap local repair from being mistaken for a funded recursive improvement. + +The next useful larger-gain question is therefore whether the three actual +head-conjugated interval products can share a correction, or whether a changed +local family makes those products cancel at their physical owner charts. +Unconstrained scalar cancellation or repeated same-chart kernels is not +sufficient. The fixed data geometry, independent inputs, borrowed state and +all endpoint repairs must be included in that next probe's objective. + +## Reproduction + +All new generators use Python's standard library and retained repository code. +From the repository root: + +```sh +python3 -B research/kappa-rebuild/bit/reconstruct_deferred.py +python3 -B research/kappa-rebuild/bit/mixed_productive_bit.py --max-p 5 +python3 -B research/kappa-rebuild/bit/check_mixed_controller.py +python3 -B research/kappa-rebuild/bit/bit_native_binding.py +python3 -B research/kappa-rebuild/bit/delayed_cover_bridge.py +python3 -B research/kappa-rebuild/bit/bit_allsize_parameters.py +python3 -B research/kappa-rebuild/bit/pr589_state_sharing_probe.py +python3 -B research/kappa-rebuild/bit/pr589_h_completion.py +python3 -B research/kappa-rebuild/bit/pr589_h_global_probe.py +python3 -B research/kappa-rebuild/bit/pr589_minimum_completion.py +``` + +The literature-to-probe map is in [LITERATURE.md](LITERATURE.md). The source and +artifact closure is recorded in `SOURCE-MANIFEST.json`. 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"diagonal_spectral_blocks": [ + { + "multiplicity": 3, + "q": -1, + "stages": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6 + ] + }, + { + "multiplicity": 3, + "q": 1, + "stages": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6 + ] + } + ], + "extra_rank8_fields": 28, + "field_masks": [ + 63, + 63, + 63, + 63, + 63, + 63, + 63 + ], + "margin_at_target": -190.29362498018236, + "targets": { + "57": 672 + } + } + ], + "enumeration_complete": true, + "limit": 200000, + "new_kappa": false, + "pairs": 3, + "runtime_seconds": 7.834249624000222, + "schema": "independent-stage-field-projector-probe/1", + "scope": "Exact Q-projector compatibility; floating complete all-fresh helper screen only. Native idle placement, explicit L/R, hosting and transfer remain open.", + "source_sha256": { + "stage_projector_probe.py": "64e45c5c3a3ddde3a5d590706dff56c544f039357ca544d95a566bdaae96e491" + }, + "target_C": "0.003" +} diff --git a/research/kappa-rebuild/bit/bit_allsize_parameters.py b/research/kappa-rebuild/bit/bit_allsize_parameters.py new file mode 100644 index 0000000000..aee5c2b85a --- /dev/null +++ b/research/kappa-rebuild/bit/bit_allsize_parameters.py @@ -0,0 +1,80 @@ +#!/usr/bin/env python3 +"""Exact parameters for the delayed-cover BIT ordinary interface. + +This records conditional all-size implications, not a materialized winning +uniform mixed-flow template. Apache-2.0; substantial OpenAI assistance. +""" +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import json +import mixed_productive_bit as B + +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[2] + +def run(): + profile_path=HERE/'MIXED-PRODUCTIVE-BIT.json' + profile=json.loads(profile_path.read_text()) + case=next(c for c in profile['candidates']if c['pairs']==3) + paid={int(k):v for k,v in case['paid'].items()} + targets={int(k):v for k,v in case['targets'].items()} + before,after=Q(69,25000),Q(11,4000) + gap=before-after + delta,old=Q(3,1000),Q(384599,10**10) + raw=(1-delta)*after+delta*old + admitted=Q(27,10000) + margin=B.interval_row(B.margin_interval(paid,targets,before)) + B.need(margin['strict_positive'],'The exact mixed rule supports C0=.00276') + B.need(gap==Q(1,100000) and before>after>0,'Strict precompile-to-cover gap') + B.need(delta>raw>admitted>0 and delta<1-raw,'Adapter and borrowing powers fit') + prime_coefficient=256/gap + fallback_bound,power_lower=1+gap/4,1+gap/2 + B.need(fallback_bound=2,B>=1,P>=1 and a whole-cover native schematic uniform word, with P/B < N^(1-C0), final paid leaves of rank1, arbitrary productive inputs and separately owned storage.', + 'prime_rule':{'bound':'q > '+str(prime_coefficient)+' * N^5', + 'also_required':'Odd prime avoiding every fixed rational denominator, matcher determinant and chosen rank witness of the entire compiled word.', + 'good_moment_upper':'N^(-d), d=C0-C1', + 'fallback_multiplier_upper':str(fallback_bound), + 'N_to_d_lower':str(power_lower), + 'complete_moment_upper':str(fallback_bound/power_lower), + 'strict_moment_slack_lower':str(1-fallback_bound/power_lower)}, + 'atom_power':str(delta),'old_ordinary_saving':str(old), + 'ordinary_saving_before_final_relaxation':str(raw), + 'ordinary_saving_display':float(raw),'ordinary_saving_admitted':str(admitted), + 'strict_power_gaps':{'linear_adapter':str(delta-raw), + 'borrowed_stock_log_absorption':str(1-raw-delta), + 'final_relaxation':str(raw-admitted)}, + 'root_cost_terms':['e^(1-delta)', + 'e^((1-delta)*(1-C1)+delta*(1-a_old))', + 'e^delta * log(e)'], + 'stock':{'role_divisor':'W(w)=B*|GL_N(F_q)|*q^(N^2*(w-1))', + 'depth_bound':'D_N(n)=floor(log_N n) for rank<=floor(n/N) children; any larger explicit integer bound is safe.', + 'reserve':'S(e,w)=W(w)^D_N(ceil(e/w))*S_old(e,w)', + 'borrowed_digits':'k <= D_N(ceil(e/w))*(N^2*(w-1)+ceil(log_q(B*|GL_N(F_q)|)))+ceil(log_q S_old)+1', + 'asymptotic':'Fixed compiled N,B,P: k=O_(N,B,q)(w*log e)=o(e), w=ceil(e^(3/1000)); pad high rows once by factor<2.', + 'descendant_weights':'Only final leaves are lowered. This outer cover row and role type become ancestor data; every next ordinary invocation has fresh disjoint role stock.'}, + 'finite_constants_to_replace':{'productive_roles':'B, not3739 or661', + 'ambient_dimension':'N, not50 or64', + 'paid_leaf_count':'P, including copied-center descendants', + 'owned_storage':'C, bounded by one owned bank per allocated finite unit instance', + 'pointwise_and_copy_count':'L, literal compiled scalar/copy/erase records', + 'fixed_row_routes':'R, original port charts and every internal matcher/inverse', + 'generic_affine_work':'At most4*N^2-2*N basic shears per rank1 normalizer together with its inverse', + 'bad_fallback_children':'At most32*N^2 per bad final leaf', + 'coefficient_data':'Every fixed rational entry, inverse, determinant, prefix descriptor and height'}, + 'new_kappa':False, + 'source_sha256':{str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest()for p in sources}} + +if __name__=='__main__': + result=run() + (HERE/'BIT-ALLSIZE-PARAMETERS.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({k:result[k]for k in ('status','C0_before_cover','C1_after_cover','ordinary_saving_before_final_relaxation','ordinary_saving_admitted')})) diff --git a/research/kappa-rebuild/bit/bit_native_binding.py b/research/kappa-rebuild/bit/bit_native_binding.py new file mode 100644 index 0000000000..f4c4082d89 --- /dev/null +++ b/research/kappa-rebuild/bit/bit_native_binding.py @@ -0,0 +1,156 @@ +#!/usr/bin/env python3 +"""Bind the p3 BIT physical controller to exact local rational port charts. + +The cover is a formal GL_N row until a complete finite uniform program is +compiled. This verifier checks every fixed port map and actual local data +endpoint, plus the native helper ownership and complete recursive invoice. +Apache-2.0. Substantial OpenAI assistance; retained source attribution is +in the immutable PR589/PR555 notices and the companion README. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import gzip +import json +import struct +import time + +import mixed_productive_bit as B + +HERE=Path(__file__).resolve().parent +ROOT=HERE.parents[2] + + +def ident(n): + return [[Q(i==j) for j in range(n)] for i in range(n)] + + +def multiply(a,b): + out=[[Q(0)]*len(b[0]) for _ in a] + for i,row in enumerate(a): + for k,value in enumerate(row): + if value: + for j,second in enumerate(b[k]): + if second: + out[i][j]+=value*second + return out + + +def transpose(a): + return list(map(list,zip(*a))) + + +def subtract(a,b): + return [[x-y for x,y in zip(row,second)] for row,second in zip(a,b)] + + +def serialize(a): + return [[str(x) for x in row] for row in a] + + +def partial_swap(p): + n=len(p) + return [[(int(i==j)-p[i][j] if i=8) for j in range(15)]for i in range(15)] + B.need(multiply(multiply(rho,local8),rho)==active,'Local helper rank8 map under the port chart') + B.need(projectors[word['start'][port]]==p and projectors[word['start'][111+port]]==zero, + 'Actual local data entrance frames') + B.need(projectors[word['final'][port]]==eye and projectors[word['final'][111+port]]==k, + 'Actual local data exit frames') + B.need(multiply(full_swap,partial_swap(p))==partial_swap(k),'Exact physical local factorD_I D_P=D_K') + ports.append(dict(port=port,root=[str(x) for x in q], + central_projector=serialize(p),complement_columns=serialize(a), + chart_15=serialize(rho), + lifted_routes=[dict(stage=s,axes=list(range(8))+list(range(8+7*(s-1),8+7*s))) + for s in range(1,7)])) + copy_checks=0 + for op,x,y,c,f,z in records: + if op==2: + a,b=projectors[c],projectors[f] + low,high=(a,b) if sum(a[i][i]for i in range(8))>(s+1)&1] + rank=(term&1)+7*len(stages) + targets[rank]+=111 + endpoint.append(dict(data_role=role,central_q=bool(term&1),stages=stages,rank=rank, + projector='P_q + selected pairwise perpendicular K_q/B_s blocks')) + B.need(dict(sorted(paid.items()))=={int(r):n for r,n in case['paid'].items()},'Literal complete paid ledger') + B.need(dict(sorted(targets.items()))=={int(r):n for r,n in case['targets'].items()},'Actual endpoint rank credit') + B.need(max(paid) 0, "Boundary adapter omission rejected") + controls[f"omit_input_adapter_{index}"] = dict(wrong_coefficients=wrong[0], wrong_rows=wrong[1]) + for stage in range(case["stages"]): + wrong = execute(reverse_local=stage) + require(wrong[0] > 0, "Local orientation corruption rejected") + controls[f"reverse_local_orientation_{stage}"] = dict(wrong_coefficients=wrong[0], wrong_rows=wrong[1]) + for index, event in enumerate(events): + if event[0] == "IDLE": + wrong = execute(omitted_idle=index) + require(wrong[0] > 0, "Remaining idle field must be paid") + controls[f"omit_remaining_idle_{index}"] = dict(wrong_coefficients=wrong[0], wrong_rows=wrong[1]) + local = {1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19} + paid = Counter({r:case["stages"]*count for r, count in local.items()}) + targets = Counter({8:439*case["stages"]}) + for event in events: + if event[0] == "IDLE": + paid[7] += 2*111*(event[2] >> 1).bit_count() + for term in endpoint: + targets[(term & 1)+7*(term >> 1).bit_count()] += 111 + require(dict(sorted(paid.items())) == {int(k):v for k, v in case["paid"].items()}, "Literal whole paid profile") + require(dict(sorted(targets.items())) == {int(k):v for k, v in case["targets"].items()}, "Actual endpoint job credit") + require(max(paid) < min(targets), "Strict contraction") + require(sum(r*n for r, n in targets.items())-sum(r*n for r, n in paid.items()) == 225, + "Independent complete rank balance") + return dict(schema="independent-bit-regular-representation/1", status="PASS", + input_receipt_sha256=sha256(receipt.read_bytes()).hexdigest(), + checker_sha256=sha256(Path(__file__).read_bytes()).hexdigest(), + physical_data_fields=n, regular_group_order=group_size, + arbitrary_function_input_columns=n*group_size, + scalar_matrix_coefficients_compared=(n*group_size)**2, + nonzero_wrong_coefficients=0, paid=dict(sorted(paid.items())), + targets=dict(sorted(targets.items())), controls=controls, + interpretation="Faithful regular representation validates the exact formal operator, not a sample of characters.", + not_checked=["Full GL cover routing", "Weighted output normalizers", "All-size recursion", "Outer multiplication"], + new_kappa=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--input", type=Path, default=HERE/"MIXED-PRODUCTIVE-BIT.json") + parser.add_argument("--output", type=Path, default=HERE/"MIXED-CONTROLLER-INDEPENDENT.json") + args = parser.parse_args() + start = time.monotonic() + result = run(args.input) + result["seconds"] = time.monotonic()-start + args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+"\n") + print(json.dumps({k:result[k] for k in ("status", "arbitrary_function_input_columns", "scalar_matrix_coefficients_compared", "seconds")})) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-rebuild/bit/complex_controller_factor.py b/research/kappa-rebuild/bit/complex_controller_factor.py new file mode 100644 index 0000000000..cb0cff7c8b --- /dev/null +++ b/research/kappa-rebuild/bit/complex_controller_factor.py @@ -0,0 +1,270 @@ +#!/usr/bin/env python3 +"""Exact complex controller factorizations over an integral Laurent ring. + +The native local operator is an EXPLICIT INPUT CONTRACT supplied by the +complex reconstruction: F_s=D_s S_y(Q^-1), B_s=D_s S_x(-Q). +No order-two relation, numerical Fourier characters, phase erasure, or +F2-to-complex lift is used in the acceptance check. All Laurent +coefficients are reconstructed and multiplied over Q. Binary candidates +are only a finite discovery filter, not an exhaustive rational census. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import time + +import mixed_productive_bit as B + +HERE = Path(__file__).resolve().parent + + +def plus(a, b, coefficient=1): + result = dict(a) + for term, value in b.items(): + result[term] = result.get(term, 0)+coefficient*value + if not result[term]: + del result[term] + return result + + +def monomial_times(polynomial, exponents, coefficient=1): + return {tuple(a+b for a, b in zip(term, exponents)):coefficient*value + for term, value in polynomial.items()} + + +def controller(p, keep, wrong_reverse_sign=False): + n, stages = 2*p, 2*p+1 + zero = (0,)*(stages+1) + matrix = [[{zero:1} if i == j else {} for j in range(n)] for i in range(n)] + events, idle = [], [] + def term(q, selected): + return (q,)+tuple(int(s in selected) for s in range(stages)) + def idle_call(pair, selected, kind): + index = len(idle) + retained = bool(keep >> index & 1) + idle.append(dict(index=index, kind=kind, pair=pair, stages=list(selected), + rank=8*len(selected), retained=retained)) + if retained: + for role in (2*pair, 2*pair+1): + matrix[role] = [monomial_times(x, term(0, selected)) for x in matrix[role]] + events.append(["IDLE", pair, list(selected)]) + def local(stage, pair, reverse): + x, y = 2*pair, 2*pair+1 + if reverse: + x, y = y, x + sign = -1 if reverse and not wrong_reverse_sign else 1 + exponent = 1 if reverse else -1 + matrix[y] = [plus(a, monomial_times(b, term(exponent, []), sign)) + for a, b in zip(matrix[y], matrix[x])] + for role in (x, y): + matrix[role] = [monomial_times(x, term(0, [stage])) for x in matrix[role]] + events.append(["LOCAL", stage, pair, reverse]) + if stage % 2 and stage <= 2*p-3: + idle_call((stage+1)//2, list(range(stage+1)), "initial") + elif stage % 2 == 0 and 2 <= stage <= 2*p-2: + idle_call(stage//2, [stage], "middle") + elif stage == 2*p: + for j in range(p-1): + idle_call(j, list(range(2*j+3, 2*p+1)), "terminal") + def bridge(dst, src, coefficient): + for d, s, c in ((2*dst, 2*src, coefficient), (2*src+1, 2*dst+1, -coefficient)): + matrix[d] = [plus(a, b, c) for a, b in zip(matrix[d], matrix[s])] + events.append(["ADD", d, s, c]) + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1, 1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1, -1) + for j in reversed(range(p-1)): + bridge(j+1, j, 1) + for j in range(p): + matrix[2*j], matrix[2*j+1] = matrix[2*j+1], [monomial_times(x, zero, -1) for x in matrix[2*j]] + events.append(["SIGNED_SWAP", 2*j, 2*j+1]) + coefficients = {} + for i, row in enumerate(matrix): + for j, polynomial in enumerate(row): + for key, value in polynomial.items(): + coefficients.setdefault(key, [[Q(0)]*n for _ in range(n)])[i][j] += value + coefficients = {key:matrix for key, matrix in coefficients.items() if any(any(row) for row in matrix)} + return coefficients, events, idle + + +def matrix_rank(rows): + basis = {} + for row in rows: + row = list(map(Q, row)) + for j in sorted(basis): + if row[j]: + scale = row[j] + row = [a-scale*b for a, b in zip(row, basis[j])] + pivot = next((j for j, x in enumerate(row) if x), None) + if pivot is not None: + scale = row[pivot] + basis[pivot] = [x/scale for x in row] + return len(basis) + + +def scalar_program(matrix): + n = len(matrix) + work = [row[:] for row in matrix] + elimination = [] + for j in range(n): + pivot = next(i for i in range(j, n) if work[i][j]) + if pivot != j: + work[j], work[pivot] = work[pivot], work[j] + elimination.append(["SWAP", j, pivot]) + if work[j][j] != 1: + scale = 1/work[j][j] + work[j] = [scale*x for x in work[j]] + elimination.append(["SCALE", j, str(scale)]) + for i in range(n): + if i != j and work[i][j]: + scale = -work[i][j] + work[i] = [a+scale*b for a, b in zip(work[i], work[j])] + elimination.append(["ADD", i, j, str(scale)]) + program = [] + for op in reversed(elimination): + if op[0] == "SWAP": + program.append(op) + elif op[0] == "SCALE": + program.append(["SCALE", op[1], str(1/Q(op[2]))]) + else: + program.append(["ADD", op[1], op[2], str(-Q(op[3]))]) + replay = [[Q(i == j) for j in range(n)] for i in range(n)] + for op in program: + if op[0] == "SWAP": + replay[op[1]], replay[op[2]] = replay[op[2]], replay[op[1]] + elif op[0] == "SCALE": + replay[op[1]] = [Q(op[2])*x for x in replay[op[1]]] + else: + replay[op[1]] = [a+Q(op[3])*b for a, b in zip(replay[op[1]], replay[op[2]])] + B.need(replay == matrix, "Exact signed scalar boundary program") + coefficients = [Q(x) for row in matrix for x in row] + B.need(all(q.denominator & (q.denominator-1) == 0 for q in coefficients), "Dyadic boundary coefficients") + return program + + +def factor(coefficients): + n = len(next(iter(coefficients.values()))) + total = [[sum(matrix[i][j] for matrix in coefficients.values()) for j in range(n)] for i in range(n)] + inverse = B.rational_inverse(total) + projectors = {key:B.matrix_multiply(inverse, matrix) for key, matrix in coefficients.items()} + for key, matrix in projectors.items(): + if B.matrix_multiply(matrix, matrix) != matrix: + return None + for other, second in projectors.items(): + if key != other and any(any(row) for row in B.matrix_multiply(matrix, second)): + return None + columns, diagonal = [], [] + for key, matrix in sorted(projectors.items()): + for column in zip(*matrix): + if matrix_rank(columns+[column]) > len(columns): + columns.append(list(column)) + diagonal.append(key) + B.need(len(columns) == n, "All rational projector images") + right = [list(row) for row in zip(*columns)] + left = B.matrix_multiply(B.rational_inverse(right), inverse) + transformed = {key:B.matrix_multiply(B.matrix_multiply(left, matrix), right) + for key, matrix in coefficients.items()} + expected = {key:[[Q(i == j and diagonal[i] == key) for j in range(n)] for i in range(n)] + for key in set(diagonal)} + B.need(transformed == expected, "Every Laurent coefficient equals the advertised diagonal") + return dict(left=left, right=right, diagonal=diagonal, + left_program=scalar_program(left), right_program=scalar_program(right)) + + +def serialize_matrix(matrix): + return [[str(x) for x in row] for row in matrix] + + +def serialize_case(p, keep, coefficients, events, idle, result): + # Rank normalization is an input contract of the complex native source: + # D0=F9 Q^-1; Ds for s>0 transform fresh eight-axis blocks; Q^2 and + # Ds^2 are explicitly free Pauli/translation operators. The EXACT + # Laurent monomials remain in this receipt for a separate phase audit. + jobs = Counter() + for term in result["diagonal"]: + B.need(term[0] in (-1, 1), "Only stated central-Q endpoint normalization") + B.need(all(x in (0, 1) for x in term[1:]), "No hidden squared stage transforms") + jobs[1+8*sum(term[1:])] += 112 + scalar_coefficients = [x for matrix in (result["left"], result["right"]) for row in matrix for x in row] + return dict(pairs=p, stages=2*p+1, keep_idle_mask=keep, + coefficient_variables=["Q"]+[f"D{s}" for s in range(2*p+1)], + coefficient_matrices=[dict(exponents=list(key), matrix=serialize_matrix(matrix)) + for key, matrix in sorted(coefficients.items())], + left=serialize_matrix(result["left"]), right=serialize_matrix(result["right"]), + left_program=result["left_program"], right_program=result["right_program"], + exact_diagonal_monomials=[list(term) for term in result["diagonal"]], + all_boundary_entries_integral=all(x.denominator == 1 for x in scalar_coefficients), + maximum_boundary_absolute_coefficient=str(max(map(abs, scalar_coefficients))), + endpoint_after_stated_free_Pauli_normalization=dict(sorted(jobs.items())), + retained_idle_windows=[row for row in idle if row["retained"]], + omitted_idle_windows=[row for row in idle if not row["retained"]], + physical_events=events, + full_Laurent_identity=True, native_operator_contract_independently_replayed=False) + + +def run(max_p): + start = time.monotonic() + cases, census = [], {} + for p in range(1, max_p+1): + binary_candidates = [] + for keep in range(1 << (3*(p-1))): + binary_coefficients, _, _ = B.physical_controller(p, keep) + if B.simultaneous_monomial_factor(binary_coefficients) is not None: + binary_candidates.append(keep) + passed = 0 + for keep in binary_candidates: + coefficients, events, idle = controller(p, keep) + result = factor(coefficients) + if result is not None: + cases.append(serialize_case(p, keep, coefficients, events, idle, result)) + passed += 1 + census[p] = dict(binary_discovery_trials=1 << (3*(p-1)), + binary_candidates=len(binary_candidates), exact_Laurent_factorizations=passed) + print(json.dumps(dict(pairs=p, **census[p])), flush=True) + chosen = next(row for row in cases if row["pairs"] == 3 and row["keep_idle_mask"] == 43) + left = [[Q(x) for x in row] for row in chosen["left"]] + right = [[Q(x) for x in row] for row in chosen["right"]] + wrong, _, _ = controller(3, 43, True) + transformed = {key:B.matrix_multiply(B.matrix_multiply(left, matrix), right) + for key, matrix in wrong.items()} + expected = {tuple(key):[[Q(i == j and tuple(chosen["exact_diagonal_monomials"][i]) == tuple(key)) + for j in range(6)] for i in range(6)] + for key in chosen["exact_diagonal_monomials"]} + B.need(transformed != expected, "Incorrect backward sign rejected over Q") + return dict(schema="complex-controller-integral-Laurent-factorization/1", + status="PASS_EXACT_LAURENT_IDENTITIES_CONDITIONAL_ON_NATIVE_LOCAL_CONTRACT", + local_contract=dict(forward="D_s * [[1,0],[Q^-1,1]]", + backward="D_s * [[1,-Q],[0,1]]", + commutation="Q,D0,...,D(2p) commute; no finite-order relation used"), + census=census, cases=cases, + selected_p3_keep43=chosen, + controls=dict(wrong_backward_sign_rejected=True), + source_sha256={Path(__file__).name:sha256(Path(__file__).read_bytes()).hexdigest(), + "mixed_productive_bit.py":sha256((HERE/"mixed_productive_bit.py").read_bytes()).hexdigest()}, + runtime_seconds=time.monotonic()-start, + scope="Exact formal operator identity and emitted constant dyadic adapters. Native source binding, free-phase normalization, helper ownership, complete tariffs and all-size transfer belong to separate certificates.", + new_kappa=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--max-p", type=int, default=5) + parser.add_argument("--output", type=Path, default=HERE/"COMPLEX-CONTROLLER-FACTOR.json") + args = parser.parse_args() + B.need(3 <= args.max_p <= 6, "Bounded candidate menu") + data = run(args.max_p) + args.output.write_text(json.dumps(data, indent=2, sort_keys=True)+"\n") + print(json.dumps(dict(status=data["status"], seconds=data["runtime_seconds"], cases=len(data["cases"])))) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-rebuild/bit/delayed_cover_bridge.py b/research/kappa-rebuild/bit/delayed_cover_bridge.py new file mode 100644 index 0000000000..c541a7ea45 --- /dev/null +++ b/research/kappa-rebuild/bit/delayed_cover_bridge.py @@ -0,0 +1,154 @@ +#!/usr/bin/env python3 +"""Exact local-ring audit of delayed BIT cover matching and weight propagation. + +This checks the routing lemma used by DELAYED-COVER-BRIDGE.md. It enumerates +every GL_2(Z/9Z) row, two different role matchers, a common nonconstant-by-axis +weight, and every address for the scalar-gauge commutation check. It is a +finite audit of the bridge, not a full mixed-flow or multiplication theorem. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from hashlib import sha256 +from itertools import product +from pathlib import Path +import json +import time + +HERE=Path(__file__).resolve().parent +MOD=9 +PRIME=3 + + +def need(condition,message): + if not condition: + raise AssertionError(message) + + +def mul(a,b,mod=MOD): + return [[sum(x*y for x,y in zip(row,column))%mod for column in zip(*b)] for row in a] + + +def inv2(a): + determinant=(a[0][0]*a[1][1]-a[0][1]*a[1][0])%MOD + inverse=pow(determinant,-1,MOD) + return [[inverse*a[1][1]%MOD,-inverse*a[0][1]%MOD],[-inverse*a[1][0]%MOD,inverse*a[0][0]%MOD]] + + +def key(a): + return tuple(x for row in a for x in row) + + +def conjugate(a,b): + return mul(mul(a,b),inv2(a)) + + +def d_projector(p): + c=[[0,1],[1,0]] + pc,cp=mul(p,c),mul(c,p) + cpc=mul(cp,c) + return [[((int(i==j)-p[i][j])%MOD if i<2 and j<2 else + pc[i][j-2] if i<2 else + cp[i-2][j] if j<2 else + (int(i==j)-cpc[i-2][j-2])%MOD) + for j in range(4)] for i in range(4)] + + +def weight(p,a): + gauge=[[a[i] if i==j and i<2 else int(i==j) for j in range(4)] for i in range(4)] + inverse=[[pow(a[i],-1,MOD) if i==j and i<2 else int(i==j) for j in range(4)] for i in range(4)] + return mul(mul(gauge,d_projector(p)),inverse) + + +def act(a,x): + return tuple(sum(v*w for v,w in zip(row,x))%MOD for row in a) + + +def run(): + start=time.monotonic() + cover=[] + for entries in product(range(MOD),repeat=4): + a=[list(entries[:2]),list(entries[2:])] + if (a[0][0]*a[1][1]-a[0][1]*a[1][0])%PRIME: + cover.append(a) + need(len(cover)==3888,'All GL_2(Z/9Z) elements') + child=[[[1,0],[0,0]],[[1,0],[1,0]]] + matchers=[[[1,1],[1,0]],[[0,1],[1,-1]]] + parent=[conjugate(s,q) for s,q in zip(matchers,child)] + common=(2,4) + matches,affine_checks=0,0 + role_fibers=[] + wrong_matcher=None + changing_weight=None + leaf_weights=Counter() + good=bad=0 + for role,(q,p,s) in enumerate(zip(child,parent,matchers)): + need(mul(q,q)==q and mul(p,p)==p,'Both endpoints are split projectors') + routed=set() + for g in cover: + h=mul(g,s) + routed.add(key(h)) + incoming=conjugate(g,p) + outgoing=conjugate(h,q) + need(incoming==outgoing,'Fixed right routing matches the full physical projector') + need(weight(incoming,common)==weight(outgoing,common),'Common external gauge survives matching') + matches+=1 + if wrong_matcher is None and conjugate(g,q)!=incoming: + wrong_matcher={'role':role,'g':g,'expected':incoming,'unrouted':conjugate(g,q)} + g0=[[x%PRIME for x in row] for row in g] + high=[[(x-y)//PRIME for x,y in zip(row,low)] for row,low in zip(g,g0)] + raw=[[sum(x*y for x,y in zip(row,column)) for column in zip(*s)] for row in g0] + h0=[[x%PRIME for x in row] for row in raw] + carry=[[(x-y)//PRIME for x,y in zip(row,low)] for row,low in zip(raw,h0)] + linear=mul(high,s,PRIME) + actual_high=[[(x-y)//PRIME for x,y in zip(row,low)] for row,low in zip(h,h0)] + predicted=[[(x+y)%PRIME for x,y in zip(row,carry_row)] for row,carry_row in zip(linear,carry)] + need(actual_high==predicted,'Low-type carry plus fixed affine high-row route') + affine_checks+=1 + # Deliberately illegal: A depends on the current fresh GL row. + a_g=(2 if g[0][0]%PRIME==1 else 1,4) + a_h=(2 if h[0][0]%PRIME==1 else 1,4) + if changing_weight is None and weight(incoming,a_g)!=weight(outgoing,a_h): + changing_weight={'role':role,'g':g,'h':h,'A_g':a_g,'A_h':a_h} + # For a rank-one residual the generic northeast pivot is A[0]*P[0,0]. + if incoming[0][0]%PRIME: + good+=1 + leaf_weights[common[0]*incoming[0][0]%MOD]+=1 + else: + bad+=1 + need(len(routed)==len(cover),'Every target role routes a whole cover fiber bijectively') + role_fibers.append(len(routed)) + need(wrong_matcher is not None,'Omitting the chart route fails') + need(changing_weight is not None,'Fresh-cover-dependent weights are rejected') + gauge=[[common[i] if i==j and i<2 else int(i==j) for j in range(4)] for i in range(4)] + address_images=set() + for address in product(range(MOD),repeat=4): + mapped=act(gauge,address) + address_images.add(mapped) + # Two orders of pointwise XOR versus one common address permutation. + before={(0,mapped),(1,mapped)} + after={(role,act(gauge,address)) for role in (0,1)} + need(before==after,'Common gauge commutes with pointwise scalar ADD on all input columns') + need(len(address_images)==MOD**4,'The common gauge is a complete address permutation') + return {'schema':'delayed-bit-cover-weight-bridge/1','status':'PASS_EXACT_LOCAL_RING_AUDIT', + 'ring':{'prime':PRIME,'atom_digits':2,'modulus':MOD}, + 'ambient_rank':2,'cover_rows':len(cover),'independent_target_roles':2, + 'child_projectors':child,'parent_projectors':parent,'fixed_matchers':matchers, + 'common_external_weight':common,'exact_matches':matches, + 'row_routing_bijections':role_fibers,'affine_high_row_checks':affine_checks, + 'scalar_gauge_address_checks':MOD**4, + 'last_step_leaf_normalization':{'good_rows':good,'fallback_rows':bad, + 'actual_weight_histogram':dict(sorted(leaf_weights.items())), + 'dependency':'A_child is evaluated from the outer ancestor A and this outer cover row g; neither is a fresh field of the leaf invocation.'}, + 'controls':{'omitted_matcher_counterexample':wrong_matcher, + 'illegal_fresh_row_weight_counterexample':changing_weight}, + 'source_sha256':sha256(Path(__file__).read_bytes()).hexdigest(), + 'runtime_seconds':time.monotonic()-start, + 'proved_in_companion_note':'General rational projector matching, one global cover, common-gauge commutation, delayed weighted lowering and fixed routing costs.', + 'not_checked':['The complete mixed-flow precompiler','Materialization of a winning enormous uniform template','Final multiplication transfer'], + 'new_kappa':False} + + +if __name__=='__main__': + result=run() + (HERE/'DELAYED-COVER-BRIDGE.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({k:result[k] for k in ('status','cover_rows','exact_matches','affine_high_row_checks','runtime_seconds')})) diff --git a/research/kappa-rebuild/bit/mixed_productive_bit.py b/research/kappa-rebuild/bit/mixed_productive_bit.py new file mode 100644 index 0000000000..9c7a8bd5d3 --- /dev/null +++ b/research/kappa-rebuild/bit/mixed_productive_bit.py @@ -0,0 +1,617 @@ +#!/usr/bin/env python3 +"""Exact finite BIT operator factorization and heterogeneous recursive invoice. + +The PR589 word is regenerated from its frozen source, byte-for-byte. This +new file changes its global controller and its declared physical endpoints. +It does not admit a mixed-target all-size BIT compiler or a new kappa. + +Apache-2.0. Substantial OpenAI assistance. Retained local-word attribution: +DreamingOfClouds / PR589, eumemic / PR555 and PR543, and their cited lineage. +The compressed controller is from PR386/436/443/463/510 and the general +global_k.py implementation. Mixed target accounting is motivated by the +PR4268 proposal, whose transfer hypotheses are not asserted here. +""" +from collections import Counter +from fractions import Fraction as Q +from functools import lru_cache, reduce +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json +import math +import platform +import struct +import sys +import time + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +REFERENCE = ROOT / "research/kappa-next-regime/bit-compression/carrier-reuse/reference" +sys.path.insert(0, str(REFERENCE)) +from joint_pair_dag import log_interval, exp_interval, quotient + +LOCAL = Counter({1:1323, 2:1342, 3:79, 4:44, 5:102, 6:11, 7:19}) +V, R, H = 111, 439, 8 +WORD_HASH = "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f" +WORD_CONTENT_HASH = "5642fd0e35a1acc11acff8d0c08350c1617e95f310d4d35ad63a651f9eeeba0b" + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def canonical(data): + return json.dumps(data, sort_keys=True, separators=(",", ":")).encode() + + +def digest(data): + return sha256(canonical(data)).hexdigest() + + +def bit_rank(rows, n): + basis = {} + for row in rows: + for pivot in sorted(basis, reverse=True): + if row >> pivot & 1: + row ^= basis[pivot] + if row: + need(row.bit_length() <= n, "Binary matrix width") + basis[row.bit_length()-1] = row + return len(basis) + + +def bit_mul(a, b): + return [reduce(int.__xor__, (b[j] for j in range(len(b)) if row >> j & 1), 0) + for row in a] + + +def bit_inverse(a): + n = len(a) + aug = [row | (1 << (n+i)) for i, row in enumerate(a)] + for j in range(n): + k = next((i for i in range(j, n) if aug[i] >> j & 1), None) + need(k is not None, "Invertible scalar boundary") + aug[j], aug[k] = aug[k], aug[j] + for i in range(n): + if i != j and aug[i] >> j & 1: + aug[i] ^= aug[j] + result = [row >> n for row in aug] + need(bit_mul(a, result) == [1 << i for i in range(n)], "Inverse replay") + return result + + +def bit_factor(a): + """Literal ADD/SWAP program for a constant F2 matrix, checked exactly.""" + n = len(a) + cur = list(a) + elimination = [] + for j in range(n): + k = next(i for i in range(j, n) if cur[i] >> j & 1) + if k != j: + cur[j], cur[k] = cur[k], cur[j] + elimination.append(["SWAP", j, k]) + for i in range(n): + if i != j and cur[i] >> j & 1: + cur[i] ^= cur[j] + elimination.append(["ADD", i, j]) + need(cur == [1 << i for i in range(n)], "Scalar elimination endpoint") + program = list(reversed(elimination)) + cur = [1 << i for i in range(n)] + for op, d, s in program: + if op == "SWAP": + cur[d], cur[s] = cur[s], cur[d] + else: + cur[d] ^= cur[s] + need(cur == a, "Scalar boundary synthesis") + # A SWAP may be emitted as three ADDs; the invoice charges that option. + return program, sum(3 if op == "SWAP" else 1 for op, _, _ in program) + + +def shift(row, monomial): + return [{term ^ monomial for term in entry} for entry in row] + + +def add(a, b): + return [x ^ y for x, y in zip(a, b)] + + +def polynomial_coefficients(matrix): + n = len(matrix) + coefficients = {} + for i, row in enumerate(matrix): + for j, entry in enumerate(row): + for term in entry: + coefficients.setdefault(term, [0]*n)[i] ^= 1 << j + return {term: rows for term, rows in coefficients.items() if any(rows)} + + +def physical_controller(p, keep, omit_bridge=None): + """Literal physical operator over F2[(Z/2)^(2p+2)]. + + Bit zero is D_P (rank one); bit s+1 is D_Bs (rank seven). + A complete local invocation on a data pair is D_Bs S(D_P), + where S is one scalar shear. No scalar bridge is assumed to join + equal decoded frames: its physical action is multiplied literally. + """ + n = 2*p + matrix = [[{0} if i == j else set() for j in range(n)] for i in range(n)] + events, idle, bridge_number = [], [], 0 + + def idle_call(pair, stages, kind): + monomial = sum(1 << (s+1) for s in stages) + index = len(idle) + retained = bool(keep >> index & 1) + idle.append(dict(index=index, pair=pair, stages=stages, kind=kind, + rank=7*len(stages), retained=retained)) + if retained: + for role in (2*pair, 2*pair+1): + matrix[role] = shift(matrix[role], monomial) + events.append(["IDLE", pair, monomial]) + + def local(stage, pair, reverse): + x, y = 2*pair, 2*pair+1 + if reverse: + x, y = y, x + matrix[y] = add(matrix[y], shift(matrix[x], 1)) + matrix[x] = shift(matrix[x], 1 << (stage+1)) + matrix[y] = shift(matrix[y], 1 << (stage+1)) + events.append(["LOCAL", stage, pair, reverse]) + if stage % 2 and stage <= 2*p-3: + idle_call((stage+1)//2, list(range(stage+1)), "initial") + elif stage % 2 == 0 and 2 <= stage <= 2*p-2: + idle_call(stage//2, [stage], "middle") + elif stage == 2*p: + for j in range(p-1): + idle_call(j, list(range(2*j+3, 2*p+1)), "terminal") + + def bridge(dst, src): + nonlocal bridge_number + for d, s in ((2*dst, 2*src), (2*src+1, 2*dst+1)): + if bridge_number != omit_bridge: + matrix[d] = add(matrix[d], matrix[s]) + events.append(["ADD", d, s]) + bridge_number += 1 + + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1) + for j in reversed(range(p-1)): + bridge(j+1, j) + for j in range(p): + matrix[2*j], matrix[2*j+1] = matrix[2*j+1], matrix[2*j] + events.append(["SWAP", 2*j, 2*j+1]) + need(len(idle) == 3*(p-1), "Idle enumeration") + need(bridge_number == 6*(p-1), "Bridge count") + return polynomial_coefficients(matrix), events, idle + + +def simultaneous_monomial_factor(coefficients): + """Return explicit L A R = diagonal monomials, or reject. + + The coefficient matrices of A(1)^-1 A must be pairwise orthogonal + idempotents. This is a sufficient and necessary constant-basis test + for this exact formal diagonal-monomial endpoint. + """ + n = len(next(iter(coefficients.values()))) + total = [0]*n + for matrix in coefficients.values(): + total = [a ^ b for a, b in zip(total, matrix)] + if bit_rank(total, n) != n: + return None + inverse = bit_inverse(total) + projectors = {term: bit_mul(inverse, matrix) for term, matrix in coefficients.items()} + for term, matrix in projectors.items(): + if bit_mul(matrix, matrix) != matrix: + return None + for other, second in projectors.items(): + if term != other and any(bit_mul(matrix, second)): + return None + columns, diagonal = [], [] + for term, matrix in sorted(projectors.items()): + for j in range(n): + column = sum(((row >> j) & 1) << i for i, row in enumerate(matrix)) + if bit_rank(columns+[column], n) > len(columns): + columns.append(column) + diagonal.append(term) + need(len(columns) == n, "Projector images span") + right = [sum(((column >> i) & 1) << j for j, column in enumerate(columns)) + for i in range(n)] + left = bit_mul(bit_inverse(right), inverse) + exact = {term: bit_mul(bit_mul(left, matrix), right) + for term, matrix in coefficients.items()} + expected = {term: [((1 << i) if diagonal[i] == term else 0) for i in range(n)] + for term in set(diagonal)} + need(exact == expected, "Full formal operator diagonalization") + left_program, left_adds = bit_factor(left) + right_program, right_adds = bit_factor(right) + return dict(left_rows=left, right_rows=right, diagonal_monomials=diagonal, + coefficient_projectors={str(t): m for t, m in sorted(projectors.items())}, + left_program=left_program, right_program=right_program, + boundary_unit_ADDs_per_port=left_adds+right_adds, + full_coefficient_endpoint_sha256=digest({str(t): x for t, x in sorted(exact.items())})) + + +def rational_inverse(matrix): + n = len(matrix) + work = [[Q(x) for x in row]+[Q(i == j) for j in range(n)] + for i, row in enumerate(matrix)] + for j in range(n): + pivot = next(i for i in range(j, n) if work[i][j]) + work[j], work[pivot] = work[pivot], work[j] + scale = work[j][j] + work[j] = [x/scale for x in work[j]] + for i in range(n): + if i != j and work[i][j]: + scale = work[i][j] + work[i] = [a-scale*b for a, b in zip(work[i], work[j])] + return [row[n:] for row in work] + + +def matrix_multiply(a, b): + return [[sum((x*y for x, y in zip(row, column)), Q(0)) + for column in zip(*b)] for row in a] + + +def frame_projectors(frame_data): + result = {} + for name, frame in frame_data.items(): + basis = [[Q(x) for x in row] for row in frame["B"]] + if not basis: + projector = [[Q(0)]*8 for _ in range(8)] + else: + transpose = list(map(list, zip(*basis))) + gram = matrix_multiply(basis, transpose) + projector = matrix_multiply(matrix_multiply(transpose, rational_inverse(gram)), basis) + need(matrix_multiply(projector, projector) == projector, "Exact frame projector") + need(sum(projector[i][i] for i in range(8)) == frame["dim"], "Projector trace") + result[int(name)] = projector + return result + + +def audit_local(word_path): + need(sha256(word_path.read_bytes()).hexdigest() == WORD_HASH, "PR589 gzip source pin") + raw = gzip.decompress(word_path.read_bytes()) + need(sha256(raw).hexdigest() == WORD_CONTENT_HASH, "PR589 canonical word pin") + word = json.loads(raw) + folder = word_path.parent / "records" + frame_data = json.loads((folder/"frames.json").read_text())["frames"] + records = list(struct.iter_unpack("<6i", (folder/"249-records.bin").read_bytes())) + need((word["h"], word["v"], word["R"]) == (8, V, R), "Pinned local type") + n = 2*V+R + projectors = frame_projectors(frame_data) + dimensions = {int(f): z["dim"] for f, z in frame_data.items()} + pairs = set() + for op, role, old, new, rank, _ in records: + if op == 0: + pairs.add((old, new, rank)) + for old, new, rank in pairs: + p, q = projectors[old], projectors[new] + need(matrix_multiply(p, q) == p and matrix_multiply(q, p) == p, + "Nested exact projectors") + need(dimensions[new]-dimensions[old] == rank >= 0, "Literal child rank") + for role in range(2*V, n): + need(dimensions[word["start"][role]] == 0 and dimensions[word["final"][role]] == 8, + "Every helper is physically D_I8, not a reset or an unearned job") + checks = [] + streams = {} + for reverse in (False, True): + current = list(word["final"] if reverse else word["start"])+[-1] + values = [1 << i for i in range(n)]+[0] + norm = [1]*n+[0] + maximum = 1 + profile = Counter() + stream = [] + owned = None + adds = copies = 0 + for op, x, y, c, f, z in (reversed(records) if reverse else records): + if op == 0: + old, new = (c, y) if reverse else (y, c) + need(current[x] == old, "Local transition entrance") + current[x] = new + if f: + profile[f] += 1 + stream.append(["MOVE", x, old, new, f]) + elif op == 1: + need(current[x] == current[y] == f and x != y, "Native common-frame ADD") + need(owned is None or x != owned, "Owned source remains immutable") + values[x] ^= values[y] if c & 1 else 0 + norm[x] += abs(c)*norm[y] + maximum = max(maximum, norm[x]) + adds += 1 + stream.append(["ADD", x, y]) + elif op == (3 if reverse else 2): + need(owned is None and y == n and values[n] == 0 and current[x] == c, + "Private COPY entrance") + owned = x + current[n] = f + values[n] = values[x] + norm[n] = norm[x] + rank = abs(dimensions[c]-dimensions[f]) + profile[rank] += 1 + copies += 1 + stream.append(["COPY", x, n]) + else: + need(op == (2 if reverse else 3) and owned == x and y == n, + "Private COPY close") + need(values[n] == values[x], "Copied field exact cleanup") + values[n], norm[n], current[n], owned = 0, 0, -1, None + stream.append(["ERASE", x, n]) + need(owned is None and copies == 9 and adds == 24305, "Complete local word") + need(current[:n] == (word["start"] if reverse else word["final"]), "All local endpoints") + expected = [(1 << i) ^ ((1 << (i-V)) if V <= i < 2*V else 0) for i in range(n)] + need(values == expected+[0], "All arbitrary input columns") + need(profile == LOCAL, "Complete local bill, including copied centers") + key = "reverse" if reverse else "forward" + streams[key] = stream + checks.append(dict(direction=key, arbitrary_columns=n, dirty_helper_columns=R, + owned_COPY_episodes=copies, unit_ADDs=adds, + positive_majorant_maximum=maximum, positive_majorant_bits=maximum.bit_length(), + profile=dict(sorted(profile.items())), stream_sha256=digest(stream))) + serial_projectors = {str(f): [[str(x) for x in row] for row in p] + for f, p in sorted(projectors.items())} + return word, streams, dict(status="PASS_FULL_LOCAL_COLUMNS_AND_EXACT_PROJECTORS", + frames=len(projectors), nested_transition_pairs=len(pairs), + projector_sha256=digest(serial_projectors), signed_directions=checks, + helper_endpoint="D_I8 on all 439 arbitrary original helper inputs", + private_COPY_contract="Inherited source-owned immutable-copy contract; all nine episodes paid", + full_GL_cover_executed=False) + + +def explicit_unbanked_scalar_replay(p, streams): + """Fresh physical fields for every local invocation; all dirt columns.""" + calls = 2*p+1 + n = 2*p*V+calls*R + values = [1 << i for i in range(n)]+[0] + norms = [1]*n+[0] + maximum = 1 + adds = 0 + def local(stage, pair, reverse): + nonlocal maximum, adds + def role(r): + if r == 2*V+R: + return n + if r < 2*V: + side, port = divmod(r, V) + side = 1-side if reverse else side + return (2*pair+side)*V+port + return 2*p*V+stage*R+r-2*V + for event in streams["reverse" if reverse else "forward"]: + if event[0] == "MOVE": + continue + op, x, y = event + x, y = role(x), role(y) + if op == "ADD": + values[x] ^= values[y] + norms[x] += norms[y] + maximum = max(maximum, norms[x]) + adds += 1 + elif op == "COPY": + need(values[y] == 0, "Global private slot empty") + values[y], norms[y] = values[x], norms[x] + else: + need(values[y] == values[x], "Global private slot restored") + values[y], norms[y] = 0, 0 + def bridge(dst, src): + nonlocal maximum, adds + for port in range(V): + for d, s in ((2*dst*V+port, 2*src*V+port), + ((2*src+1)*V+port, (2*dst+1)*V+port)): + values[d] ^= values[s] + norms[d] += norms[s] + maximum = max(maximum, norms[d]) + adds += 1 + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1) + for j in reversed(range(p-1)): + bridge(j+1, j) + for pair in range(p): + for port in range(V): + x, y = 2*pair*V+port, (2*pair+1)*V+port + values[x], values[y] = values[y], values[x] + norms[x], norms[y] = norms[y], norms[x] + need(values == [1 << i for i in range(n)]+[0], "Every global decoded column restored") + need(adds == calls*24305+6*(p-1)*V, "Literal global scalar bill") + return dict(all_arbitrary_columns=n, independent_dirty_helpers=calls*R, + COPY_episodes=9*calls, private_COPY_closed=True, + unit_ADDs_before_new_boundary_adapters=adds, + final_free_pair_swaps=p*V, + positive_majorant_maximum=maximum, positive_majorant_bits=maximum.bit_length(), + notes="Decoded scalar audit; physical monomial endpoint is checked separately on every coefficient.") + + +@lru_cache(None) +def h_interval(rank, exponent): + lo, hi = log_interval(rank) + elo, ehi = exp_interval(exponent*lo, exponent*hi) + return Q(rank)/ehi, Q(rank)/elo + + +def margin_interval(paid, targets, exponent): + signed = Counter(targets) + signed.subtract(paid) + lo = hi = Q(0) + for rank, count in signed.items(): + a, b = h_interval(rank, exponent) + lo += count*(a if count >= 0 else b) + hi += count*(b if count >= 0 else a) + return lo, hi + + +def interval_row(bounds): + lo, hi = bounds + scale = 1 << 128 + lo = Q(lo.numerator*scale//lo.denominator, scale) + hi = Q(-((-hi.numerator*scale)//hi.denominator), scale) + return dict(lower=quotient(lo), upper=quotient(hi), + display=float((lo+hi)/2), strict_positive=lo > 0, strict_negative=hi < 0) + + +def price_candidate(p, keep, factor, idle): + calls, m = 2*p+1, 14*p+8 + paid = Counter({rank:calls*n for rank, n in LOCAL.items()}) + # All idle blocks are explicitly split into perpendicular rank-seven + # subcalls. No rank14/28/42 child is hidden under a rank8 target. + paid[7] += 2*V*sum(len(row["stages"]) for row in idle if row["retained"]) + targets = Counter({8:calls*R}) + for monomial in factor["diagonal_monomials"]: + targets[(monomial & 1)+7*(monomial >> 1).bit_count()] += V + need(max(paid) < min(targets), "Every child strictly smaller than every job") + credit = sum(rank*n for rank, n in targets.items()) + bill = sum(rank*n for rank, n in paid.items()) + need(credit-bill == 2*p*V-calls*63, "Full rank credit identity") + def floating(exponent): + return sum(n*r**(1-exponent) for r, n in targets.items())-sum(n*r**(1-exponent) for r, n in paid.items()) + lo, hi = 0.0, 0.01 + for _ in range(64): + middle = (lo+hi)/2 + if floating(middle) > 0: + lo = middle + else: + hi = middle + lower = Q(math.floor(lo*10**12), 10**12) + upper = lower+Q(1, 10**12) + lower_bounds = margin_interval(paid, targets, lower) + upper_bounds = margin_interval(paid, targets, upper) + need(lower_bounds[0] > 0 > upper_bounds[1], "Exact root bracket") + prices = {str(a):interval_row(margin_interval(paid, targets, a)) + for a in (Q("0.001745"), Q("0.002"), Q("0.004"), Q("0.006"))} + # The old raw-price bad-residue envelope is displayed conservatively. + # It is not a proof that the new mixed output normalizers inherit it. + old_residue_envelope = Q(32*m*m*sum(paid.values()), 10**16) + return dict(pairs=p, stages=calls, keep_idle_mask=keep, ambient_rank=m, + paid=dict(sorted(paid.items())), targets=dict(sorted(targets.items())), + total_paid_queries=sum(paid.values()), total_productive_fields=sum(targets.values()), + rank_bill=bill, rank_credit=credit, rank_margin=credit-bill, + strict_separation=dict(maximum_child=max(paid), minimum_target=min(targets)), + bare_mixed_moment_root_bracket=dict(lower=quotient(lower), upper=quotient(upper), + lower_display=float(lower), upper_display=float(upper), + lower_margin=interval_row(lower_bounds), upper_margin=interval_row(upper_bounds)), + target_margins=prices, old_raw_bad_residue_envelope=quotient(old_residue_envelope), + envelope_transfer_proved=False, kappa_claim=False) + + +def run(max_p=5): + start = time.monotonic() + source = HERE / "sources/pr589-regenerated/WORD.json.gz" + word, streams, local = audit_local(source) + cases, counts, winners = [], {}, {} + for p in range(1, max_p+1): + valid = [] + for keep in range(1 << (3*(p-1))): + coefficients, events, idle = physical_controller(p, keep) + factor = simultaneous_monomial_factor(coefficients) + if factor is None: + continue + # Discovery ranks by a floating root; winners receive exact + # rational price brackets below. All accepted factorizations + # have already passed exact finite operator identities. + paid = Counter({r:(2*p+1)*n for r, n in LOCAL.items()}) + paid[7] += 2*V*sum(len(row["stages"]) for row in idle if row["retained"]) + targets = Counter({8:(2*p+1)*R}) + for term in factor["diagonal_monomials"]: + targets[(term & 1)+7*(term >> 1).bit_count()] += V + def value(a): + return sum(n*r**(1-a) for r, n in targets.items())-sum(n*r**(1-a) for r, n in paid.items()) + lo, hi = 0.0, 0.01 + for _ in range(48): + mid = (lo+hi)/2 + if value(mid) > 0: + lo = mid + else: + hi = mid + valid.append((lo, keep, factor, coefficients, events, idle)) + counts[p] = dict(enumerated=1 << (3*(p-1)), exact_factorizations=len(valid)) + winning = max(valid, key=lambda row:(row[0], row[1])) + _, keep, factor, coefficients, events, idle = winning + case = price_candidate(p, keep, factor, idle) + case.update(factorization=factor, idle_events=idle, physical_events=events, + coefficient_matrices={str(term):rows for term, rows in sorted(coefficients.items())}, + decoded_global_replay=explicit_unbanked_scalar_replay(p, streams)) + case["unit_ADDs_including_constant_boundaries"] = ( + case["decoded_global_replay"]["unit_ADDs_before_new_boundary_adapters"]+ + V*factor["boundary_unit_ADDs_per_port"]) + case["unit_ADDs_if_all_final_SWAPs_use_three_ADDs"] = ( + case["unit_ADDs_including_constant_boundaries"]+3*p*V) + # Conservative norm multiplier for literal boundary ADDs. This + # input is offered to a precision compiler, not admitted as one. + base = case["decoded_global_replay"]["positive_majorant_maximum"] + norm_bound = base*(1 << (factor["boundary_unit_ADDs_per_port"]+3*p)) + case["positive_majorant_with_boundary_bound"] = str(norm_bound) + case["positive_majorant_with_boundary_bits"] = norm_bound.bit_length() + cases.append(case) + winners[p] = case["bare_mixed_moment_root_bracket"]["lower_display"] + strongest = max(cases, key=lambda row:row["bare_mixed_moment_root_bracket"]["lower_display"]) + p, keep = strongest["pairs"], strongest["keep_idle_mask"] + factor = strongest["factorization"] + # A literal omitted bridge is compared with the ACTUAL advertised + # target using the same boundary matrices. Mere diagonalizability is + # deliberately not enough for this negative control. + rejected = [] + expected = {int(t):bit_mul(bit_mul(factor["left_rows"], rows), factor["right_rows"]) + for t, rows in strongest["coefficient_matrices"].items()} + for missing in range(6*(p-1)): + coefficients, _, _ = physical_controller(p, keep, missing) + wrong = {t:bit_mul(bit_mul(factor["left_rows"], rows), factor["right_rows"]) + for t, rows in coefficients.items()} + need(wrong != expected, "Omitted physical bridge rejected") + rejected.append(missing) + controls = dict(omitted_bridges_rejected=rejected, + rank14_child_under_rank8_job_rejected=not(14 < 8), + full_rank8_helper_downgrade_to_rank7_rejected=(8 != 7), + all_idle_removed_p2_is_not_constant_monomial_diagonal=( + simultaneous_monomial_factor(physical_controller(2, 0)[0]) is None)) + pins = [Path(__file__), source, source.parent/"BASE.json.gz", + source.parent/"XCHECK.json", source.parent/"records/249-records.bin", + source.parent/"records/249-states.json", source.parent/"records/frames.json", + ROOT/"research/e8-bit-p111-tuned/code/gen_p3.py", + ROOT/"research/e8-bit-p111-tuned/code/xcheck_p.py", + ROOT/"research/e8-bit-p111-tuned/tune/apply.py", + ROOT/"research/e8-bit-p111-tuned/tune/wlib.py", + ROOT/"research/e8-bit-p111-tuned/tune/EXPECTED.json", + ROOT/"research/e8-bit-carrier-skeleton/code/bit/export_records.py", + REFERENCE/"joint_pair_dag.py"] + return dict(schema="finite-bit-mixed-productive-controller/1", + status="PASS_EXACT_FINITE_OPERATOR_AND_DISPLAYED_MIXED_INVOICE", + source_recovery="PR589 source regenerated byte-for-byte; not the missing stronger PR4085 BIT word", + local=local, finite_idle_menu=counts, candidates=cases, + strongest_pairs=strongest["pairs"], controls=controls, + exact_algebra="F2[(Z/2)^(2p+2)], represented by every monomial coefficient; no numerical character sampling", + physical_generator_ranks="D_P rank1, mutually perpendicular D_Bs rank7; helper endpoint D_(P+Bs) rank8", + not_proved=["Cover routing for these changed independent target projectors", + "Canonical weighted BIT output normalizers and their incoming-weight compatibility", + "Mixed-target all-size recursion or its fixed uniform precompilation", + "Physical tape, row stock, rounding/precision and complete outer multiplication transfer"], + source_sha256={str(path.relative_to(ROOT)):sha256(path.read_bytes()).hexdigest() for path in pins}, + runtime=dict(python=platform.python_version(), seconds=time.monotonic()-start), + new_multiplication_kappa=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--max-p", type=int, default=5) + parser.add_argument("--output", type=Path, default=HERE/"MIXED-PRODUCTIVE-BIT.json") + args = parser.parse_args() + need(1 <= args.max_p <= 6, "Bounded finite controller menu") + result = run(args.max_p) + args.output.write_text(json.dumps(result, indent=2, sort_keys=True)+"\n") + print(json.dumps(dict(status=result["status"], strongest_pairs=result["strongest_pairs"], + roots={c["pairs"]:c["bare_mixed_moment_root_bracket"]["lower_display"] + for c in result["candidates"]}, runtime=result["runtime"]), sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-rebuild/bit/pr589_h_completion.py b/research/kappa-rebuild/bit/pr589_h_completion.py new file mode 100644 index 0000000000..03f973b8ef --- /dev/null +++ b/research/kappa-rebuild/bit/pr589_h_completion.py @@ -0,0 +1,252 @@ +#!/usr/bin/env python3 +"""Actual PR589 defect-nine involution, with native rank-one completion. + +The old nine copied rank-seven centers are removed. The dirty-input +compensation is recomputed, preserving every source and helper. Nine row +corrections are gathered at the completed frame and transported into selected +target holes through one explicitly owned, reused field. All arithmetic here +is native XOR over F2; no complex rational coefficient is imported. + +Apache-2.0; substantial OpenAI assistance. The unmodified local body retains +the eumemic/Claude attribution and predecessor lineage in its NOTICE.md. +""" +from collections import Counter +from hashlib import sha256 +from pathlib import Path +import gzip +import json +import time + +import mixed_productive_bit as B +import pr589_state_sharing_probe as P + +HERE = Path(__file__).resolve().parent + + +def geometry(w): + v, n = w['v'], 2*w['v']+w['R'] + rows = [1 << i for i in range(n)] + for _, d, ss in w['A']: + for s in ss: + rows[d] ^= rows[s] + mask = (1 << v)-1 + V = [rows[r] & mask for r in w['ret']] + U = [sum(1 << k for k in row) for row in w['scatter']] + B.need(B.bit_rank(V, v) == len(V) == 9, 'Nine independent actual producer signals') + selected, columns = [], [] + for j in range(v): + c = sum(((r >> j) & 1) << i for i, r in enumerate(V)) + if B.bit_rank(columns+[c], 9) > len(columns): + selected.append(j) + columns.append(c) + if len(selected) == 9: + break + minor = [sum(((row >> a) & 1) << j for j, a in enumerate(selected)) for row in V] + VU = B.bit_mul(V, U) + M = B.bit_mul(B.bit_inverse(minor), VU) + correction = B.bit_mul(M, V) + Up = list(U) + for a, r in zip(selected, M): + Up[a] ^= r + B.need(not any(B.bit_mul(V, Up)), 'Completed defect is square-zero') + J = [r ^ (1 << i) for i, r in enumerate(B.bit_mul(U, V))] + H = [r ^ (1 << i) for i, r in enumerate(B.bit_mul(Up, V))] + B.need(B.bit_mul(H, H) == [1 << i for i in range(v)], 'Actual 111-port H squared is identity') + changed = list(J) + for a, r in zip(selected, correction): + changed[a] ^= r + B.need(changed == H, 'Only nine declared target rows change') + gaussian_program, gaussian_add_count = B.bit_factor(H) + # In the (ordinary, selected) split write V=V_A [L,I]. Because VU'=0, + # U'V_A=[K;LK], so H=S_L S_K S_L over F2. This literal three-shear + # program is much smaller than generic elimination and uses no scratch. + ordinary = [i for i in range(v) if i not in selected] + normalized = B.bit_mul(B.bit_inverse(minor), V) + L = [row & ~sum(1 << a for a in selected) for row in normalized] + K = B.bit_mul([Up[i] for i in ordinary], minor) + L_program = [['ADD', a, s] for a, row in zip(selected, L) + for s in ordinary if row >> s & 1] + K_program = [['ADD', u, a] for u, row in zip(ordinary, K) + for j, a in enumerate(selected) if row >> j & 1] + program = L_program+K_program+L_program + exact = [1 << i for i in range(v)] + for _, d, s in program: + exact[d] ^= exact[s] + B.need(exact == H, 'All columns of the three-shear common-frame H') + add_count = len(program) + return dict(V_rows=V, U_rows=U, VU_rows=VU, selected=selected, + selected_minor_rows=minor, M_rows=M, correction_rows=correction, + Uprime_rows=Up, J_rows=J, H_rows=H, L_full_port_rows=L, + ordinary_ports=ordinary, K_selected_coordinate_rows=K, + H_constant_program=program, H_unit_ADDs_including_three_ADD_SWAP=add_count, + H_gaussian_fallback_ADDs=gaussian_add_count, + H_three_shear_ADDs=[len(L_program), len(K_program), len(L_program)], + correction_source_row_weights=[r.bit_count() for r in correction]) + + +def clean_body(w, geo, masks, dims): + v, n = w['v'], 2*w['v']+w['R'] + rows = [1 << i for i in range(n)] + for name in ('A', 'B'): + for _, d, ss in w[name]: + for s in ss: + rows[d] ^= rows[s] + low = (1 << (2*v))-1 + D = [] + for t in range(v): + B.need(rows[v+t] & low == (1 << (v+t)) | geo['J_rows'][t], + 'Actual no-scatter raw body has the claimed J endpoint') + dirt = rows[v+t] >> (2*v) + srcs = [2*v+i for i in range(w['R']) if dirt >> i & 1] + if srcs: + D.append([0, v+t, srcs]) + current = list(w['start']) + events = [] + def climb(r, new): + old = current[r] + if old == new: + return + B.need(masks[old] & ~masks[new] == 0 and dims[old] < dims[new], 'Body nested frame') + events.append(['MOVE', r, old, new, dims[new]-dims[old]]) + current[r] = new + for section in (D, w['A'], w['B'], w['C']): + for f, d, ss in section: + for r in [d]+ss: + climb(r, f) + for s in ss: + events.append(['ADD', d, s, f]) + for r, f in enumerate(w['final']): + climb(r, f) + return events, D + + +def completion(w, geo, masks, dims): + v, n = w['v'], 2*w['v']+w['R'] + full = next(i for i, d in enumerate(dims) if d == 8) + events = [] + for a, row in zip(geo['selected'], geo['correction_rows']): + target = w['final'][v+a] + B.need(dims[target] == 7 and masks[target] & ~masks[full] == 0, 'Rank-one target hole') + events.append(['OWNED_CLEAR', n, full]) + for s in range(v): + if row >> s & 1: + events.append(['ADD', n, s, full]) + events += [['MOVE', n, full, target, 1], ['ADD', v+a, n, target], ['OWNED_ERASE', n, target]] + return events + + +def reverse_body(events): + return [[op, r, new, old, rank] if op == 'MOVE' else list(event) + for event in reversed(events) + for op, r, *tail in [event] + for old, new, rank in ([tail] if op == 'MOVE' else [[0, 0, 0]])] + + +def audit(w, geo, events, forward, completion_enabled=True): + v, n = w['v'], 2*w['v']+w['R'] + current = list(w['start'] if forward else w['final'])+[None] + rows = [1 << i for i in range(n)]+[0] + norms = [1]*n+[0] + maximum = 1 + invoice, counts = Counter(), Counter() + for event in events: + op = event[0] + counts[op] += 1 + if op == 'MOVE': + _, r, old, new, rank = event + B.need(current[r] == old, 'Exact physical transition entrance') + current[r] = new + invoice[rank] += 1 + elif op == 'ADD': + _, d, s, frame = event + B.need(current[d] == current[s] == frame and d != s, 'Literal common-frame XOR') + rows[d] ^= rows[s] + norms[d] += norms[s] + maximum = max(maximum, norms[d]) + elif op == 'OWNED_CLEAR': + _, r, frame = event + B.need(r == n and current[r] is None and rows[r] == 0, 'Owned slot is not independent input credit') + rows[r], norms[r], current[r] = 0, 0, frame + else: + B.need(op == 'OWNED_ERASE', 'Known primitive') + _, r, frame = event + B.need(r == n and current[r] == frame, 'Erase only the owned completed stream') + rows[r], norms[r], current[r] = 0, 0, None + target = geo['H_rows'] if completion_enabled else geo['J_rows'] + expected = [(1 << i) ^ (target[i-v] if v <= i < 2*v else 0) for i in range(n)]+[0] + B.need(rows == expected, 'All arbitrary sources, targets and helper input columns') + B.need(current == list(w['final'] if forward else w['start'])+[None], 'Every actual endpoint frame') + expected_bill = Counter(B.LOCAL) + expected_bill[7] -= 9 + if completion_enabled: + expected_bill[1] += 9 + B.need(invoice == expected_bill, 'Complete literal invoice including owned completion') + return dict(direction='forward' if forward else 'reverse', + arbitrary_columns=n, arbitrary_dirty_helpers=w['R'], + owned_fields=1 if completion_enabled else 0, + owned_field_cleared_after_every_use=current[n] is None, + paid=dict(sorted(invoice.items())), paid_rank_mass=sum(r*c for r, c in invoice.items()), + primitive_counts=dict(counts), prefix_positive_majorant=str(maximum), + prefix_positive_majorant_bits=maximum.bit_length(), + no_permanent_helper_query_defect=True) + + +def run(): + start = time.monotonic() + w, masks, dims = P.load() + geo = geometry(w) + body, dirt = clean_body(w, geo, masks, dims) + complete = completion(w, geo, masks, dims) + backward_body = reverse_body(body) + forward, backward = body+complete, complete+backward_body + checks = [audit(w, geo, forward, True), audit(w, geo, backward, False)] + no_completion = audit(w, geo, body, True, False) + B.need(geo['H_rows'] != geo['J_rows'], 'Omitting completion changes the advertised endpoint') + # New descents are literal complement projectors from the full frame. + metadata = json.loads((HERE/'sources/pr589-regenerated/records/frames.json').read_text())['frames'] + projectors = B.frame_projectors(metadata) + physical = [] + for a in geo['selected']: + line = projectors[w['start'][a]] + hyp = projectors[w['final'][w['v']+a]] + B.need(all(line[i][j]+hyp[i][j] == int(i == j) for i in range(8) for j in range(8)), + 'Actual q-line complements the actual target hyperplane') + physical.append(dict(selected_port=a, source_frame=1, target_frame=w['final'][w['v']+a], + rank=1, physical_operator='D_(I−P_qperp)=D_Pq', + exact_rational_complement_checked=True)) + artifact = dict(schema='actual-PR589-H-involution-native-word/1', + source_sha256=B.WORD_HASH, start=w['start'], final=w['final'], + geometry=geo, new_D=dirt, forward=forward, reverse=backward, + additional_owned_field=2*w['v']+w['R'], + owned_erase_contract='Owned workstream may be cleared in linear stream cost; it is never credited as an independent arbitrary productive input.') + raw = B.canonical(artifact) + data = gzip.compress(raw, compresslevel=9, mtime=0) + out = HERE/'PR589-H-NATIVE.json.gz' + out.write_bytes(data) + return dict(schema='actual-PR589-H-involution-native-receipt/1', + status='PASS_EXACT_NATIVE_H_COMPLETION', source_word_sha256=B.WORD_HASH, + generator_sha256=sha256(Path(__file__).read_bytes()).hexdigest(), + artifact_sha256=sha256(data).hexdigest(), canonical_artifact_sha256=sha256(raw).hexdigest(), + scalar_field='F2, literal XOR ADDs and three-ADD SWAP lowering', + geometry=dict(selected=geo['selected'], defect_rank=9, + old_VU_rank=B.bit_rank(geo['VU_rows'], 9), completed_VU_zero=True, + H_squared_identity=True, H_equals_J=False, + correction_source_row_weights=geo['correction_source_row_weights'], + common_frame_H_ADDs=geo['H_unit_ADDs_including_three_ADD_SWAP'], + common_frame_H_three_shear_counts=geo['H_three_shear_ADDs'], + common_frame_H_gaussian_fallback_ADDs=geo['H_gaussian_fallback_ADDs']), + full_arbitrary_input_checks=checks, + omitted_completion_negative_control=dict(expected_endpoint='H', actual_endpoint='J', + unequal_rows=sum(a != b for a, b in zip(geo['H_rows'], geo['J_rows'])), + paid=no_completion['paid']), + physical_completion_contacts=physical, + changed_local_invoice=dict(rank1_increment=9, rank7_decrement=9, + net_rank_mass_saved=54, owned_field_count=1), + global_monodromy_closed=False, new_kappa=False, + runtime_seconds=time.monotonic()-start) + + +if __name__ == '__main__': + result = run() + (HERE/'PR589-H-COMPLETION.json').write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps(result, indent=2)) diff --git a/research/kappa-rebuild/bit/pr589_h_global_probe.py b/research/kappa-rebuild/bit/pr589_h_global_probe.py new file mode 100644 index 0000000000..60bfc974da --- /dev/null +++ b/research/kappa-rebuild/bit/pr589_h_global_probe.py @@ -0,0 +1,166 @@ +#!/usr/bin/env python3 +"""Noncommuting p3 BIT H endpoint and complete direct-normalization invoice. + +This proves the useful factorization, then refuses the unearned global gain: +three independent interval banks retain a Q-conjugated free monodromy. The +direct endpoint repair costs six times111 rank-one queries, not twice111. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import gzip +import json +import time + +import bit_native_binding as N +import mixed_productive_bit as B + +HERE = Path(__file__).resolve().parent + + +def controller(p=3, keep=43): + n = 2*p + a = [[{()} if i == j else set() for j in range(n)] for i in range(n)] + idle, events = [], [] + def add(d, s): + a[d] = [x ^ y for x, y in zip(a[d], a[s])] + events.append(['ADD', d, s]) + def shift(r, word): + a[r] = [{word+term for term in entry} for entry in a[r]] + def idle_call(pair, stages): + index = len(idle) + retained = bool(keep >> index & 1) + idle.append(dict(index=index, pair=pair, stages=stages, retained=retained)) + if retained: + for r in (2*pair, 2*pair+1): + shift(r, tuple(reversed(stages))) + events.append(['IDLE_T', pair, stages]) + def local(stage, pair, backward): + x, y = 2*pair, 2*pair+1 + if backward: + x, y = y, x + add(y, x) + shift(x, (stage,)) + shift(y, (stage,)) + events.append(['LOCAL_T', stage, pair, backward]) + if stage % 2 and stage <= 2*p-3: + idle_call((stage+1)//2, list(range(stage+1))) + elif stage % 2 == 0 and 2 <= stage <= 2*p-2: + idle_call(stage//2, [stage]) + elif stage == 2*p: + for j in range(p-1): + idle_call(j, list(range(2*j+3, 2*p+1))) + def bridge(d, s): + add(2*d, 2*s) + add(2*s+1, 2*d+1) + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1) + for j in reversed(range(p-1)): + bridge(j+1, j) + for j in range(p): + a[2*j], a[2*j+1] = a[2*j+1], a[2*j] + coefficients = B.polynomial_coefficients(a) + factor = B.simultaneous_monomial_factor(coefficients) + B.need(factor is not None, 'Exact noncommuting interval-word diagonalization') + return coefficients, factor, idle, events + + +def run(): + start = time.monotonic() + native_path = HERE/'BIT-P3-NATIVE-BINDING.json' + h_path = HERE/'PR589-H-NATIVE.json.gz' + h_receipt_path = HERE/'PR589-H-COMPLETION.json' + native = json.loads(native_path.read_text()) + hword = json.loads(gzip.decompress(h_path.read_bytes())) + hreceipt = json.loads(h_receipt_path.read_text()) + native_checks = [] + for port in native['ports']: + p = [[Q(x) for x in row] for row in port['central_projector']] + rho = [[Q(x) for x in row] for row in port['chart_15']] + k = [[Q(int(i == j))-p[i][j] for j in range(8)] for i in range(8)] + old_hole = [[k[i][j] if i < 8 and j < 8 else Q(0) for j in range(15)] for i in range(15)] + fresh = [[Q(i == j and i >= 8) for j in range(15)] for i in range(15)] + active = [[p[i][j] if i < 8 and j < 8 else Q(i == j and i >= 8) + for j in range(15)] for i in range(15)] + old_full = [[Q(i == j and i < 8) for j in range(15)] for i in range(15)] + B.need(N.multiply(N.multiply(rho, old_hole), rho) == fresh, + 'Every other-stage old-hole transform is common at this local vertex') + B.need(N.multiply(N.multiply(rho, active), rho) == old_full, + 'Own Q b_s becomes common full I8') + native_checks.append(dict(port=port['port'], old_hole_to_fresh=True, + own_active_to_old_full=True, + all_other_fresh_blocks_fixed=True)) + coefficients, factor, idle, events = controller() + expected = [(2, 1, 0)]*2+[(6, 5, 4)]*2+[(6, 5, 4, 3, 2, 1, 0)]*2 + B.need(factor['diagonal_monomials'] == expected, 'All six exact interval words') + left, right = factor['left_rows'], factor['right_rows'] + parity = [0, 1, 0, 1, 1, 0] + for i in range(6): + for j in range(6): + if left[i] >> j & 1: + B.need(parity[i] == j % 2, 'L S = Sstar L, with no Q-valued coefficient') + if right[i] >> j & 1: + B.need(i % 2 == parity[j], 'S R = R Sstar, with no Q-valued coefficient') + local = Counter({int(r): n for r, n in hreceipt['full_arbitrary_input_checks'][0]['paid'].items()}) + paid = Counter({r: 7*n for r, n in local.items()}) + idle_stages = sum(len(row['stages']) for row in idle if row['retained']) + paid[7] += 222*idle_stages + targets = Counter({8: 3073, 22: 444, 50: 222}) + B.need(sum(r*n for r, n in targets.items())-sum(r*n for r, n in paid.items()) == 603, + 'All actual H calls and productive helpers counted') + corrected = paid.copy() + corrected[1] += 666 + B.need(sum(r*n for r, n in targets.items())-sum(r*n for r, n in corrected.items()) == -63, + 'Direct Q conjugation does not have a positive rank margin') + rows = [] + for role, word in enumerate(expected): + rows.append(dict(role=role, T_word=list(word), H_monodromy=list(word), + endpoint='K F' if parity[role] else '(Q K Q) F', + interval_target_rank=1+7*len(word), physical_fields=111, + free_K_inverse=bool(parity[role]), + direct_rank1_normalizer_queries=0 if parity[role] else 222)) + def margins(profile): + return {str(c): B.interval_row(B.margin_interval(profile, targets, c)) + for c in (Q(0), Q('0.003'), Q('0.0035'), Q('0.004'))} + return dict(schema='actual-BIT-H-noncommuting-p3-global-probe/1', + status='PASS_EXACT_ENDPOINT_FACTOR_WITH_UNFUNDED_DIRECT_HEAD_REPAIR', + physical_local_contract='F_s/B_s = diag(T_s,Q T_s Q) times the Q-valued elementary shear; T_s=H_s b_s, H_s^2=I, [H_s,Q b_s]=0.', + gauge='S=diag(I,Q) on each pair. Native scalar bridges commute with S. Final pair swaps contribute one common Q on the left.', + stage_commutation='[H_s,b_t]=0 for s!=t, by the exact native chart identities. Hence each interval product M=K B, and [K,Q B]=0.', + noncommuting_factor=factor, + exact_free_word_coefficients={str(k): v for k, v in coefficients.items()}, + boundary_Q_parity=parity, endpoints=rows, + native_chart_checks=native_checks, + helper_owners=native['helper_owners'], + raw_paid=dict(sorted(paid.items())), targets=dict(sorted(targets.items())), + raw_rank_margin=603, raw_margins=margins(paid), + direct_normalization=dict(rank1_children=666, paid=dict(sorted(corrected.items())), + rank_margin=-63, margins=margins(corrected)), + scalar_costs=dict(local_H_unit_ADDs=11955, + common_frame_H_ADDs=1452, + local_common_frame_wrappers=14, + idle_common_frame_H_uses=2*idle_stages, + endpoint_inverse_monodromy_H_uses=26, + global_unit_ADDs_with_direct_normalization=7*11955+(14+2*idle_stages+26)*1452+22*111+12*111+9*111, + owned_clear_or_erase_records=126, + controller_boundary_ADDs_per_port=factor['boundary_unit_ADDs_per_port'], + extra_fixed_cover_routes='Every H use and inverse has its literal per-port right routes; no physical H use is free of linear/routing tariff.'), + owned_COPY_episodes=63, + conclusion='The local54-rank saving is real. A full H-only p3 supplier with direct head normalization fails already at C=0. A new way to cancel/share the three interval head monodromies is required.', + source_sha256={p.name: sha256(p.read_bytes()).hexdigest() for p in + (Path(__file__), native_path, h_path, h_receipt_path)}, + runtime_seconds=time.monotonic()-start, new_kappa=False) + + +if __name__ == '__main__': + result = run() + (HERE/'PR589-H-GLOBAL-PROBE.json').write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps({k: result[k] for k in ('status', 'raw_paid', 'targets', 'raw_rank_margin', 'direct_normalization', 'runtime_seconds')}, indent=2)) diff --git a/research/kappa-rebuild/bit/pr589_minimum_completion.py b/research/kappa-rebuild/bit/pr589_minimum_completion.py new file mode 100644 index 0000000000..f230b4f3e0 --- /dev/null +++ b/research/kappa-rebuild/bit/pr589_minimum_completion.py @@ -0,0 +1,188 @@ +#!/usr/bin/env python3 +"""Minimum three-row invertible completion of the actual no-COPY PR589 J. + +This strengthens the local H experiment: invertibility, with opposite scalar +wrappers in the two orientations, suffices for the noncommuting controller. +Three new rank-one completed-frame gathers are optimal for row-supported +completion because the actual J has corank three. The full direct endpoint +bill still fails, and is recorded without an exponent claim. + +Apache-2.0; substantial OpenAI assistance. Immutable local-body attribution +is the retained PR589 eumemic/Claude notice and its cited predecessors. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import gzip +import json +import time + +import mixed_productive_bit as B +import pr589_h_completion as H +import pr589_state_sharing_probe as P + +HERE = Path(__file__).resolve().parent + + +def completion_geometry(J): + v = len(J) + basis, dependent = [], [] + for i, row in enumerate(J): + if B.bit_rank(basis+[row], v) > len(basis): + basis.append(row) + else: + dependent.append(i) + original_rank = len(basis) + B.need(original_rank == 108 and len(dependent) == 3, 'Actual J corank is three') + G, correction, choices = list(J), [], [] + for a in dependent: + j = next(j for j in range(v) if B.bit_rank(basis+[1 << j], v) > len(basis)) + basis.append(1 << j) + correction.append(J[a] ^ (1 << j)) + choices.append(j) + G[a] = 1 << j + inverse = B.bit_inverse(G) + B.need(B.bit_rank(G, v) == v, 'Three literal rows complete invertibility') + B.need(B.bit_rank(correction, v) == 3, 'Correction has exact rank three') + forward_program, forward_adds = B.bit_factor(G) + inverse_program, inverse_adds = B.bit_factor(inverse) + return dict(J_rows=J, H_rows=G, G_rows=G, G_inverse_rows=inverse, + J_rank=original_rank, selected=dependent, replacement_source_columns=choices, + correction_rows=correction, correction_source_row_weights=[r.bit_count() for r in correction], + forward_constant_program=forward_program, inverse_constant_program=inverse_program, + forward_constant_ADDs=forward_adds, inverse_constant_ADDs=inverse_adds) + + +def scalar_program(program, offset, frame): + out = [] + for op, d, s in program: + d, s = offset+d, offset+s + if op == 'ADD': + out.append(['ADD', d, s, frame]) + else: + B.need(op == 'SWAP', 'Constant program has binary ADD/SWAP only') + out += [['ADD', d, s, frame], ['ADD', s, d, frame], ['ADD', d, s, frame]] + return out + + +def audit(w, geo, events, forward, wrapped=False): + v, n = w['v'], 2*w['v']+w['R'] + cur = list(w['start'] if forward else w['final'])+[None] + values = [1 << i for i in range(n)]+[0] + norms, maximum = [1]*n+[0], 1 + paid, counts = Counter(), Counter() + for event in events: + op = event[0] + counts[op] += 1 + if op == 'MOVE': + _, r, old, new, rank = event + B.need(cur[r] == old, 'Literal frame entrance') + cur[r] = new + paid[rank] += 1 + elif op == 'ADD': + _, d, s, f = event + B.need(cur[d] == cur[s] == f and d != s, 'Literal common-frame XOR') + values[d] ^= values[s] + norms[d] += norms[s] + maximum = max(maximum, norms[d]) + elif op == 'OWNED_CLEAR': + _, r, f = event + B.need(r == n and values[r] == 0 and cur[r] is None, 'One separately owned field') + values[r], norms[r], cur[r] = 0, 0, f + else: + B.need(op == 'OWNED_ERASE', 'Known native primitive') + _, r, f = event + B.need(r == n and cur[r] == f, 'Owned transformed stream may be erased') + values[r], norms[r], cur[r] = 0, 0, None + expected = [1 << i for i in range(n)]+[0] + if wrapped: + matrix = geo['G_rows'] if forward else geo['G_inverse_rows'] + for i, row in enumerate(matrix): + expected[i] = row + expected[v+i] = row ^ (row << v) + else: + for i, row in enumerate(geo['G_rows']): + expected[v+i] ^= row + B.need(values == expected, 'All 661 arbitrary original data/helper columns') + B.need(cur == list(w['final'] if forward else w['start'])+[None], 'All native endpoint frames') + want = Counter(B.LOCAL) + want[7] -= 9 + want[1] += 3 + B.need(paid == want, 'Three new rank-one requests and no hidden center return') + return dict(direction='forward' if forward else 'reverse', wrapped=wrapped, + arbitrary_columns=n, restored_dirty_helpers=w['R'], paid=dict(sorted(paid.items())), + paid_rank_mass=sum(r*c for r, c in paid.items()), primitive_counts=dict(counts), + prefix_positive_majorant=str(maximum), prefix_positive_majorant_bits=maximum.bit_length(), + owned_fields=1, owned_clear_erase_episodes=3) + + +def run(): + start = time.monotonic() + w, masks, dims = P.load() + h_path = HERE/'PR589-H-NATIVE.json.gz' + h = json.loads(gzip.decompress(h_path.read_bytes())) + geo = completion_geometry(h['geometry']['J_rows']) + body, D = H.clean_body(w, geo, masks, dims) + correction = H.completion(w, geo, masks, dims) + forward = body+correction + reverse = correction+H.reverse_body(body) + wrapped_forward = (scalar_program(geo['forward_constant_program'], 111, 0)+ + forward+scalar_program(geo['forward_constant_program'], 0, 1)) + wrapped_reverse = (scalar_program(geo['inverse_constant_program'], 0, 1)+ + reverse+scalar_program(geo['inverse_constant_program'], 111, 0)) + checks = [audit(w, geo, forward, True), audit(w, geo, reverse, False), + audit(w, geo, wrapped_forward, True, True), audit(w, geo, wrapped_reverse, False, True)] + # Removing even one completion row leaves rank at most110. This is a + # concrete negative control as well as the corank lower bound. + controls = [] + for a in geo['selected']: + wrong = list(geo['G_rows']) + wrong[a] = geo['J_rows'][a] + rank = B.bit_rank(wrong, 111) + B.need(rank == 110, 'Every one of the three completions is necessary here') + controls.append(dict(omitted_selected_port=a, resulting_rank=rank)) + paid = Counter({int(r): 7*n for r, n in checks[0]['paid'].items()}) + paid[7] += 1332 + targets = Counter({8: 3073, 22: 444, 50: 222}) + repaired = paid.copy() + repaired[1] += 666 + margin = lambda profile: sum(r*n for r, n in targets.items())-sum(r*n for r, n in profile.items()) + B.need(margin(paid) == 645 and margin(repaired) == -21, 'Full source/helper/output invoice') + artifact = dict(schema='actual-PR589-minimum-invertible-completion-native-word/1', + geometry=geo, start=w['start'], final=w['final'], new_D=D, + forward=forward, reverse=reverse, + wrapped_forward=wrapped_forward, wrapped_reverse=wrapped_reverse, + owned_field=2*w['v']+w['R'], + source_word_sha256=B.WORD_HASH) + raw = B.canonical(artifact) + data = gzip.compress(raw, compresslevel=9, mtime=0) + (HERE/'PR589-MINIMUM-COMPLETION-NATIVE.json.gz').write_bytes(data) + return dict(schema='actual-PR589-minimum-invertible-completion-receipt/1', + status='PASS_MINIMUM_THREE_ROW_NATIVE_COMPLETION_GLOBAL_DEBT_UNFUNDED', + source_word_sha256=B.WORD_HASH, generator_sha256=sha256(Path(__file__).read_bytes()).hexdigest(), + H_source_artifact_sha256=sha256(h_path.read_bytes()).hexdigest(), + artifact_sha256=sha256(data).hexdigest(), canonical_artifact_sha256=sha256(raw).hexdigest(), + J_rank=108, completed_rank=111, minimum_row_supported_correction_rank=3, + minimum_proof='rank(J+E)-rank(J)<=rank(E)<=number of modified rows; the three displayed row replacements attain the bound.', + selected=geo['selected'], replacement_source_columns=geo['replacement_source_columns'], + correction_source_row_weights=geo['correction_source_row_weights'], + common_frame_G_ADDs=geo['forward_constant_ADDs'], common_frame_G_inverse_ADDs=geo['inverse_constant_ADDs'], + full_arbitrary_input_checks=checks, omitted_row_negative_controls=controls, + physical_completion='Each selected target is its native q-perp; the completed source bank is fullI8. Their exact complement is native rank-one Pq, as in the independently audited H completion.', + global_contract='In the pair gauge S=diag(I,Q), forward wrapped multiplier is G_s b_s and reverse wrapped multiplier is G_s^-1 b_s. Both multiply the two fields equally, so the same noncommuting interval factorization applies.', + global_raw_paid=dict(sorted(paid.items())), targets=dict(sorted(targets.items())), raw_rank_margin=645, + direct_head_repair_rank1_calls=666, direct_repaired_paid=dict(sorted(repaired.items())), + direct_repaired_rank_margin=-21, + exact_raw_margin_at_004=B.interval_row(B.margin_interval(paid, targets, Q(1, 250))), + exact_repaired_margin_at_004=B.interval_row(B.margin_interval(repaired, targets, Q(1, 250))), + no_new_global_closure=True, new_kappa=False, + runtime_seconds=time.monotonic()-start) + + +if __name__ == '__main__': + result = run() + (HERE/'PR589-MINIMUM-COMPLETION.json').write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps({k: result[k] for k in ('status', 'selected', 'replacement_source_columns', 'correction_source_row_weights', + 'common_frame_G_ADDs', 'common_frame_G_inverse_ADDs', 'full_arbitrary_input_checks', + 'raw_rank_margin', 'direct_repaired_rank_margin', 'runtime_seconds')}, indent=2)) diff --git a/research/kappa-rebuild/bit/pr589_state_sharing_probe.py b/research/kappa-rebuild/bit/pr589_state_sharing_probe.py new file mode 100644 index 0000000000..9fc16cda8e --- /dev/null +++ b/research/kappa-rebuild/bit/pr589_state_sharing_probe.py @@ -0,0 +1,247 @@ +#!/usr/bin/env python3 +"""Exact, fully invoiced helper-coalescing probe on the actual PR589 word. + +The pinned predecessors are only read. This searches every pair of helper +roles whose complete forward frame timelines can share one monotone physical +register. Scalar endpoint validity is a separate exact gate; successful +candidates have their old-dirt compensation rebuilt and every arbitrary input +column replayed. Owned COPY ranks remain paid. Apache-2.0; substantial OpenAI +assistance. The retained PR589 source has its own eumemic/Claude attribution. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import gzip +import json +import time + +import mixed_productive_bit as B + +HERE = Path(__file__).resolve().parent + + +def load(): + path = HERE/'sources/pr589-regenerated/WORD.json.gz' + data = path.read_bytes() + B.need(sha256(data).hexdigest() == B.WORD_HASH, 'Exact PR589 source') + w = json.loads(gzip.decompress(data)) + metadata = json.loads((path.parent/'records/frames.json').read_text())['frames'] + dimensions = [metadata[str(i)]['dim'] for i in range(len(w['frames']))] + masks = [sum(1 << i for i in frame) for frame in w['frames']] + return w, masks, dimensions + + +def alias_gates(gates, a, b): + out = [] + for f, d, ss in gates: + d = a if d == b else d + src = set() + for s in ss: + s = a if s == b else s + if s in src: + src.remove(s) + else: + src.add(s) + if d in src: + return None + if src: + out.append([f, d, sorted(src)]) + return out + + +def raw_replay(w, A, BB, ret, val): + for _, d, ss in A: + for s in ss: + val[d] ^= val[s] + snap = [val[r] for r in ret] + for t, ks in enumerate(w['scatter']): + for k in ks: + val[w['v']+t] ^= snap[k] + for _, d, ss in BB: + for s in ss: + val[d] ^= val[s] + return val + + +def rebuild(w, a, b, A, BB, ret): + v, n = w['v'], 2*w['v']+w['R'] + full = next(i for i, f in enumerate(w['frames']) if len(f) == v) + zero = next(i for i, f in enumerate(w['frames']) if not f) + val = raw_replay(w, A, BB, ret, [1 << i for i in range(n)]) + helper_mask = ((1 << n)-1) ^ ((1 << (2*v))-1) + D = [] + for t in range(v): + B.need(val[v+t] & ((1 << (2*v))-1) == (1 << (v+t)) | (1 << t), + 'Raw source and target response') + dirt = val[v+t] & helper_mask + src = [i for i in range(2*v, n) if dirt >> i & 1] + if src: + D.append([zero, v+t, src]) + C = [[full, d, list(ss)] for f, d, ss in reversed(A+BB) + if not v <= d < 2*v] + rid = lambda r: r-int(r > b) + def compact(gates): + return [[f, rid(d), [rid(s) for s in ss]] for f, d, ss in gates] + out = {k: value for k, value in w.items() if k not in ('A', 'B', 'C', 'D')} + out.update(R=w['R']-1, A=compact(A), B=compact(BB), C=compact(C), D=compact(D), + ret=[rid(r) for r in ret], start=w['start'][:b]+w['start'][b+1:], + final=w['final'][:b]+w['final'][b+1:]) + return out + + +def exact_audit(w, masks, dims): + v, n = w['v'], 2*w['v']+w['R'] + current = list(w['start']) + values = [1 << i for i in range(n)] + invoice, adds = Counter(), 0 + def climb(r, f): + old = current[r] + B.need(masks[old] & ~masks[f] == 0, 'Every frame transition remains nested') + if old != f: + B.need(dims[f] > dims[old], 'Every nontrivial transition increases rank') + invoice[dims[f]-dims[old]] += 1 + current[r] = f + for section in ('D', 'A', 'scatter', 'B', 'C'): + if section == 'scatter': + snapshot = [values[r] for r in w['ret']] + invoice.update(dims[current[r]] for r in w['ret']) + for t, ks in enumerate(w['scatter']): + for k in ks: + values[v+t] ^= snapshot[k] + adds += 1 + continue + for f, d, ss in w[section]: + B.need(d not in ss and len(ss) == len(set(ss)), 'Invertible primitive operands') + for r in [d]+ss: + climb(r, f) + for s in ss: + values[d] ^= values[s] + adds += 1 + for r, f in enumerate(w['final']): + climb(r, f) + expected = [(1 << i) ^ ((1 << (i-v)) if v <= i < 2*v else 0) + for i in range(n)] + B.need(values == expected, 'All arbitrary original/source/helper columns') + return dict(paid=dict(sorted(invoice.items())), unit_ADDs=adds, + arbitrary_columns=n, dirty_helpers=w['R'], + owned_COPY_episodes=len(w['ret']), + all_original_helpers_restored=True) + + +def price(audit, case): + paid = Counter({int(r): 7*n for r, n in audit['paid'].items()}) + paid[7] += 2*B.V*sum(len(row['stages']) for row in case['idle_events'] if row['retained']) + targets = Counter({8: 7*audit['dirty_helpers']}) + for term in case['factorization']['diagonal_monomials']: + targets[(term & 1)+7*(term >> 1).bit_count()] += B.V + B.need(max(paid) < min(targets), 'Strict child/job separation') + def margin(c): + return sum(n*r**(1-c) for r, n in targets.items())-sum(n*r**(1-c) for r, n in paid.items()) + lo, hi = 0., .01 + for _ in range(60): + mid = (lo+hi)/2 + if margin(mid) > 0: + lo = mid + else: + hi = mid + return dict(paid=dict(sorted(paid.items())), targets=dict(sorted(targets.items())), + rank_margin=sum(r*n for r, n in targets.items())-sum(r*n for r, n in paid.items()), + floating_root_for_screen=lo, + exact_margin_at_004=B.interval_row(B.margin_interval(paid, targets, Q(1, 250)))) + + +def run(): + start = time.monotonic() + w, masks, dims = load() + v, n = w['v'], 2*w['v']+w['R'] + case = next(c for c in json.loads((HERE/'MIXED-PRODUCTIVE-BIT.json').read_text())['candidates'] + if c['pairs'] == 3) + baseline = exact_audit(w, masks, dims) + B.need(baseline['paid'] == dict(B.LOCAL), 'Pinned complete original tariff') + uses = {r: [] for r in range(2*v, n)} + forward = w['A']+w['B'] + for k, (f, d, ss) in enumerate(forward): + for r in [d]+ss: + if r in uses: + uses[r].append((k, f)) + # Include the snapshot cut as a use, without fabricating a new frame. + for r in w['ret']: + prior = [f for k, f in uses[r] if k < len(w['A'])] + uses[r].append((len(w['A'])-0.5, prior[-1] if prior else w['start'][r])) + for r in uses: + uses[r].sort() + candidates, disjoint = [], [] + for a in uses: + for b in range(a+1, n): + previous = 0 + ok = True + timeline = sorted(uses[a]+uses[b]) + for _, f in timeline: + if previous & ~masks[f]: + ok = False + break + previous = masks[f] + if ok: + candidates.append((a, b)) + if not uses[a] or not uses[b] or uses[a][-1][0] < uses[b][0][0] or uses[b][-1][0] < uses[a][0][0]: + disjoint.append((a, b)) + counts = Counter(total_helper_pairs=w['R']*(w['R']-1)//2, + monotone_frame_candidates=len(candidates), + disjoint_forward_lifetimes=len(disjoint)) + accepted, rejections = [], [] + for a, b in candidates: + A, BB = alias_gates(w['A'], a, b), alias_gates(w['B'], a, b) + if A is None or BB is None: + counts['self_gate_reject'] += 1 + continue + ret = [a if r == b else r for r in w['ret']] + out = raw_replay(w, A, BB, ret, [1 << r if r < v else 0 for r in range(n)]) + failed = [t for t in range(v) if out[v+t] != 1 << t] + if failed: + counts['source_response_reject'] += 1 + if len(rejections) < 12: + rejections.append(dict(pair=[a, b], wrong_targets=failed, + wrong_source_coefficients=sum((out[v+t] ^ (1 << t)).bit_count() for t in failed))) + continue + new = rebuild(w, a, b, A, BB, ret) + audit = exact_audit(new, masks, dims) + counts['full_arbitrary_input_pass'] += 1 + invoice = price(audit, case) + accepted.append(dict(pair=[a, b], audit=audit, global_p3=invoice, + canonical_word_sha256=B.digest(new))) + # Consumer-only and producer-only rank lower bounds for the owned totals. + values = [1 << r for r in range(n)] + for _, d, ss in w['A']: + for s in ss: + values[d] ^= values[s] + centers = [values[r] for r in w['ret']] + source_mask = (1 << v)-1 + scatter = [sum(1 << k for k in row) for row in w['scatter']] + center_audit = dict(owned_center_count=len(centers), + consumer_matrix_rank=B.bit_rank(scatter, len(centers)), + center_source_response_rank=B.bit_rank([x & source_mask for x in centers], v), + center_full_arbitrary_response_rank=B.bit_rank(centers, n), + current_center_rank_bill=63, + conclusion='The nine actual center signals are independent, and their consumer map is injective. A same-cut linear compression to eight signals cannot preserve all responses even before its frame-conversion and cleanup cost is charged.') + accepted.sort(key=lambda row: row['global_p3']['floating_root_for_screen'], reverse=True) + return dict(schema='actual-PR589-monotone-helper-state-sharing/1', + status='PASS_EXHAUSTIVE_FINITE_PROBE', + source_word_sha256=B.WORD_HASH, + source_generator_sha256=sha256(Path(__file__).read_bytes()).hexdigest(), + baseline=baseline, baseline_global_p3=price(baseline, case), + coverage=dict(counts), accepted=accepted, sample_response_rejections=rejections, + owned_center_independence=center_audit, + not_claimed=['No unrestricted scheduling lower bound.', + 'No decoder-COPY charge removed without a new operator identity.', + 'No exponent gain from a finite helper count alone.'], + runtime_seconds=time.monotonic()-start) + + +if __name__ == '__main__': + result = run() + out = HERE/'PR589-STATE-SHARING-PROBE.json' + out.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps({k: result[k] for k in ('status', 'coverage', 'owned_center_independence', 'runtime_seconds')}, indent=2)) + if result['accepted']: + print(json.dumps(result['accepted'][0], indent=2)) diff --git a/research/kappa-rebuild/bit/reconstruct_deferred.py b/research/kappa-rebuild/bit/reconstruct_deferred.py new file mode 100644 index 0000000000..474970f9c7 --- /dev/null +++ b/research/kappa-rebuild/bit/reconstruct_deferred.py @@ -0,0 +1,299 @@ +#!/usr/bin/env python3 +"""Fresh reconstruction: deferred original functions and nested dirty pools. + +The October 21 pre-reset sources did not survive. This is a newly specified +construction, not a reconstruction of those source bytes or historical counts. +Frozen October 20 implementations are imported read only. Apache-2.0; +substantial OpenAI assistance, with predecessor attribution retained in README. +""" +from collections import Counter, defaultdict +from fractions import Fraction as Q +from hashlib import sha256 +from itertools import combinations +from pathlib import Path +import argparse +import json +import platform +import sys +import time + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +FROZEN = ROOT / "research/kappa-function-composition/bit" +sys.path.insert(0, str(FROZEN)) +import native_functions as N +import staged_continuation as S +C = N.C + + +def minimum_chains(patterns): + """Exact minimum path cover of the strict reverse-inclusion DAG.""" + ps = sorted(patterns, key=lambda p: (-p.bit_count(), p)) + matched_right = {} + + def augment(i, seen): + for j in range(i + 1, len(ps)): + if j in seen or ps[i] & ps[j] != ps[j]: + continue + seen.add(j) + if j not in matched_right or augment(matched_right[j], seen): + matched_right[j] = i + return True + return False + + for i in range(len(ps)): + augment(i, set()) + successor = {i: j for j, i in matched_right.items()} + chains = [] + for i in range(len(ps)): + if i in matched_right: + continue + chain = [ps[i]] + while i in successor: + i = successor[i] + chain.append(ps[i]) + chains.append(chain) + C.require(sorted(p for chain in chains for p in chain) == sorted(ps), + "Chain cover dropped or duplicated a function") + C.require(len(chains) == len(ps) - len(matched_right), "Wrong matching bound") + return chains + + +def fresh_instance(q): + targets = N.grid_targets() + targets += [C.mask((0, 1, 2, *edge)) for edge in combinations(range(3, 9), 2) + if C.mask((0, 1, 2, *edge)) not in targets] + if q == 16: + targets.append(C.mask((0, 1, 3, 5, 6))) + C.require(q in (15, 16) and len(targets) == q, "Unsupported fresh fixture") + forest = [N.old_grid_tree(), C.tree(range(8, 15))] + if q == 16: + forest.append((1 << 15, ())) + g = N.instance(19, targets, forest, f"fresh_{q}_static") + stages = {node: stage for stage, root in enumerate(forest) + for node, _ in C.walk(root)} + # Make the static comparison's original assignment compatible with the + # frozen restored-original cut contract; this changes no helper count. + helpers, queries = [], defaultdict(list) + for item in g["patterns"]: + nodes, pattern, pivot = item["nodes"], item["pattern"], item["pivot"] + original = min(nodes, key=lambda node: (stages[node], -node.bit_count(), node)) + copies = [] + for node in nodes: + if node == original: + role = pivot + else: + role = g["V"] + q + len(helpers) + helper = (role, pivot, node, pattern) + helpers.append(helper) + copies.append(helper) + queries[node].append((role, pivot, pattern)) + item["copies"] = copies + item["original_node"] = original + g.update(helpers=helpers, queries=dict(queries), A=len(helpers), + W=g["V"] + q + len(helpers)) + return g, stages + + +def pooled_instance(base): + by_node = defaultdict(list) + for _, _, node, pattern in base["helpers"]: + by_node[node].append(pattern) + pools = [] + for node, patterns in sorted(by_node.items()): + for chain in minimum_chains(patterns): + pools.append(dict(role=base["V"] + base["q"] + len(pools), + node=node, patterns=chain)) + g = dict(base, A=len(pools), W=base["V"] + base["q"] + len(pools), + design=f"fresh_{base['q']}_deferred_nested_pools", pools=pools) + return g + + +def events(g, sign, omit_cleanup=False, early_original_cleanup=False): + V = g["V"] + items = {item["pattern"]: item for item in g["patterns"]} + for pool in g["pools"]: + for target in C.bits(pool["node"]): + yield "ADD", V + target, pool["role"], sign, "old_pool_compensation" + # Original pivots remain aggregate functions until every pool is clean. + for item in g["patterns"]: + if len(item["nodes"]) == 1: + continue + pattern, pivot = item["pattern"], item["pivot"] + for source in item["members"]: + yield "MOVE", source, ("E", pattern) + for source in item["members"][1:]: + yield "ADD", pivot, source, 1, "original_function_gather" + yield "MOVE", source, "I" + for pool in g["pools"]: + for pattern in pool["patterns"]: + yield "MOVE", pool["role"], ("E", pattern) + yield "ADD", pool["role"], items[pattern]["pivot"], 1, "nested_pool_acquire" + node_pools = defaultdict(list) + node_originals = defaultdict(list) + for pool in g["pools"]: + node_pools[pool["node"]].append(pool) + for item in g["patterns"]: + node_originals[item["original_node"]].append(item) + + def process(node): + group, children = node + leaves = C.bits(group) + pivot = V + leaves[0] + if node_pools[group] or node_originals[group]: + for target in leaves[1:]: + yield "ADD", V + target, pivot, -1, "output_precode" + for target in leaves: + yield "MOVE", V + target, ("E", group) + for item in node_originals[group]: + if len(item["nodes"]) == 1: + for source in item["members"]: + yield "MOVE", source, ("E", group) + yield "ADD", pivot, source, -sign, "single_use_original_read" + yield "MOVE", source, "I" + else: + yield "MOVE", item["pivot"], ("E", group) + yield "ADD", pivot, item["pivot"], -sign, "original_function_read" + yield "MOVE", item["pivot"], "I" + for pool in node_pools[group]: + yield "MOVE", pool["role"], ("E", group) + yield "ADD", pivot, pool["role"], -sign, "pooled_function_read" + yield "MOVE", pool["role"], "I" + for target in leaves[1:]: + yield "ADD", V + target, pivot, 1, "output_decode" + if children: + for child in children: + yield from process(child) + else: + yield "MOVE", pivot, ("E", group) + for stage, root in enumerate(g["forest"]): + yield from process(root) + yield "CUT", stage + for source in range(V): + if source not in g["used"]: + yield "MOVE", source, "I" + + def restore_originals(): + for item in reversed(g["patterns"]): + if len(item["nodes"]) > 1: + for source in reversed(item["members"][1:]): + yield "ADD", item["pivot"], source, -1, "original_function_restore" + if early_original_cleanup: + yield from restore_originals() + if not omit_cleanup: + for pool in reversed(g["pools"]): + for pattern in reversed(pool["patterns"]): + yield "ADD", pool["role"], items[pattern]["pivot"], -1, "nested_pool_restore" + if not early_original_cleanup: + yield from restore_originals() + + +def cut_audit(g, stages, sign): + V, q, W = g["V"], g["q"], g["W"] + rows = [{i: 1} for i in range(W)] + frames = [("P", i) for i in range(V)] + ["zero"] * (W - V) + output_stage = {t: stage for stage, root in enumerate(g["forest"]) + for t in C.bits(root[0])} + cuts = [] + for event in events(g, sign): + if event[0] == "MOVE": + frames[event[1]] = event[2] + elif event[0] == "ADD": + _, dst, src, coefficient, _ = event + C.require(frames[dst] == frames[src], "Cut audit sees unequal-frame ADD") + rows[dst] = C.add(rows[dst], rows[src], coefficient) + else: + stage = event[1] + expected = [{i: 1} for i in range(W)] + expected_frames = [("P", i) for i in range(V)] + ["zero"] * (W - V) + for item in g["patterns"]: + consumed = stages[item["original_node"]] <= stage + if len(item["nodes"]) == 1: + if consumed: + for source in item["members"]: + expected_frames[source] = "I" + else: + expected[item["pivot"]] = {r: 1 for r in item["members"]} + for source in item["members"]: + expected_frames[source] = "I" + if not consumed: + expected_frames[item["pivot"]] = ("E", item["pattern"]) + for target in range(q): + if output_stage[target] <= stage: + for source, mask in enumerate(g["labels"]): + if (mask & g["targets"][target]).bit_count() == 2: + expected[V + target][source] = -sign + expected_frames[V + target] = ("E", 1 << target) + else: + for pool in g["pools"]: + if pool["node"] >> target & 1: + expected[V + target][pool["role"]] = sign + for pool in g["pools"]: + for pattern in pool["patterns"]: + item = next(x for x in g["patterns"] if x["pattern"] == pattern) + for source in item["members"]: + expected[pool["role"]][source] = 1 + expected_frames[pool["role"]] = ("I" if stages[pool["node"]] <= stage + else ("E", pool["patterns"][-1])) + C.require(rows == expected and frames == expected_frames, + "Fresh deferred original cut contract failed") + cuts.append(dict(stage=stage, sign=sign, + completed_outputs=[t for t in range(q) if output_stage[t] <= stage], + complete_independent_input_columns=W, + original_aggregate_pivots=sum(len(x["nodes"]) > 1 for x in g["patterns"]), + ordinary_pool_states_retained=g["A"], + future_outputs_have_only_old_pool_compensation=True, + matrix_sha256=sha256(json.dumps([sorted(r.items()) for r in rows], + separators=(",", ":")).encode()).hexdigest(), + frame_sha256=sha256(json.dumps(frames, separators=(",", ":")).encode()).hexdigest(), + mismatching_coefficients=0)) + return cuts + + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument("--receipt", type=Path, default=HERE / "DEFERRED-RECONSTRUCTION.json") + args = parser.parse_args() + started = time.monotonic() + checks, cuts, comparisons, fixtures, negatives = [], [], [], [], [] + for q in (15, 16): + base, stages = fresh_instance(q) + g = pooled_instance(base) + baseline = N.replay_word(base, (e for e in S.staged_events(base, 1) if e[0] != "CUT"), 1) + checks.append(baseline) + for sign in (1, -1): + result = N.replay_word(g, (e for e in events(g, sign) if e[0] != "CUT"), sign) + checks.append(result) + cuts.extend(cut_audit(g, stages, sign)) + for a in (Q(349, 200000), Q(1, 250)): + comparisons.append(dict(outputs=q, **N.cost_difference(baseline["local_profile"], + result["local_profile"], base["A"], g["A"], 19, a))) + for control in ("omit_pool_cleanup", "original_cleanup_too_early"): + opts = {"omit_cleanup": control == "omit_pool_cleanup", + "early_original_cleanup": control == "original_cleanup_too_early"} + r = N.replay_word(g, (e for e in events(g, 1, **opts) if e[0] != "CUT"), + 1, expect_failure=True) + negatives.append(dict(control=control, outputs=q, + wrong_coefficients=r["mismatching_coefficients"])) + fixtures.append(dict(outputs=q, target_masks=g["targets"], forest=g["forest"], + patterns=[dict(pattern=x["pattern"], original_node=x["original_node"], + members=x["members"], pivot=x["pivot"]) + for x in g["patterns"]], + pools=g["pools"], geometry=g["geometric_checks"])) + print(json.dumps(dict(outputs=q, baseline_helpers=base["A"], helpers=g["A"], + baseline_ADDs=baseline["ADDs"], ADDs=result["ADDs"], + status="PASS")), flush=True) + pins = N.pins(Path(__file__), Path(S.__file__)) + receipt = dict(schema="fresh-bit-deferred-original-nested-pools/1", status="PASS", + provenance="New post-reset construction; historical lost target/forest bytes were not recovered.", + coefficient_ring="Z", geometry_field="Q", checks=checks, cut_audits=cuts, + comparisons=comparisons, fixtures=fixtures, negative_controls=negatives, + source_sha256=pins, new_multiplication_kappa=False, + global_supplier_completed=False, + environment=dict(python=sys.version, platform=platform.platform()), + elapsed_seconds=time.monotonic() - started) + args.receipt.write_text(json.dumps(receipt, indent=2) + "\n") + + +if __name__ == "__main__": + main() diff --git a/research/kappa-rebuild/bit/single_field_controller_probe.py b/research/kappa-rebuild/bit/single_field_controller_probe.py new file mode 100644 index 0000000000..3a18a7ab09 --- /dev/null +++ b/research/kappa-rebuild/bit/single_field_controller_probe.py @@ -0,0 +1,132 @@ +#!/usr/bin/env python3 +"""Probe independent positive/negative idle windows in the physical controller. + +Binary candidates are screened exactly and then replayed over the integral +Laurent ring. This changes no native local word; it changes which physical +data fields receive idle recursive calls. No new exponent is claimed. +Apache-2.0. Substantial OpenAI assistance. +""" +from fractions import Fraction as Q +from pathlib import Path +from hashlib import sha256 +import argparse +import json +import time + +import complex_controller_factor as C +import mixed_productive_bit as B + +HERE = Path(__file__).resolve().parent + + +def controller(p, keep, binary=False): + n, stages = 2*p, 2*p+1 + zero = 0 if binary else (0,)*(stages+1) + matrix = [[({0} if i == j else set()) if binary else + ({zero:1} if i == j else {}) for j in range(n)] for i in range(n)] + events, idle = [], [] + + def term(q, selected): + return (q & 1) | sum(1 << (s+1) for s in selected) if binary else (q,)+tuple(int(s in selected) for s in range(stages)) + + def multiply(polynomial, exponents, coefficient=1): + return {x ^ exponents for x in polynomial} if binary else C.monomial_times(polynomial, exponents, coefficient) + + def plus(a, b, coefficient=1): + return a ^ b if binary else C.plus(a, b, coefficient) + + def idle_call(pair, selected, kind): + for side in (0, 1): + index = len(idle) + retained = bool(keep >> index & 1) + role = 2*pair+side + idle.append(dict(index=index, pair=pair, side=side, field=role, + stages=list(selected), rank=8*len(selected), + kind=kind, retained=retained)) + if retained: + matrix[role] = [multiply(x, term(0, selected)) for x in matrix[role]] + events.append(["IDLE_FIELD", role, list(selected)]) + + def local(stage, pair, reverse): + x, y = 2*pair, 2*pair+1 + if reverse: + x, y = y, x + sign, q = (-1, 1) if reverse else (1, -1) + matrix[y] = [plus(a, multiply(b, term(q, []), sign)) for a, b in zip(matrix[y], matrix[x])] + for role in (x, y): + matrix[role] = [multiply(a, term(0, [stage])) for a in matrix[role]] + events.append(["LOCAL", stage, pair, reverse]) + if stage % 2 and stage <= 2*p-3: + idle_call((stage+1)//2, list(range(stage+1)), "initial") + elif stage % 2 == 0 and 2 <= stage <= 2*p-2: + idle_call(stage//2, [stage], "middle") + elif stage == 2*p: + for j in range(p-1): + idle_call(j, list(range(2*j+3, 2*p+1)), "terminal") + + def bridge(dst, src, coefficient): + for d, s, c in ((2*dst, 2*src, coefficient), (2*src+1, 2*dst+1, -coefficient)): + matrix[d] = [plus(a, b, c) for a, b in zip(matrix[d], matrix[s])] + events.append(["ADD", d, s, c]) + + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1, 1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1, -1) + for j in reversed(range(p-1)): + bridge(j+1, j, 1) + for j in range(p): + matrix[2*j], matrix[2*j+1] = matrix[2*j+1], [multiply(x, zero, -1) for x in matrix[2*j]] + events.append(["SIGNED_SWAP", 2*j, 2*j+1]) + if binary: + coefficients = B.polynomial_coefficients(matrix) + else: + coefficients = {} + for i, row in enumerate(matrix): + for j, polynomial in enumerate(row): + for key, value in polynomial.items(): + coefficients.setdefault(key, [[Q(0)]*n for _ in range(n)])[i][j] += value + coefficients = {k:a for k,a in coefficients.items() if any(any(row) for row in a)} + return coefficients, events, idle + + +def run(p): + start = time.monotonic() + trials, candidates, cases = 1 << (6*(p-1)), [], [] + for mask in range(trials): + coefficients, _, _ = controller(p, mask, True) + if B.simultaneous_monomial_factor(coefficients) is not None: + candidates.append(mask) + for mask in candidates: + coefficients, events, idle = controller(p, mask) + result = C.factor(coefficients) + if result is not None: + case = C.serialize_case(p, mask, coefficients, events, idle, result) + case["idle_mask_semantics"] = "Two consecutive bits per former paired window; positive then negative field" + case["all_windows_paired"] = all(bool(mask >> (2*j)&1) == bool(mask >> (2*j+1)&1) for j in range(3*(p-1))) + cases.append(case) + return dict(schema="single-field-idle-controller/1", pairs=p, trials=trials, + binary_candidates=len(candidates), exact_Laurent_cases=len(cases), + asymmetric_cases=sum(not c["all_windows_paired"] for c in cases), + source_sha256={Path(__file__).name:sha256(Path(__file__).read_bytes()).hexdigest()}, + cases=cases, runtime_seconds=time.monotonic()-start, new_kappa=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--pairs", type=int, default=3) + parser.add_argument("--output", type=Path) + args = parser.parse_args() + B.need(1 <= args.pairs <= 4, "Bounded probe") + result = run(args.pairs) + path = args.output or HERE/f"SINGLE-FIELD-CONTROLLER-P{args.pairs}.json" + path.write_text(json.dumps(result, indent=2, sort_keys=True)+"\n") + print(json.dumps({k:v for k,v in result.items() if k != "cases"}, sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/research/kappa-rebuild/bit/sources/pr589-regenerated/BASE.json.gz b/research/kappa-rebuild/bit/sources/pr589-regenerated/BASE.json.gz new file mode 100644 index 0000000000..bdbb95aa97 Binary files /dev/null and b/research/kappa-rebuild/bit/sources/pr589-regenerated/BASE.json.gz differ diff --git a/research/kappa-rebuild/bit/sources/pr589-regenerated/WORD.json.gz b/research/kappa-rebuild/bit/sources/pr589-regenerated/WORD.json.gz new file mode 100644 index 0000000000..85a4499bf9 Binary files /dev/null and b/research/kappa-rebuild/bit/sources/pr589-regenerated/WORD.json.gz differ diff --git a/research/kappa-rebuild/bit/sources/pr589-regenerated/XCHECK.json b/research/kappa-rebuild/bit/sources/pr589-regenerated/XCHECK.json new file mode 100644 index 0000000000..8d324bde35 --- /dev/null +++ b/research/kappa-rebuild/bit/sources/pr589-regenerated/XCHECK.json @@ -0,0 +1,65 @@ +{ + "status": "PASS_EXACT_REPLAY_AND_INTERLEAVED_INTERFACE", + "hypotheses": { + "H1_no_target_feeds_z": true, + "H2_D_frame0_targets_from_helpers": true, + "H3_A_z_only_scatter_targets_B_split": true, + "H4_C_literal_reverse_at_full": true, + "H5_V_identity": true + }, + "conclusion": { + "U_block_form": true, + "J_identity": true, + "K_equals_D": true, + "C_is_U_inverse": true + }, + 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\ No newline at end of file diff --git a/research/kappa-rebuild/bit/stage_projector_probe.py b/research/kappa-rebuild/bit/stage_projector_probe.py new file mode 100644 index 0000000000..2bcd71bf22 --- /dev/null +++ b/research/kappa-rebuild/bit/stage_projector_probe.py @@ -0,0 +1,226 @@ +#!/usr/bin/env python3 +"""Independent-stage physical idle allocation in a fixed scalar controller. + +Each local stage may apply its D_s to any additional physical fields at +the same time cut. Conjugating this field projector by the exact Q-Laurent +prefix exposes whether it is constant and compatible with the final Q +eigenspaces. Compatible stage projectors are simultaneously refined. +This is a broader discovery probe, with native hosting and transfer open. +Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import json +import math +import time + +import complex_controller_factor as C +import mixed_productive_bit as B + +HERE = Path(__file__).resolve().parent + + +def addpoly(a, b, shift=0, coefficient=1): + result = dict(a) + for power, value in b.items(): + key = power+shift + result[key] = result.get(key, 0)+coefficient*value + if not result[key]: + del result[key] + return result + + +def mulpoly(a, b): + out = {} + for power, value in a.items(): + out = addpoly(out, b, power, value) + return out + + +def scalar_prefixes(p): + n = 2*p + identity = [[{0:1} if i == j else {} for j in range(n)] for i in range(n)] + matrix = [[dict(x) for x in row] for row in identity] + inverse = [[dict(x) for x in row] for row in identity] + cuts, events = [], [] + + def add(d, s, coefficient, q=0): + matrix[d] = [addpoly(a, b, q, coefficient) for a, b in zip(matrix[d], matrix[s])] + for row in inverse: + row[s] = addpoly(row[s], row[d], q, -coefficient) + events.append(["ADD", d, s, coefficient, q]) + + def local(stage, pair, reverse): + x, y = 2*pair, 2*pair+1 + if reverse: + x, y = y, x + add(y, x, -1 if reverse else 1, 1 if reverse else -1) + transported = [] + for k in range(n): + polynomial = {} + for i in range(n): + for j in range(n): + for power, value in mulpoly(inverse[i][k], matrix[k][j]).items(): + polynomial.setdefault(power, [[0]*n for _ in range(n)])[i][j] += value + transported.append({power:a for power,a in polynomial.items() if any(any(row) for row in a)}) + cuts.append(dict(stage=stage, pair=pair, transported=transported)) + + def bridge(d, s, coefficient): + add(2*d, 2*s, coefficient) + add(2*s+1, 2*d+1, -coefficient) + + local(0, 0, False) + for j in range(p): + local(2*j+1, j, True) + if j < p-1: + bridge(j, j+1, 1) + local(2*j+2, j, False) + if j < p-1: + bridge(j, j+1, -1) + for j in reversed(range(p-1)): + bridge(j+1, j, 1) + for j in range(p): + matrix[2*j], matrix[2*j+1] = matrix[2*j+1], [{q:-v for q,v in entry.items()} for entry in matrix[2*j]] + coefficients = {} + for i, row in enumerate(matrix): + for j, entry in enumerate(row): + for q, value in entry.items(): + coefficients.setdefault((q,), [[Q(0)]*n for _ in range(n)])[i][j] += value + coefficients = {q:a for q,a in coefficients.items() if any(any(row) for row in a)} + factor = C.factor(coefficients) + B.need(factor is not None, "Scalar Q controller diagonalizes") + total = [[sum(a[i][j] for a in coefficients.values()) for j in range(n)] for i in range(n)] + total_inverse = B.rational_inverse(total) + end = {q[0]:B.matrix_multiply(total_inverse,a) for q,a in coefficients.items()} + return cuts, end, factor, events + + +def commute(a, b): + return multiply(a,b) == multiply(b,a) + + +def multiply(a, b): + n = len(a) + output = [[0]*n for _ in range(n)] + for i,row in enumerate(a): + for k,value in enumerate(row): + if value: + for j,second in enumerate(b[k]): + if second: + output[i][j] += value*second + return output + + +def candidates(cut, end, p): + n = 2*p + active = 3 << (2*cut["pair"]) + result = [] + for mask in range(1 << n): + if mask & active != active: + continue + coefficients = {} + for k in range(n): + if mask >> k & 1: + for q, a in cut["transported"][k].items(): + target = coefficients.setdefault(q, [[0]*n for _ in range(n)]) + for i in range(n): + for j in range(n): + target[i][j] += a[i][j] + coefficients = {q:a for q,a in coefficients.items() if any(any(row) for row in a)} + if set(coefficients) != {0}: + continue + a = coefficients[0] + if not all(commute(a,b) for b in end.values()): + continue + result.append(dict(field_mask=mask, rank=mask.bit_count(), projector=a)) + return result + + +def refine(joint, projector, stage): + result = {} + for (q, mask), a in joint.items(): + yes = multiply(a, projector) + no = [[x-y for x,y in zip(row, second)] for row,second in zip(a,yes)] + if any(any(row) for row in yes): + result[(q,mask | (1 << stage))] = yes + if any(any(row) for row in no): + result[(q,mask)] = no + return result + + +def search(p, Ctarget=0.003, limit=200000): + start = time.monotonic() + cuts, end, factor, events = scalar_prefixes(p) + choices = [sorted(candidates(cut,end,p),key=lambda c:(c['rank'],c['field_mask'])) for cut in cuts] + print(json.dumps(dict(pairs=p, stage_candidate_counts=list(map(len,choices)))),flush=True) + n = 2*p + end = {q:[[int(x) if x.denominator==1 else x for x in row] for row in a] for q,a in end.items()} + compatibility = {} + for s, menu in enumerate(choices): + for t in range(s): + compatibility[(s,t)] = [[commute(a['projector'],b['projector']) for b in choices[t]] for a in menu] + leaves, best, compatible = 0, None, [] + local = {1:2487,2:713,3:224,4:102,5:20,8:8} + helpers = {5:28,6:18,7:56,8:93,9:213} + padding = sum(count*(9-rank)**(1-Ctarget) for rank,count in helpers.items() if rank<9) + base_paid = (2*p+1)*(sum(count*rank**(1-Ctarget) for rank,count in local.items())+padding) + helper_credit = (2*p+1)*sum(helpers.values())*9**(1-Ctarget) + histograms = {} + + def visit(stage, selected, joint, extra): + nonlocal leaves, best + if leaves >= limit: + return + if stage == len(choices): + leaves += 1 + targets = Counter() + for (q, mask), a in joint.items(): + rank = sum(a[i][i] for i in range(n)) + B.need(rank.denominator == 1 if isinstance(rank,Q) else True, "Projector rank integral") + targets[1+8*mask.bit_count()] += 112*int(rank) + margin = helper_credit+sum(count*r**(1-Ctarget) for r,count in targets.items())-base_paid-extra*112*8**(1-Ctarget) + case = dict(field_masks=[choices[s][j]["field_mask"] for s,j in enumerate(selected)], + extra_rank8_fields=extra, targets=dict(sorted(targets.items())), + margin_at_target=margin, + diagonal_spectral_blocks=[dict(q=q, stages=[s for s in range(2*p+1) if mask>>s&1], + multiplicity=int(sum(a[i][i] for i in range(n)))) + for (q,mask),a in sorted(joint.items())]) + signature=json.dumps(case['targets'],sort_keys=True)+':'+str(extra) + histograms.setdefault(signature,case) + if best is None or margin > best["margin_at_target"]: + best=case + return + for index,candidate in enumerate(choices[stage]): + if all(compatibility[(stage,s)][index][j] for s,j in enumerate(selected)): + visit(stage+1, selected+[index], refine(joint,candidate["projector"],stage), extra+candidate["rank"]-2) + + visit(0, [], {(q,0):a for q,a in end.items()}, 0) + return dict(schema="independent-stage-field-projector-probe/1", pairs=p, + candidate_counts=list(map(len,choices)), compatible_leaves=leaves, + limit=limit, enumeration_complete=leaves> i) & 1: + out ^= columns[i] + return out + + +def normalize(terms): + counts = Counter() + for axis, coefficient in terms: + counts[axis] += coefficient + return tuple(sorted((a, c % 4) for a, c in counts.items() if a and c % 4)) + + +def evaluate(terms, address): + return sum(coefficient*((axis & address).bit_count() & 1) for axis, coefficient in terms) % 4 + + +def discrepancy(old, new, columns, shift): + return normalize([(act(columns, a), c) for a, c in old]+[(shift, 2)]+[(a, -c) for a, c in new]) + + +def main(): + started = time.monotonic() + receipt = json.loads(gzip.decompress(INPUT.read_bytes())) + source = HERE.parent/'controller/frame_exchange.py' + need(digest(source) == receipt['generator_sha256'], 'Actual contact-generator source pin') + for path, expected in receipt['dependency_hashes'].items(): + need(digest(ROOT/path) == expected, 'Actual contact dependency pin '+path) + phase_path = ROOT/'research/kappa-next-regime/direct-decoder/sources/DATA-PHASES.json' + need(digest(phase_path) == receipt['immutable_source_pins']['DATA-PHASES.json'], 'Actual phase packet pin') + phases = json.loads(phase_path.read_text()) + points = (0,)+AXES+tuple(a ^ b for a, b in combinations(AXES, 2)) + need(len(points) == 2702, 'Degree-two determining address set') + phase_patterns = {}; count = 0; witnesses = [] + for stage in range(9): + row = receipt['H_wrapper_contacts'][str(stage)] + pre_common = () if stage == 0 else normalize([(a, 1) for a in AXES[9:9+8*(stage-1)]]+ + [(a, 1) for a in AXES[9+8*(stage-1):9+8*stage]]) + post_common = normalize((a, 1) for a in AXES[:9+8*stage]) + need(normalize(row['preY_common']) == pre_common and normalize(row['postX_common']) == post_common, + 'Claimed common birth and completed frames') + need(len(row['preY_views']) == len(row['postX_views']) == 112, 'Every port has both contacts') + for kind in ('preY', 'postX'): + for port, view in enumerate(row[kind+'_views']): + g = tuple(view['owner_columns']); inv = tuple(view['inverse_owner_columns']) + need(len(g) == len(inv) == 73 and all(0 <= x < (1 << 73) for x in g+inv), 'Actual73-axis owner maps') + need(all(act(g, g[i]) == AXES[i] for i in range(73)), 'Each owner route is an involution') + need(all(act(g, inv[i]) == act(inv, g[i]) == AXES[i] for i in range(73)), 'Serialized inverse owner') + need(view['inverse_physical_shift'] == act(inv, view['physical_shift']), 'Inverse affine translation') + if kind == 'preY': + old = () if stage == 0 else normalize([(a, 1) for a in AXES[:9+8*(stage-1)]]+ + [(phases['recipes'][port]['q'], -1)]) + new = pre_common + else: + old = new = post_common + delta = discrepancy(old, new, g, view['physical_shift']) + if delta not in phase_patterns: + bad = next((address for address in points if evaluate(delta, address)), None) + need(bad is None, 'Direct Fourier-symbol phase mismatch '+repr((stage, kind, port, bad))) + phase_patterns[delta] = len(points) + inverse_delta = discrepancy(new, old, inv, view['inverse_physical_shift']) + if inverse_delta not in phase_patterns: + bad = next((address for address in points if evaluate(inverse_delta, address)), None) + need(bad is None, 'Inverse direct-symbol phase mismatch') + phase_patterns[inverse_delta] = len(points) + need(view['global_phase_Z4'] == 0 and evaluate(delta, 0) == 0, 'Absolute zero-frequency phase') + if not witnesses and stage == 1: + wrong = discrepancy(old, new, g, view['physical_shift'] ^ 1) + address = next(x for x in points if evaluate(wrong, x)) + witnesses.append(dict(stage=stage, contact=kind, port=port, mutated_shift_xor=1, + rejecting_address=address, nonzero_phase_Z4=evaluate(wrong, address))) + count += 1 + need(count == 2016 and witnesses, 'Complete9-stage112-port two-contact inventory') + result = dict(schema='independent-direct-symbol-H-contact-audit/1', + status='PASS_ALL_2016_CONTACTS_AND_INVERSES_WITH_GLOBAL_PHASE', + generator_sha256=digest(Path(__file__)), input_receipt_sha256=digest(INPUT), + input_generator_sha256=receipt['generator_sha256'], phase_packet_sha256=digest(phase_path), + contact_count=count, determining_addresses_per_unique_phase_pattern=len(points), + unique_phase_patterns=len(phase_patterns), all_owner_routes_involutive=True, + all_affine_maps_have_explicit_checked_inverses=True, + backward_orientation='The physical views are real permutations/translations; conjugating the checked identity proves the dual orientation.', + H_stage_identification='At both contacts the normalized logical map is D_s H D_s, with D_s diagonal in port index and each owner route squaring to identity.', + negative_control=witnesses[0], seconds=time.monotonic()-started, + proof_scope='Exact finite frame/phase contact admission only. It does not remove terminal monodromy or establish a complete caller.', + no_new_kappa_claimed=True) + output = HERE/'FRAME-CONTACT-INDEPENDENT-AUDIT.json' + output.write_text(json.dumps(result, indent=2, sort_keys=True)+'\n') + print(json.dumps(result, sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-FORWARD-LOGGER.json.gz b/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-FORWARD-LOGGER.json.gz new file mode 100644 index 0000000000..c833421e34 Binary files /dev/null and b/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-FORWARD-LOGGER.json.gz differ diff --git a/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-H112-NATIVE.json.gz b/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-H112-NATIVE.json.gz new file mode 100644 index 0000000000..bcde0b5049 Binary files /dev/null and b/research/kappa-rebuild/complex/balanced-certificates/FOUR-base-BALANCED-H112-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-FORWARD-LOGGER.json.gz b/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-FORWARD-LOGGER.json.gz new file mode 100644 index 0000000000..c833421e34 Binary files /dev/null and b/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-FORWARD-LOGGER.json.gz differ diff --git a/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-H112-NATIVE.json.gz b/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-H112-NATIVE.json.gz new file mode 100644 index 0000000000..bcde0b5049 Binary files /dev/null and b/research/kappa-rebuild/complex/balanced-certificates/FOUR-extra-BALANCED-H112-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/complex/balanced-certificates/RESULT.json b/research/kappa-rebuild/complex/balanced-certificates/RESULT.json new file mode 100644 index 0000000000..9d5830cbf3 --- /dev/null +++ b/research/kappa-rebuild/complex/balanced-certificates/RESULT.json @@ -0,0 +1 @@ +{"no_new_kappa_claimed":true,"providers":{"FOUR-base":{"H":{"artifact":"FOUR-base-BALANCED-H112-NATIVE.json.gz","checks":{"backward":{"all_408_helpers_exactly_restored_in_logical_coordinates":true,"all_new_coefficients_dyadic":true,"all_scalar_frames_checked":true,"direction":"backward","endpoint":"H","event_counts":{"adapter":978,"center":8,"climb":3517,"phase":11,"residual_adapter":390,"scalar":14322,"warp":240},"exact_endpoint":true,"independent_persistent_input_fields":632,"independently_initialized_owned_C_fields":1,"mismatch_examples":[],"owned_C_fields_zero_at_exit":true,"paid_profile":{"1":2463,"2":716,"3":227,"4":71,"5":48},"paid_rank_mass":5100,"prefix":{"l1_witness":{"event":17089,"role":["L",108]},"max_abs_coefficient":"48","max_abs_numerator":67,"max_denominator":64,"max_formal_row_l1":"2034","scope":"Exact logical ideal prefixes; no rounded recursive error bound."},"real_scalar_allowance":118864},"forward":{"all_408_helpers_exactly_restored_in_logical_coordinates":true,"all_new_coefficients_dyadic":true,"all_scalar_frames_checked":true,"direction":"forward","endpoint":"H","event_counts":{"adapter":978,"center":8,"climb":3517,"phase":11,"residual_adapter":390,"scalar":14322,"warp":240},"exact_endpoint":true,"independent_persistent_input_fields":632,"independently_initialized_owned_C_fields":1,"mismatch_examples":[],"owned_C_fields_zero_at_exit":true,"paid_profile":{"1":2463,"2":716,"3":227,"4":71,"5":48},"paid_rank_mass":5100,"prefix":{"l1_witness":{"event":16396,"role":["L",224]},"max_abs_coefficient":"32","max_abs_numerator":32,"max_denominator":32,"max_formal_row_l1":"457","scope":"Exact logical ideal prefixes; no rounded recursive error bound."},"real_scalar_allowance":118864}},"increment":{"owned_C_fields":1,"profile":{"1":8},"real_scalar_allowance":3504},"new_child_recipes":{"backward":[{"event":56,"exact_inverse_port_gate_including_phase":true,"physical_reference":"original external endpoint gauge; 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Substantial OpenAI assistance. No native body is regenerated here; +this binds the complete native receipt and checks the new matrix implication. +""" +from pathlib import Path +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +import rebuild_native as N + + +def main(): + p, v, ce, predecessor, pins = N.authenticated() + geo = N.geometry(v.PACKET) + queries = [{a: N.Q(1), **geo['L'][k]} for k, a in enumerate(geo['selected'])] + for i, query in enumerate(queries): + for j in range(112): + value = sum((coefficient*geo['H'][r].get(j, 0) for r, coefficient in query.items()), N.Q(0)) + N.need(value == -query.get(j, 0), 'Exact C H = -C at row/column '+repr((i, j))) + receipt_path = HERE/'extension-certificates/RESULT.json' + receipt = N.read(receipt_path) + for provider in receipt['providers'].values(): + info = provider['embedded_logger'] + N.need(N.digest(receipt_path.parent/info['artifact']) == info['sha256'], 'Existing complete native embedded-logger artifact') + result = dict(schema='fresh-native-query-composition-contracts/1', + generator_sha256=N.digest(Path(__file__)), native_extension_receipt_sha256=N.digest(receipt_path), + reconstruction_source_sha256=N.digest(HERE/'rebuild_native.py'), + status='PASS_EXACT_C_H_NEGATIVE_C_AND_SOURCE_BOUND_NATIVE_COMPOSITION', + C='[L,I8] in ordinary/selected port order', C_H_equals_negative_C=True, + checked_matrix_coefficients=8*112, + embedded_logger_body='Balanced H forward; X is unchanged before the logging suffix.', + embedded_logger_endpoint='X_out=X_in; Y_out=Y_in+H X_in; d_out=d_in+C X_in.', + append_postX_H_after_logging='X_new=H X_in; d_out=d_in+C X_in=d_in-C X_new.', + logging_after_postX_H='d_out=d_in+C H X_in=d_in-C X_in.', + wrapped_H_then_logging='X_out=H X_in; Y_out=H(Y_in+X_in); d_out=d_in-C X_in.', + helper_output_credit='The eight dirty cache outputs require explicit whole-caller closure before independent helper credit.', + no_new_native_children_for_any_of_these_scalar_compositions=True, + no_new_kappa_claimed=True) + N.save(HERE/'COMPOSITION-CONTRACTS.json', result) + print(result['status']) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/complex/extension-certificates/FOUR-base-EMBEDDED-LOGGER.json.gz b/research/kappa-rebuild/complex/extension-certificates/FOUR-base-EMBEDDED-LOGGER.json.gz new file mode 100644 index 0000000000..b441289b0d Binary files /dev/null and b/research/kappa-rebuild/complex/extension-certificates/FOUR-base-EMBEDDED-LOGGER.json.gz differ diff --git a/research/kappa-rebuild/complex/extension-certificates/FOUR-base-SCALAR-WRAPPED-H.json.gz b/research/kappa-rebuild/complex/extension-certificates/FOUR-base-SCALAR-WRAPPED-H.json.gz new file mode 100644 index 0000000000..40c352ab91 Binary files /dev/null and b/research/kappa-rebuild/complex/extension-certificates/FOUR-base-SCALAR-WRAPPED-H.json.gz differ diff --git a/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-EMBEDDED-LOGGER.json.gz b/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-EMBEDDED-LOGGER.json.gz new file mode 100644 index 0000000000..b441289b0d Binary files /dev/null and b/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-EMBEDDED-LOGGER.json.gz differ diff --git a/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-SCALAR-WRAPPED-H.json.gz b/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-SCALAR-WRAPPED-H.json.gz new file mode 100644 index 0000000000..40c352ab91 Binary files /dev/null and b/research/kappa-rebuild/complex/extension-certificates/FOUR-extra-SCALAR-WRAPPED-H.json.gz differ diff --git a/research/kappa-rebuild/complex/extension-certificates/RESULT.json b/research/kappa-rebuild/complex/extension-certificates/RESULT.json new file mode 100644 index 0000000000..756c3d97f6 --- /dev/null +++ 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diff --git a/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz b/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz new file mode 100644 index 0000000000..4a8c4ed26e Binary files /dev/null and b/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-CANONICAL.json.gz differ diff --git a/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz b/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz new file mode 100644 index 0000000000..b46d44c6ee Binary files /dev/null and b/research/kappa-rebuild/complex/full-identity-certificates/THREE-base-FULL-IDENTITY-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-CANONICAL.json.gz b/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-CANONICAL.json.gz new file mode 100644 index 0000000000..d4b93d28ba Binary files /dev/null and b/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-CANONICAL.json.gz differ diff --git a/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-NATIVE.json.gz b/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-NATIVE.json.gz new file mode 100644 index 0000000000..b46d44c6ee Binary files /dev/null and b/research/kappa-rebuild/complex/full-identity-certificates/THREE-extra-FULL-IDENTITY-NATIVE.json.gz differ diff --git a/research/kappa-rebuild/complex/full_identity_rebuild.py b/research/kappa-rebuild/complex/full_identity_rebuild.py new file mode 100644 index 0000000000..7cff9bff0f --- /dev/null +++ b/research/kappa-rebuild/complex/full_identity_rebuild.py @@ -0,0 +1,173 @@ +#!/usr/bin/env python3 +"""Restore a full-identity native supplier with zero external cleanup bank. + +Apache-2.0. Substantial OpenAI assistance. The full-identity ancestor supplies +the original center cuts and compensation; the independently retained later +cleanup supplies a legal scheduled alias of all91 old buffers. A fresh native +replay checks their actual composition. Eight sequential center fields reuse +one explicitly cleared C0. Each rank8 center is optionally split into rank5 +and rank3 with exact phase decoding, keeping a strict child-rank ceiling5. +""" +from collections import Counter +from copy import deepcopy +from pathlib import Path +import argparse +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +import rebuild_native as N +import native_extensions as X + +INTEGRATED = N.ROOT/'research/kappa-global-routing/integrated-complex' +INTEGRATED_RECEIPT_SHA = '70c3d5b1c1d6a18afb29ea1febec3c619ff2b1e38a09c44db7828794ea095429' + + +def reuse_centers(words): + words = deepcopy(words) + for program in words.values(): + for event in program: + if event[0] == 'center': + event[1] = 0 + elif event[0] == 'scalar': + g = event[1] + for role in (g[1:3] if g[0] in ('add', 'copy') else g[1:2]): + if role[0] == 'C': + role[1] = 0 + return words + + +def compose_full_identity(v, tag, integrated_receipt): + source = v.CORE/'sources'/(tag+'-CANONICAL.json.gz') + scheduled = INTEGRATED/(tag+'-CANONICAL.json.gz') + N.need(N.digest(scheduled) == integrated_receipt['providers'][tag]['artifacts'][scheduled.name], + 'Authenticated scheduled cleanup input') + original = N.read(source); newer = N.read(scheduled) + changes = {k for k in set(original) | set(newer) if original.get(k) != newer.get(k)} + N.need(changes == {'centers', 'center_frames', 'cut', 'compensation', 'cleanup', 'local_profile', 'zero_buffers'}, + 'Exact ancestor-to-clean-J change inventory') + c = deepcopy(original) + c['cleanup'] = deepcopy(newer['cleanup']) + c['zero_buffers'] = [] + N.need(len(c['centers']) == 8 and len(c['cut']) == 176, 'Retained full-identity center response') + words = reuse_centers(v.emit_original(c)) + return c, words, dict(original_canonical_sha256=N.digest(source), scheduled_canonical_sha256=N.digest(scheduled), + changed_fields_from_original=['cleanup', 'zero_buffers'], + ordinary_compensation_and_all_eight_center_cuts_retained=True) + + +def split_centers(p, ce, canonical, words): + c = deepcopy(canonical); result = {}; phases = {}; profile = Counter({int(k): n for k, n in c['local_profile'].items()}) + N.need(profile[8] == 8, 'Eight old rank8 children') + del profile[8]; profile[3] += 8; profile[5] += 8 + c['local_profile'] = {str(k): n for k, n in sorted(profile.items())} + for direction, program in words.items(): + out = []; detail = [] + for event in program: + if event[0] != 'center': + out.append(deepcopy(event)); continue + _, role, old, new, rank = event + N.need(role == 0 and rank == 8 and {len(c['frames'][old]), len(c['frames'][new])} == {0, 8}, + 'Single eight-dimensional old center transport') + high = old if len(c['frames'][old]) == 8 else new + basis = c['frames'][high][:3] + if basis not in c['frames']: + c['frames'].append(basis) + mid = c['frames'].index(basis) + path = [old, mid, new]; pieces = []; recipes = [] + for a, b in zip(path, path[1:]): + width = abs(len(c['frames'][b])-len(c['frames'][a])) + part = ce.canonical_frame(9, c['frames'][a]).adjoint().then(ce.canonical_frame(9, c['frames'][b])) + if direction == 'backward': + part = p.conjugate(ce, part) + recipe = N.lower_exact(p, ce, part) + N.need(recipe['rank'] == width and width in (3, 5), 'Split child exact rank') + out.append(['center', 0, a, b, width]); pieces.append(part) + recipes.append(dict(old_frame=a, new_frame=b, recipe=recipe)) + whole = ce.canonical_frame(9, c['frames'][old]).adjoint().then(ce.canonical_frame(9, c['frames'][new])) + if direction == 'backward': + whole = p.conjugate(ce, whole) + product = pieces[0].then(pieces[1]) + N.need(product.tab == whole.tab and product.zero_state() == whole.zero_state(), + 'Two split children equal old center including absolute phase') + detail.append(dict(old_frame=old, intermediate_frame=mid, new_frame=new, + intermediate_basis=basis, pieces=recipes, product_exact_including_phase=True)) + N.need(len(detail) == 8, 'All eight center paths split') + result[direction] = out; phases[direction] = detail + return c, result, phases + + +def bind_helper_boundaries(c, tag): + source = HERE/'fresh-certificates'/(tag+'-HELPER-BOUNDARIES.json.gz') + receipt = N.read(HERE/'fresh-certificates/RESULT-helpers.json') + expected = receipt['providers'][tag]['helper_boundaries']['sha256'] + N.need(N.digest(source) == expected, 'Fresh helper boundary certificate pin') + helper = N.read(source) + for r in helper['recipes']: + role = r['role'] + N.need(c['frames'][c['start'][role]] == r['certificate']['start_basis'] and + c['frames'][c['final'][role]] == r['certificate']['end_basis'], + 'Full-identity helper has identical exact endpoint references') + return dict(certificate_sha256=expected, certificate=str(source.relative_to(N.ROOT)), + count=408, widths=helper['widths'], total_width=helper['total_width'], + both_orientations_and_global_phases_bound=True) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--out', type=Path, default=HERE/'full-identity-certificates') + args = ap.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + p, v, ce, predecessor, inherited = N.authenticated(); geo = N.geometry(v.PACKET) + N.need(N.digest(INTEGRATED/'RESULT.json') == INTEGRATED_RECEIPT_SHA, 'Pinned scheduled cleanup receipt') + scheduled_receipt = N.read(INTEGRATED/'RESULT.json') + result = dict(schema='fresh-full-identity-zero-bank-native-supplier/1', + generator_sha256=N.digest(Path(__file__)), reconstruction_source_sha256=N.digest(HERE/'rebuild_native.py'), + independent_interpreter_sha256=p.CHECKER_SHA, + independent_prefix_evaluator_sha256=N.digest(HERE/'native_extensions.py'), + integrated_cleanup_receipt_sha256=INTEGRATED_RECEIPT_SHA, + source_closure_verified=inherited, no_new_kappa_claimed=True, providers={}) + for tag in N.TAGS: + c, words, composition = compose_full_identity(v, tag, scheduled_receipt) + unsplit = {direction: v.replay(program, c, geo['R'], direction == 'backward', False) + for direction, program in words.items()} + c, split, phases = split_centers(p, ce, c, words) + checks = {} + for direction, program in split.items(): + checked = v.replay(program, c, geo['R'], direction == 'backward', False) + rows, prefixes = X.literal_replay(v, c, program, direction == 'backward') + N.need(all(rows['L', i] == v.expected(('L', i), geo['R'], direction == 'backward', False) for i in range(632)), + 'Second all632-column endpoint implementation') + N.need(checked['conservative_real_scalar_primitive_allowance'] == + unsplit[direction]['conservative_real_scalar_primitive_allowance'], 'Splitting adds no scalar macro') + N.need(max(checked['paid_profile']) == 5 and checked['explicitly_owned_C_fields_cleared'] == 1, + 'Child ceiling5 and one reusable owned C0') + checks[direction] = dict(independent_native_replay=checked, independent_prefix_and_endpoint=prefixes) + bad = deepcopy(split['forward']) + remove = next(i for i, e in enumerate(bad) if e[0] == 'scalar' and e[1][0] == 'add' and e[1][2] == ['C', 0]) + bad.pop(remove) + rejected = False + try: + v.replay(bad, c, geo['R'], False, False) + except ValueError as error: + rejected = 'operator mismatch' in str(error) + N.need(rejected, 'Dropped center response negative control rejected') + cp = args.out/(tag+'-FULL-IDENTITY-CANONICAL.json.gz') + wp = args.out/(tag+'-FULL-IDENTITY-NATIVE.json.gz') + result['providers'][tag] = dict(composition=composition, unsplit_replays=unsplit, split_replays=checks, + split_center_phase_recipes=phases, helper_boundary=bind_helper_boundaries(c, tag), + artifacts={cp.name:N.save(cp, c), wp.name:N.save(wp, split)}, + all_632_persistent_inputs_arbitrary=True, all_408_helpers_autonomous=True, + external_cleanup_fields=0, owned_C_fields=1, owned_C_input_arbitrary_and_erased=True, + max_child_rank=5, no_H_involution_or_query_controller_needed=True, + dropped_center_response_negative_control_rejected=True) + N.save(args.out/'RESULT.json', result) + print('PASS full identity', tag, checks['forward']['independent_native_replay']['paid_profile'], flush=True) + result['status'] = 'PASS_FULL_IDENTITY_ALL_FOUR_BOTH_DIRECTIONS_632_ARBITRARY_FIELDS_ZERO_EXTERNAL_BANK_CHILD_CEILING5' + result['scope'] = ('Complete ideal local native supplier and helper boundaries. Outer productive job ownership, ' + 'complete recurrence, scalar and storage charges, and rounded induction remain caller obligations.') + N.save(args.out/'RESULT.json', result) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/complex/native_extensions.py b/research/kappa-rebuild/complex/native_extensions.py new file mode 100644 index 0000000000..7f84894efc --- /dev/null +++ b/research/kappa-rebuild/complex/native_extensions.py @@ -0,0 +1,233 @@ +#!/usr/bin/env python3 +"""Native embedded query logger and common-frame H conjugation contacts. + +Apache-2.0. Substantial OpenAI assistance. These are fresh constructions with +explicit arbitrary-input endpoints. An embedded cache output is not claimed +as an independent clean helper output until a whole caller erases its defect. +""" +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path +import argparse +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +import rebuild_native as N +import power_two_balance as B + +B_SHA = '76929bb3887911ccf100c4593574b8a1c82ee64e96f939a8aca196869438b424' + + +def common_frame_h(geo, offset, frame): + """H = diag(I,-I) S_L S_-K S_L, with all shear ADDs having coefficient ±1. + + Balancing only the eight selected coordinates avoids scaling104 ordinary + coordinates. No private field and no recursive transport are introduced. + Precondition: ALL112 physical fields share the stated actual frame gauge. + """ + A = geo['selected'] + out = [['phase', 'fresh_common_frame_H']] + def scale(n, d): + out.extend(N.scalar(['scale', ['L', offset+a], n, d], frame) for a in A) + def left(): + for k, a in enumerate(A): + for u, value in sorted(geo['L'][k].items()): + coefficient = 2*value + N.need(abs(coefficient) == 1, 'Balanced unit L coefficients') + out.append(N.scalar(['add', ['L', offset+a], ['L', offset+u], int(coefficient), 1], frame)) + scale(2, 1); left(); scale(1, 4) + for k, a in enumerate(A): + for u, value in sorted(geo['L'][k].items()): + coefficient = -2*value + out.append(N.scalar(['add', ['L', offset+u], ['L', offset+a], int(coefficient), 1], frame)) + scale(4, 1); left(); scale(-1, 2) + N.need(B.scalar_sum(out) == 10096, 'Exact common-frame H scalar allowance') + return out + + +def embedded_cache_roles(canonical): + roles = [i for i in range(224, 632) if canonical['start'][i] == 0 and canonical['final'][i] == 1] + N.need(len(roles) == 213, '213 genuinely cold, full-ended native helper roles') + return roles[:8] + + +def embedded_logger_suffix(canonical, geo, roles=None): + roles = embedded_cache_roles(canonical) if roles is None else list(roles) + N.need(len(roles) == len(set(roles)) == 8, 'Eight distinct embedded caches') + N.need(all(224 <= i < 632 and canonical['start'][i] == 0 and canonical['final'][i] == 1 for i in roles), + 'Embedded logger only uses cold full-ended physical helper references') + out = [['phase', 'fresh_embedded_dirty_logger']] + for k, a in enumerate(geo['selected']): + r = roles[k] + out += [N.scalar(['scale', ['L', r], 2, 1], 1), + N.scalar(['add', ['L', r], ['L', a], 2, 1], 1)] + out += [N.scalar(['add', ['L', r], ['L', u], int(2*q), 1], 1) + for u, q in sorted(geo['L'][k].items())] + out.append(N.scalar(['scale', ['L', r], 1, 2], 1)) + N.need(B.scalar_sum(out) == 3440, 'Embedded logger exact scalar bill') + return out + + +def make_embedded_logger(canonical, base_word, geo): + word = B.make_balanced_h(canonical, base_word, geo)['forward'] + return word+embedded_logger_suffix(canonical, geo) + + +def make_wrapped_h(canonical, base_word, geo): + words = B.make_balanced_h(canonical, base_word, geo) + return {direction: common_frame_h(geo, 112, 0)+word+common_frame_h(geo, 0, 1) + for direction, word in words.items()} + + +def literal_replay(v, c, program, backward=False): + """Second literal evaluator for new endpoints; all632 inputs arbitrary. + + This checker has no endpoint-specific shortcut. It applies every scalar + and free-warp event and checks each contact's frame before comparing rows. + """ + rows = {('L', i): v.identity(('L', i)) for i in range(632)} + rows['C', 0] = v.identity(('C', 0)) + current = {('L', i): f for i, f in enumerate(c['start'])}; current['C', 0] = None + spaces = [v.span(frame) for frame in c['frames']] + profile = Counter(); cost = 0; max_l1 = Q(1); witness = None; max_denominator = 1 + for j, event in enumerate(program): + kind = event[0]; touched = None + if kind == 'scalar': + g, frame = event[1:]; target = tuple(g[1]); touched = target + cost += N.primitive_allowance(g) + if g[0] == 'clear': + N.need(target == ('C', 0), 'Only owned C0 can be cleared') + rows[target] = {}; current[target] = frame + else: + N.need(current[target] == frame, 'Exact scalar target frame') + if g[0] in ('add', 'copy'): + source = tuple(g[2]); N.need(current[source] == frame, 'Exact scalar source frame') + if g[0] == 'copy': + N.need(target == ('C', 0), 'Only owned field overwritten by copy') + rows[target] = dict(rows[source]) + else: + N.need(target != source, 'Distinct reversible ADD arguments') + v.add(rows[target], rows[source], Q(g[3], g[4])) + else: + N.need(g[0] == 'scale', 'Known scalar action') + q = Q(g[2], g[3]); rows[target] = {i: q*x for i, x in rows[target].items()} + elif kind == 'warp': + _, r, op, old, new = event; target = tuple(r); touched = target + N.need(current[target] == old, 'Warp input frame') + rows[target] = v.warp(rows[target], op, backward); current[target] = new + elif kind in ('climb', 'center'): + if kind == 'climb': + _, part, i, old, new, rank = event; target = ('L', i) + N.need(spaces[old] < spaces[new], 'Ascending borrowed climb') + else: + _, i, old, new, rank = event; target = ('C', i) + N.need(spaces[old] < spaces[new] or spaces[new] < spaces[old], 'Nested center transport') + N.need(current[target] == old and rank == abs(len(c['frames'][new])-len(c['frames'][old])), 'Paid rank and source frame') + current[target] = new; profile[rank] += 1 + else: + N.need(kind in ('phase', 'adapter', 'residual_adapter'), 'Known inert native marker') + if touched is not None: + norm = sum(map(abs, rows[touched].values()), Q(0)) + if norm > max_l1: + max_l1 = norm; witness = dict(event=j, role=list(touched)) + max_denominator = max(max_denominator, max((x.denominator for x in rows[touched].values()), default=1)) + N.need([current['L', i] for i in range(632)] == c['final'], 'Final native frames') + N.need(not rows['C', 0], 'Owned C0 cleared at return') + return rows, dict(independent_persistent_columns=632, independent_owned_entry_columns=1, + scalar_frames_checked=True, arbitrary_owned_C0_cleared=True, + paid_profile=dict(sorted(profile.items())), paid_rank_mass=sum(k*x for k, x in profile.items()), + real_scalar_allowance=cost, max_formal_prefix_l1=str(max_l1), max_denominator=max_denominator, + l1_witness=witness) + + +def check_rows(v, rows, geo, kind, backward=False, cache_roles=()): + mismatch = [] + for r in range(632): + actual = rows['L', r] + if kind == 'wrapped' and r < 224: + port = r % 112; wanted = {} + for j, q in geo['H'][port].items(): + v.add(wanted, v.identity(('L', (112 if r >= 112 else 0)+j)), q) + if backward and r < 112: + v.add(wanted, v.identity(('L', 112+j)), -q) + if not backward and r >= 112: + v.add(wanted, v.identity(('L', j)), q) + else: + wanted = v.identity(('L', r)) + if 112 <= r < 224: + for j, q in geo['H'][r-112].items(): + v.add(wanted, v.identity(('L', j)), q) + if r in cache_roles: + k = list(cache_roles).index(r) + for j, q in {geo['selected'][k]: Q(1), **geo['L'][k]}.items(): + v.add(wanted, v.identity(('L', j)), q) + if actual != wanted: + mismatch.append(r) + N.need(not mismatch, 'Fresh exact endpoint mismatch '+repr(mismatch[:5])) + + +def check_scalar_h(v, geo): + """Independent112-column execution without native body or Clifford data.""" + rows = {i: {i: Q(1)} for i in range(112)} + program = common_frame_h(geo, 0, 0) + for event in program: + if event[0] != 'scalar': + continue + g = event[1]; target = g[1][1] + if g[0] == 'scale': + q = Q(g[2], g[3]); rows[target] = {i: q*x for i, x in rows[target].items()} + else: + v.add(rows[target], rows[g[2][1]], Q(g[3], g[4])) + N.need(all(rows[i] == geo['H'][i] for i in range(112)), 'Three balanced shears exactly equal H') + return dict(all_columns=112, real_scalar_allowance=B.scalar_sum(program), + additions=sum(e[0] == 'scalar' and e[1][0] == 'add' for e in program), + unit_add_coefficients=True, owned_fields=0, new_recursive_children=0, + required_physical_precondition='All112 fields occupy one actual common frame reference.') + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--out', type=Path, default=HERE/'extension-certificates') + args = ap.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + N.need(N.digest(HERE/'rebuild_native.py') == B.N_SHA and N.digest(HERE/'power_two_balance.py') == B_SHA, + 'Authenticated fresh prerequisite sources') + p, v, ce, receipt, inherited = N.authenticated(); geo = N.geometry(v.PACKET) + result = dict(schema='fresh-native-helper-log-and-common-frame-conjugation/1', + source_sha256=N.digest(Path(__file__)), reconstruction_source_sha256=B.N_SHA, + balance_source_sha256=B_SHA, scalar_H=check_scalar_h(v, geo), providers={}, no_new_kappa_claimed=True) + for tag in N.TAGS: + c = N.read(N.BASE/(tag+'-CANONICAL.json.gz')); word = N.read(N.BASE/(tag+'-NATIVE-WORD.json.gz')) + base_scalar = B.scalar_sum(word['forward']) + roles = embedded_cache_roles(c); embedded = make_embedded_logger(c, word, geo) + rows, embedded_check = literal_replay(v, c, embedded) + check_rows(v, rows, geo, 'embedded', cache_roles=roles) + N.need(embedded_check['real_scalar_allowance'] == base_scalar+3504+3440, 'Embedded complete scalar invoice') + wrapped = make_wrapped_h(c, word, geo); wrapped_checks = {} + for direction, program in wrapped.items(): + back = direction == 'backward' + rows, checked = literal_replay(v, c, program, back) + check_rows(v, rows, geo, 'wrapped', back) + N.need(checked['real_scalar_allowance'] == base_scalar+3504+2*10096, 'Wrapped H complete scalar invoice') + wrapped_checks[direction] = checked + ep = args.out/(tag+'-EMBEDDED-LOGGER.json.gz'); wp = args.out/(tag+'-SCALAR-WRAPPED-H.json.gz') + result['providers'][tag] = dict( + embedded_logger=dict(artifact=ep.name, sha256=N.save(ep, embedded), check=embedded_check, + cache_roles=roles, exactly_400_other_helpers_restored=True, + cache_output='d_k + X_A[k] + L[k] X_U', extra_children=0, + extra_persistent_fields=0, extra_owned_fields=0, extra_scalar_allowance=3440, + independent_clean_output_credit_for_eight_caches=False), + scalar_wrapped_H=dict(artifact=wp.name, sha256=N.save(wp, wrapped), checks=wrapped_checks, + endpoint='diag(H,H) times the corresponding identity shear in local logical coordinates', + new_recursive_children_beyond_H=0, extra_scalar_allowance=20192, + all_408_helpers_restored=True, + outer_intercall_frame_identification_certified=False)) + N.save(args.out/'RESULT.json', result) + print('PASS extensions', tag, 'embedded0 extra children; wrapped H20192 extra scalars', flush=True) + result['status'] = 'PASS_LOCAL_NATIVE_ENDPOINTS_WITH_EXPLICIT_OUTER_FRAME_OBLIGATION' + N.save(args.out/'RESULT.json', result) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/complex/power_two_balance.py b/research/kappa-rebuild/complex/power_two_balance.py new file mode 100644 index 0000000000..a55ae0ce78 --- /dev/null +++ b/research/kappa-rebuild/complex/power_two_balance.py @@ -0,0 +1,130 @@ +#!/usr/bin/env python3 +"""Exact power-of-two balancing of the reconstructed H112 and dirty logger. + +Apache-2.0. Substantial OpenAI assistance. This changes only explicit local +coefficient streams and dyadic unit scales; every recursive child is retained. +The exact arbitrary-input verifier and endpoint-gauge phase receipts are reused +from the separately source-bound reconstruction in rebuild_native.py. +""" +from collections import Counter +from copy import deepcopy +from pathlib import Path +import argparse +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +import rebuild_native as N + +N_SHA = 'f94c527cc02215de49350d477618188c4c0da48bc8fb05b4e37b24c482cad7d5' + + +def make_balanced_h(c, word, geo, exponent=1): + """Assemble 2^e v, transport it, deliver 12/2^e times that field.""" + N.need(isinstance(exponent, int) and -8 <= exponent <= 8, 'Bounded integral balancing exponent') + q = N.Q(2)**exponent + result = N.make_h_word(c, word, geo) + for direction, program in result.items(): + out = []; completion = False + for event in program: + if event[0] == 'phase': + completion = event[1] == 'fresh_H112_completion' + if not completion or event[0] != 'scalar': + out.append(event); continue + op, frame = event[1:] + if op[0] == 'copy' and op[1] == ['C', 0]: + out.append(event) + if q != 1: + out.append(N.scalar(['scale', ['C', 0], q.numerator, q.denominator], frame)) + elif op[0] == 'add': + factor = q if op[1] == ['C', 0] else 1/q + amount = N.Q(op[3], op[4])*factor + out.append(N.scalar(op[:3]+[amount.numerator, amount.denominator], frame)) + else: + out.append(event) + result[direction] = out + return result + + +def make_balanced_logger(c, word, geo, exponent=1): + """Scale arbitrary P by 2^e, add 2^e v, scale back; dirty P is preserved.""" + N.need(isinstance(exponent, int) and -8 <= exponent <= 8, 'Bounded integral balancing exponent') + q = N.Q(2)**exponent + suffix = [['phase', 'fresh_balanced_forward_dirty_logger']] + for k, a in enumerate(geo['selected']): + suffix.append(['cache_child', k, 0, 1, 9]) + if q != 1: + suffix.append(N.scalar(['scale', ['P', k], q.numerator, q.denominator], 1)) + suffix.append(N.scalar(['add', ['P', k], ['L', a], q.numerator, q.denominator], 1)) + for p, coefficient in sorted(geo['L'][k].items()): + x = q*coefficient + suffix.append(N.scalar(['add', ['P', k], ['L', p], x.numerator, x.denominator], 1)) + if q != 1: + suffix.append(N.scalar(['scale', ['P', k], q.denominator, q.numerator], 1)) + return word['forward']+suffix + + +def scalar_sum(program): + return sum(N.primitive_allowance(e[1]) for e in program if e[0] == 'scalar') + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--out', type=Path, default=HERE/'balanced-certificates') + args = ap.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + N.need(N.digest(HERE/'rebuild_native.py') == N_SHA, 'Pinned fresh reconstruction source') + p, v, ce, receipt, inherited = N.authenticated(); geo = N.geometry(v.PACKET) + result = dict(schema='fresh-native-power-two-balancing/1', source_sha256=N.digest(Path(__file__)), + reconstruction_source_sha256=N_SHA, no_new_kappa_claimed=True, providers={}, + scope='Finite native all-column and phase certificates; full outer recurrence and rounded induction remain separate.') + for tag in N.TAGS: + c = N.read(N.BASE/(tag+'-CANONICAL.json.gz')); word = N.read(N.BASE/(tag+'-NATIVE-WORD.json.gz')) + base_scalar = scalar_sum(word['forward']) + scan = [] + for exponent in range(-4, 5): + hw = make_balanced_h(c, word, geo, exponent) + lw = make_balanced_logger(c, word, geo, exponent) + scan.append(dict(exponent=exponent, H_increment=scalar_sum(hw['forward'])-base_scalar, + logger_increment=scalar_sum(lw)-base_scalar)) + N.need(min(scan, key=lambda r: r['H_increment'])['exponent'] == 1, 'H minimum in scanned power-two family') + N.need(min(scan, key=lambda r: r['logger_increment'])['exponent'] == 1, 'Logger minimum in scanned power-two family') + hw = make_balanced_h(c, word, geo) + hp = Counter({int(k): n for k, n in c['local_profile'].items()}); hp[1] += 8 + checks = {d: N.replay_extended(v, hw[d], c, geo, d == 'backward', expected_profile=hp) + for d in ('forward', 'backward')} + N.need(all(check['real_scalar_allowance']-base_scalar == 3504 for check in checks.values()), 'H increment3504') + phases = N.h_child_recipes(p, ce, c, hw) + lw = make_balanced_logger(c, word, geo) + lp = Counter({int(k): n for k, n in c['local_profile'].items()}); lp[9] += 8 + logger_check = N.replay_extended(v, lw, c, geo, endpoint='logger', cache_count=8, expected_profile=lp) + N.need(logger_check['real_scalar_allowance']-base_scalar == 3440, 'Logger increment3440') + wrong = deepcopy(lw) + omitted = next(i for i, e in enumerate(wrong) if e[0] == 'scalar' and e[1] == ['scale', ['P', 0], 1, 2]) + wrong.pop(omitted) + control = N.replay_extended(v, wrong, c, geo, endpoint='logger', cache_count=8, allow_mismatch=True) + dirty_gain = N.Q(1) + for event in wrong: + if event[0] == 'scalar' and event[1][:2] == ['scale', ['P', 0]]: + dirty_gain *= N.Q(event[1][2], event[1][3]) + N.need(not control['exact_endpoint'] and dirty_gain == 2 and + any(x['role'] == ['P', 0] for x in control['mismatch_examples']), + 'Missing final dirty-cache rescale exposes arbitrary P coefficient error') + hpfile = args.out/(tag+'-BALANCED-H112-NATIVE.json.gz') + lpfile = args.out/(tag+'-BALANCED-FORWARD-LOGGER.json.gz') + result['providers'][tag] = dict(power_two_scan=scan, + H=dict(artifact=hpfile.name, sha256=N.save(hpfile, hw), checks=checks, new_child_recipes=phases, + increment=dict(profile={'1': 8}, real_scalar_allowance=3504, owned_C_fields=1), saved_scalar_allowance=832), + logger=dict(artifact=lpfile.name, sha256=N.save(lpfile, lw), check=logger_check, + increment=dict(profile={'9': 8}, real_scalar_allowance=3440, arbitrary_P_fields=8), saved_scalar_allowance=784, + missing_final_rescale_negative_control=dict(dirty_P0_coefficient=str(dirty_gain), + mismatches=control['mismatch_examples']))) + N.save(args.out/'RESULT.json', result) + print('PASS balanced', tag, 'H8H1+3504; logger8H9+3440', flush=True) + result['status'] = 'PASS_BALANCED_H_BOTH_DIRECTIONS_AND_FORWARD_LOGGER_ALL_FOUR_PROVIDERS' + N.save(args.out/'RESULT.json', result) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/complex/rebuild_native.py b/research/kappa-rebuild/complex/rebuild_native.py new file mode 100644 index 0000000000..78a7b1b0e6 --- /dev/null +++ b/research/kappa-rebuild/complex/rebuild_native.py @@ -0,0 +1,453 @@ +#!/usr/bin/env python3 +"""Fresh reconstruction of H112, its dirty logger and helper boundary factors. + +Apache-2.0. Substantial OpenAI assistance. This is newly generated evidence; +it does not recreate or assert the identity of files lost in an earlier session. +The immutable predecessor is the attributed function-composition clean-J word. +Logical sparse replay extends the independently pinned verify_buffers grammar. +Physical lowering uses the attributed exact Clifford compiler and independently +decodes each new recipe, comparing tableau AND the exact zero-state phase. +""" +from collections import Counter +from copy import deepcopy +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import importlib.util +import json +import platform +import sys +import time + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +P_PATH = ROOT/'research/kappa-function-composition/complex/prune_native.py' +BASE = P_PATH.parent/'pruned-native' +P_SHA = 'f2b37c183a10922eabc8e0436b426a08866430e96bd0adc7e178951c7fbb5257' +RECEIPT_SHA = 'b62c6c123f43a697802a336ee7178942040c0249c64ebc19cd6aad7ba2728ae7' +CE_SHA = '2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962' +TAGS = ('THREE-base', 'THREE-extra', 'FOUR-base', 'FOUR-extra') +SELECTED = (0, 7, 20, 27, 36, 46, 57, 111) + + +def need(ok, why): + if not ok: + raise ValueError(why) + + +def digest(path): + return sha256(Path(path).read_bytes()).hexdigest() + + +def read(path): + path = Path(path) + raw = path.read_bytes() + return json.loads(gzip.decompress(raw) if path.suffix == '.gz' else raw) + + +def save(path, value): + path = Path(path) + data = json.dumps(value, sort_keys=True, separators=(',', ':')).encode() + path.write_bytes(gzip.compress(data, mtime=0) if path.suffix == '.gz' else data+b'\n') + return digest(path) + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + obj = importlib.util.module_from_spec(spec) + sys.modules[name] = obj + spec.loader.exec_module(obj) + return obj + + +def authenticated(): + need(digest(P_PATH) == P_SHA, 'Pinned clean-J reconstruction utility') + need(digest(BASE/'RESULT.json') == RECEIPT_SHA, 'Pinned clean-J receipt') + p = load(P_PATH, 'fresh_h112_prune') + need(digest(p.CHECKER) == p.CHECKER_SHA, 'Pinned independent native interpreter') + need(digest(p.CLIFFORD) == CE_SHA, 'Pinned phase compiler') + v = load(p.CHECKER, 'fresh_h112_independent') + inherited = v.authenticate() + ce = load(p.CLIFFORD, 'fresh_h112_clifford') + receipt = read(BASE/'RESULT.json') + for tag in TAGS: + for filename, expected in receipt['providers'][tag]['artifacts'].items(): + need(digest(BASE/filename) == expected, 'Pinned predecessor artifact '+filename) + return p, v, ce, receipt, inherited + + +def geometry(packet): + """Return exact sparse R, L and the freshly checked 112 by 112 H.""" + packet = read(packet) if isinstance(packet, (str, Path)) else packet + R = [{int(k): Q(x) for k, x in row.items()} for row in packet['R_rows']] + ordinary = tuple(i for i in range(112) if i not in SELECTED) + need(len(R) == 8, 'Eight query rows') + dot = lambda a, b: sum((x*b.get(j, Q(0)) for j, x in a.items()), Q(0)) + need(all(dot(a, b) == 56*int(i == j) for i, a in enumerate(R) for j, b in enumerate(R)), 'R R^t = 56 I') + need(all(sum((r.get(a, 0)*r.get(b, 0) for r in R), Q(0)) == 4*int(a == b) + for a in SELECTED for b in SELECTED), 'Selected Gram matrix 4 I') + L = [{p: q for p in ordinary if (q := sum((r.get(a, 0)*r.get(p, 0) for r in R), Q(0))/4)} + for a in SELECTED] + need(all(len(row) == 52 and set(map(abs, row.values())) == {Q(1, 2)} for row in L), '416 half-integral L entries') + need(all(dot(a, b) == 13*int(i == j) for i, a in enumerate(L) for j, b in enumerate(L)), 'L L^t = 13 I') + J = [] + H = [] + for i in range(112): + row = {j: q for j in range(112) + if (q := Q(i == j)-sum((r.get(i, 0)*r.get(j, 0) for r in R), Q(0))/4)} + J.append(row) + changed = dict(row) + if i in SELECTED: + k = SELECTED.index(i) + for j, x in {i: Q(1), **L[k]}.items(): + changed[j] = changed.get(j, Q(0))+12*x + if not changed[j]: + del changed[j] + H.append(changed) + need(all(sum((x*H[k].get(j, 0) for k, x in H[i].items()), Q(0)) == int(i == j) + for i in range(112) for j in range(112)), 'Exact H squared equals identity') + need(any(H[i].get(j, 0) != H[j].get(i, 0) for i in range(112) for j in range(112)), 'H is not symmetric') + return dict(selected=SELECTED, ordinary=ordinary, R=R, L=L, J=J, H=H) + + +def kernel_h(packet): + return geometry(packet)['H'] + + +def scalar(op, frame): + return ['scalar', op, frame] + + +def make_h_word(canonical, base_word, geo): + """Forward suffix and separate backward prefix; H is NOT self-transpose.""" + out = deepcopy(base_word) + for backward in (False, True): + correction = [['phase', 'fresh_H112_completion']] + common = 0 if backward else 1 + source_offset = 112 if backward else 0 + target_offset = 0 if backward else 112 + for k, a in enumerate(geo['selected']): + target_frame = canonical['start'][a] if backward else canonical['final'][112+a] + correction += [scalar(['clear', ['C', 0]], common), + scalar(['copy', ['C', 0], ['L', source_offset+a]], common)] + correction += [scalar(['add', ['C', 0], ['L', source_offset+p], q.numerator, q.denominator], common) + for p, q in sorted(geo['L'][k].items())] + correction += [['center', 0, common, target_frame, 1], + scalar(['add', ['L', target_offset+a], ['C', 0], -12 if backward else 12, 1], target_frame), + scalar(['clear', ['C', 0]], target_frame)] + name = 'backward' if backward else 'forward' + out[name] = correction+out[name] if backward else out[name]+correction + return out + + +def make_logger_word(canonical, base_word, geo): + """Eight arbitrary cache fields; the caller owns their dirty baselines.""" + suffix = [['phase', 'fresh_forward_dirty_query_logger']] + for k, a in enumerate(geo['selected']): + suffix += [['cache_child', k, 0, 1, 9], scalar(['add', ['P', k], ['L', a], 1, 1], 1)] + suffix += [scalar(['add', ['P', k], ['L', p], q.numerator, q.denominator], 1) + for p, q in sorted(geo['L'][k].items())] + return base_word['forward']+suffix + + +def primitive_allowance(op): + if op[0] in ('clear', 'copy'): + return 2 + q = Q(op[3], op[4]) if op[0] == 'add' else Q(op[2], op[3]) + n, d = abs(q.numerator), q.denominator + need(n and not d & (d-1), 'Nonzero dyadic scalar') + if op[0] == 'scale': + need(not n & (n-1), 'Dyadic unit SCALE') + return 2*(abs(n.bit_length()-d.bit_length())+int(q < 0)) + need(op[0] == 'add', 'Known scalar macro') + return 2*(3+n.bit_length()-1+n.bit_count()+d.bit_length()-1) + + +def replay_extended(v, program, canonical, geo, backward=False, endpoint='H', cache_count=0, + expected_profile=None, allow_mismatch=False): + """Literal exact all-column native replay, including arbitrary C at entry. + + Transports change canonical coordinate tags and retain logical row values; + separate phase receipts implement the exact physical coordinate change. + Warp words remain free noncommuting symbols, including dual orientation. + """ + need(endpoint in ('H', 'J', 'logger'), 'Known endpoint') + n = len(canonical['start']) + need(n == 632, '632 persistent predecessor roles') + rows = {('L', i): v.identity(('L', i)) for i in range(n)} + current = {('L', i): f for i, f in enumerate(canonical['start'])} + centers = {e[1] for e in program if e[0] == 'center'} + centers |= {r[1] for e in program if e[0] == 'scalar' + for r in (e[1][1:3] if e[1][0] in ('add', 'copy') else e[1][1:2]) + if r[0] == 'C'} + need(centers <= {0}, 'Only one owned completion field') + for i in centers: + rows['C', i] = v.identity(('C', i)); current['C', i] = None + for i in range(cache_count): + rows['P', i] = v.identity(('P', i)); current['P', i] = 0 + spaces = [v.span(vectors) for vectors in canonical['frames']] + dimensions = [len(s).bit_length()-1 for s in spaces] + profile = Counter(); counts = Counter(); scalar_cost = 0 + max_l1 = Q(1); max_coefficient = Q(1); max_numerator = 1; max_denominator = 1 + witness = {'event': -1, 'role': ['L', 0]} + for i, e in enumerate(program): + kind = e[0]; counts[kind] += 1; touched = None + if kind in ('climb', 'center', 'cache_child'): + if kind == 'climb': + _, part, j, old, new, rank = e; role = ('L', j) + need(spaces[old] < spaces[new], 'Strict ascending borrowed climb') + else: + _, j, old, new, rank = e + role = ('C' if kind == 'center' else 'P', j) + need(spaces[old] < spaces[new] or spaces[new] < spaces[old], 'Nested transport') + need(current[role] == old and rank == abs(dimensions[new]-dimensions[old]), 'Transport frame and exact rank') + current[role] = new; profile[rank] += 1 + elif kind == 'warp': + _, r, op, old, new = e; role = tuple(r) + need(current[role] == old, 'Warp source frame') + rows[role] = v.warp(rows[role], op, backward); current[role] = new; touched = role + elif kind == 'scalar': + op, frame = e[1:]; role = tuple(op[1]); touched = role + scalar_cost += primitive_allowance(op) + if op[0] == 'clear': + need(role[0] == 'C', 'No clearing arbitrary persistent inputs') + rows[role] = {}; current[role] = frame + else: + need(current[role] == frame, 'Scalar target frame at event '+str(i)) + if op[0] in ('add', 'copy'): + source = tuple(op[2]); need(current[source] == frame, 'Scalar source frame at event '+str(i)) + if op[0] == 'copy': + need(role[0] == 'C', 'Copy targets owned field') + rows[role] = dict(rows[source]) + else: + need(role != source, 'Reversible ADD distinct registers') + v.add(rows[role], rows[source], Q(op[3], op[4])) + else: + q = Q(op[2], op[3]); rows[role] = {k: q*x for k, x in rows[role].items()} + else: + need(kind in ('phase', 'adapter', 'residual_adapter'), 'Unsupported event '+kind) + if touched is not None: + values = [abs(x) for x in rows[touched].values()] + norm = sum(values, Q(0)) + if norm > max_l1: + max_l1 = norm; witness = {'event': i, 'role': list(touched)} + if values: + max_coefficient = max(max_coefficient, max(values)) + max_numerator = max(max_numerator, max(x.numerator for x in values)) + max_denominator = max(max_denominator, max(x.denominator for x in values)) + need([current['L', i] for i in range(n)] == canonical['final'], 'Exact final native frames') + need(all(current['P', i] == 1 for i in range(cache_count)), 'Logger cache terminal frame') + if expected_profile is not None: + need(dict(profile) == {int(k): x for k, x in expected_profile.items()}, 'Complete profile binding') + mismatch = [] + matrix = geo['H'] if endpoint == 'H' else geo['J'] + for role, actual in rows.items(): + wanted = {} if role[0] == 'C' else v.identity(role) + if role[0] == 'L': + port = role[1] if backward else role[1]-112 + if 0 <= port < 112: + offset, sign = (112, -1) if backward else (0, 1) + for j, q in matrix[port].items(): + v.add(wanted, v.identity(('L', offset+j)), sign*q) + elif role[0] == 'P': + need(endpoint == 'logger' and not backward, 'Only forward logger cache contract') + k = role[1] + for j, q in {geo['selected'][k]: Q(1), **geo['L'][k]}.items(): + v.add(wanted, v.identity(('L', j)), q) + if actual != wanted: + differences = {k: actual.get(k, Q(0))-wanted.get(k, Q(0)) for k in actual.keys() | wanted.keys()} + differences = {k: x for k, x in differences.items() if x} + first = min(differences) + mismatch.append({'role': list(role), 'unequal_coefficients': len(differences), + 'first_input': list(first[0]), 'first_word': list(first[1]), + 'first_coefficient': str(differences[first])}) + if not allow_mismatch: + need(not mismatch, 'Endpoint mismatch: '+repr(mismatch[:3])) + return dict(direction='backward' if backward else 'forward', endpoint=endpoint, + independent_persistent_input_fields=n+cache_count, + independently_initialized_owned_C_fields=len(centers), owned_C_fields_zero_at_exit=all(not rows['C', i] for i in centers), + all_408_helpers_exactly_restored_in_logical_coordinates=all(rows['L', i] == v.identity(('L', i)) for i in range(224, 632)), + all_scalar_frames_checked=True, all_new_coefficients_dyadic=True, + event_counts=dict(counts), paid_profile=dict(sorted(profile.items())), + paid_rank_mass=sum(k*x for k, x in profile.items()), real_scalar_allowance=scalar_cost, + prefix=dict(max_formal_row_l1=str(max_l1), max_abs_coefficient=str(max_coefficient), + max_abs_numerator=max_numerator, max_denominator=max_denominator, l1_witness=witness, + scope='Exact logical ideal prefixes; no rounded recursive error bound.'), + exact_endpoint=not mismatch, mismatch_examples=mismatch[:3]) + + +def lower_exact(p, ce, op): + recipe, _ = ce.lower_child(op) + left = p.monomial(ce, recipe['left'], op.n) + if recipe['rank']: + right = p.monomial(ce, recipe['right'], op.n) + decoded = right.then(ce.fourier(op.n, range(recipe['rank']))).then(left) + else: + decoded = left + need(decoded.tab == op.tab and decoded.zero_state() == op.zero_state(), 'Independent recipe decode including absolute phase') + return recipe + + +def normal_form(p, ce, op, rank): + recipe = lower_exact(p, ce, op) + need(recipe['rank'] == rank, 'Physical rank equals promised helper width') + left = p.monomial(ce, recipe['left'], 9) + right = p.monomial(ce, recipe['right'], 9) + pre = right.adjoint(); post = left.adjoint() + pre_recipe = lower_exact(p, ce, pre); post_recipe = lower_exact(p, ce, post) + need(pre_recipe['rank'] == post_recipe['rank'] == 0, 'Free boundary wrappers') + wrapped = pre.then(op).then(post); target = ce.fourier(9, range(rank)) + need(wrapped.tab == target.tab and wrapped.zero_state() == target.zero_state(), 'Helper boundary is exact standard Fourier including phase') + return dict(rank=rank, factorization=recipe, outer_pre_monomial=pre_recipe['left'], + outer_post_monomial=post_recipe['left'], exact_tableau_and_global_phase=True) + + +def helper_normal_forms(p, ce, canonical, cache=None): + """Certify every helper, with explicit inherited and fresh outer wrappers. + + Bare residual is F_end F_start^-1. The retained residual_adapter convention + already maps it to C_width in forward, and conjugate(C_width) in backward. + New outer normalizers below act at the WHOLE invocation boundaries. They + require no change to internal frames, helper reads, or scalar event order. + """ + cache = {} if cache is None else cache + rows = []; widths = Counter() + for role in range(224, 632): + start, end = canonical['start'][role], canonical['final'][role] + A, B = canonical['frames'][start], canonical['frames'][end] + rank = len(B)-len(A) + need(0 < rank <= 9, 'Positive productive helper width') + key = (tuple(A), tuple(B)) + if key not in cache: + physical = ce.canonical_frame(9, A).adjoint().then(ce.canonical_frame(9, B)) + inherited = lower_exact(p, ce, physical) + need(inherited['rank'] == rank, 'Native helper residual rank') + lm = p.monomial(ce, inherited['left'], 9) + rm = p.monomial(ce, inherited['right'], 9) + directions = {} + for backward in (False, True): + name = 'backward' if backward else 'forward' + op = p.conjugate(ce, physical) if backward else physical + l = p.conjugate(ce, lm) if backward else lm + r = p.conjugate(ce, rm) if backward else rm + native = r.adjoint().then(op).then(l.adjoint()) + target = ce.fourier(9, range(rank)) + if backward: + target = p.conjugate(ce, target) + need(native.tab == target.tab and native.zero_state() == target.zero_state(), 'Inherited oriented native helper boundary') + directions[name] = dict(bare_residual_normal_form=normal_form(p, ce, op, rank), + inherited_native_target='conjugate(C_width)' if backward else 'C_width', + existing_native_boundary_normal_form=normal_form(p, ce, native, rank)) + cache[key] = dict(start_basis=A, end_basis=B, inherited_forward_factorization=inherited, directions=directions) + rows.append(dict(role=role, start_frame=start, end_frame=end, width=rank, certificate=cache[key])) + widths[rank] += 1 + expected = {5: 28, 6: 18, 7: 56, 8: 93, 9: 213} if widths[7] == 56 else {5: 28, 6: 18, 7: 57, 8: 92, 9: 213} + need(dict(widths) == expected, 'Exact all-helper productive width profile') + return dict(status='PASS_ALL_408_HELPERS_BOTH_ORIENTATIONS_WITH_ABSOLUTE_PHASE', + count=len(rows), widths=dict(sorted(widths.items())), total_width=sum(k*x for k, x in widths.items()), + recipes=rows, unique_frame_pairs=len({(r['start_frame'], r['end_frame']) for r in rows}), + scope='Native ideal helper block and free boundary adapters only; productive outer ownership and full recursive bill are separate obligations.') + + +def h_child_recipes(p, ce, canonical, words): + result = {} + for direction, program in words.items(): + rows = [] + for i, e in enumerate(program): + if e[0] != 'center': + continue + _, role, old, new, rank = e + # These corrections occur OUTSIDE the inherited data adapters. + # The physical endpoint references are the original port gates, + # not the monomial-equivalent canonical representatives. Using a + # bare canonical_frame here would leave an uncharged/missing phase + # adapter on the scalar contact despite a correct logical replay. + k = len(rows); port_index = SELECTED[k]; q = canonical['ports'][port_index] + if direction == 'forward': + source = ce.fourier(9) + target = ce.cgate(9, q).adjoint().then(ce.fourier(9)) + else: + source = ce.ident(9) + target = p.conjugate(ce, ce.cgate(9, q)) + op = source.adjoint().then(target) + recipe = lower_exact(p, ce, op) + need(recipe['rank'] == rank == 1, 'Completion child rank one') + direct = ce.cgate(9, q).adjoint() + need(op.tab == direct.tab and op.zero_state() == direct.zero_state(), 'External completion is exact inverse port gate') + rows.append(dict(event=i, selected_port=port_index, port_vector=q, + source_frame=old, target_frame=new, recipe=recipe, + physical_reference='original external endpoint gauge; inherited adapters already outside the contact', + exact_inverse_port_gate_including_phase=True)) + need(len(rows) == 8, 'Exactly eight completion children') + result[direction] = rows + return result + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--out', type=Path, default=HERE/'fresh-certificates') + ap.add_argument('--stage', choices=('helpers', 'H', 'logger', 'all'), default='all') + ap.add_argument('--tags', nargs='+', choices=TAGS, default=list(TAGS)) + args = ap.parse_args(); args.out.mkdir(parents=True, exist_ok=True) + p, v, ce, receipt, inherited = authenticated() + geo = geometry(v.PACKET) + run = dict(schema='fresh-native-H112-reconstruction/1', reconstructed_after_loss=True, + source_sha256=digest(Path(__file__)), predecessor_receipt_sha256=RECEIPT_SHA, + predecessor_generator_sha256=P_SHA, independent_interpreter_sha256=p.CHECKER_SHA, + clifford_compiler_sha256=CE_SHA, packet_sha256=digest(v.PACKET), + geometry=dict(selected=list(SELECTED), ordinary_count=104, L_nonzeros=416, H_squared_identity=True, + H_not_symmetric=True, divisions_by_odd_numbers=0), + environment=dict(python=sys.version, platform=platform.platform()), + no_new_kappa_claimed=True, providers={}) + phase_cache = {} + for tag in args.tags: + started = time.monotonic() + c = read(BASE/(tag+'-CANONICAL.json.gz')); word = read(BASE/(tag+'-NATIVE-WORD.json.gz')) + row = dict(predecessor_artifacts=receipt['providers'][tag]['artifacts']) + if args.stage in ('helpers', 'all'): + helpers = helper_normal_forms(p, ce, c, phase_cache) + path = args.out/(tag+'-HELPER-BOUNDARIES.json.gz') + row['helper_boundaries'] = dict(artifact=path.name, sha256=save(path, helpers), widths=helpers['widths'], + total_width=helpers['total_width'], count=408) + print('PASS helpers', tag, helpers['widths'], flush=True) + if args.stage in ('H', 'all'): + hw = make_h_word(c, word, geo); expected_profile = Counter({int(k): x for k, x in c['local_profile'].items()}); expected_profile[1] += 8 + checks = {d: replay_extended(v, hw[d], c, geo, d == 'backward', expected_profile=expected_profile) for d in ('forward', 'backward')} + phases = h_child_recipes(p, ce, c, hw) + baseline_scalar = receipt['providers'][tag]['checks']['forward']['independent_literal_replay']['conservative_real_scalar_primitive_allowance'] + need(all(check['real_scalar_allowance']-baseline_scalar == 4336 for check in checks.values()), 'Exact 4336 scalar increment') + broken = deepcopy(hw['forward']) + omitted = next(i for i, e in enumerate(broken) if e[0] == 'scalar' and e[1][:2] == ['add', ['L', 112+SELECTED[0]]] and e[1][2] == ['C', 0]) + broken.pop(omitted) + negative = replay_extended(v, broken, c, geo, allow_mismatch=True) + need(not negative['exact_endpoint'], 'Missing completion negative control rejected') + path = args.out/(tag+'-H112-NATIVE.json.gz') + row['H112'] = dict(artifact=path.name, sha256=save(path, hw), checks=checks, new_child_recipes=phases, + increment=dict(profile={'1': 8}, real_scalar_allowance=4336, owned_C_fields=1, persistent_fields=0), + omitted_completion_negative_control=negative['mismatch_examples']) + print('PASS H112', tag, '8 H1 + 4336; both directions', flush=True) + if args.stage in ('logger', 'all'): + lw = make_logger_word(c, word, geo) + profile = Counter({int(k): x for k, x in c['local_profile'].items()}); profile[9] += 8 + check = replay_extended(v, lw, c, geo, endpoint='logger', cache_count=8, expected_profile=profile) + baseline_scalar = receipt['providers'][tag]['checks']['forward']['independent_literal_replay']['conservative_real_scalar_primitive_allowance'] + need(check['real_scalar_allowance']-baseline_scalar == 4224, 'Exact 4224 scalar logger increment') + op = ce.canonical_frame(9, c['frames'][0]).adjoint().then(ce.canonical_frame(9, c['frames'][1])) + recipe = lower_exact(p, ce, op); need(recipe['rank'] == 9, 'Logger paid rank nine') + path = args.out/(tag+'-FORWARD-LOGGER.json.gz') + row['logger'] = dict(artifact=path.name, sha256=save(path, lw), check=check, cache_child_recipe=recipe, + increment=dict(profile={'9': 8}, real_scalar_allowance=4224, owned_C_fields=0, arbitrary_cache_fields=8), + cache_restored=False, productive_cache_credit_claimed=False) + print('PASS logger', tag, '8 H9 + 4224', flush=True) + row['seconds'] = time.monotonic()-started + run['providers'][tag] = row + save(args.out/('RESULT-'+args.stage+'.json'), run) + run['status'] = 'PASS_FRESH_RECONSTRUCTION_REQUESTED_STAGES' + save(args.out/('RESULT-'+args.stage+'.json'), run) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.json b/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.json new file mode 100644 index 0000000000..19b27c6e40 --- /dev/null +++ b/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.json @@ -0,0 +1,149 @@ +{ + "cases": [ + { + "N": { + "H1": 276540184361, + "encoding": "2^H1" + }, + "delta": "1/2000000", + "eta_strict_lower": "1/4", + "exact_integer_test": { + "N_exponent": 15589, + "P_and_B_exponent": 15625 + }, + "exit_power_gap": "0", + "family": "complex", + "final_strict_power_gap": "1/2000000", + "h": "1/10", + "induction_constant_A": "n0^(1-z0)", + "k_min": 6, + "m": 57, + "m_to_h_integer_comparison": { + "left": 57, + "right": 1024 + }, + "materialized_winning_word": false, + "n0": { + "H0": 60000040, + "encoding": "2^H0" + }, + "negative_controls": { + "one_less_H0_fails_chosen_exit_bound": true, + "one_less_H1_fails_strict_final_bound": true, + "requested_004_not_supported_by_this_unit": true + }, + "paid": { + "1": 17623, + "2": 5065, + "3": 1738, + "4": 714, + "5": 254 + }, + "proven_uniform_moment": "P_N/B_N < N^z", + "r_max": 5, + "rank_one_surplus": 224, + "strict_test_margin": { + "lower": "0.00872690086259165952018340218333643", + "upper": "0.00872690086259165952018340218333644" + }, + "t": "199539/200000", + "targets": { + "6": 192, + "7": 56, + "8": 93, + "9": 1229, + "25": 448, + "57": 224 + }, + "template_definition": "Apply the integer, ordered, denominator-cleared compiler to this exact N. Keep every scalar/route operation and one owned bank per finite unit instance; only final rank-one leaves remain paid.", + "tested_mixed_saving": "461/200000", + "theta_integer_comparison": { + "left": 9765625, + "right": 10077696 + }, + "theta_upper": "9/10", + "uniform_saving": "36/15625", + "z": "15589/15625", + "z0": "1995391/2000000" + }, + { + "N": { + "H1": 6552928412, + "encoding": "2^H1" + }, + "delta": "9/2000000", + "eta_strict_lower": "1/4", + "exact_integer_test": { + "N_exponent": 24931, + "P_and_B_exponent": 25000 + }, + "exit_power_gap": "3/32000000", + "family": "BIT", + "final_strict_power_gap": "9/2000000", + "h": "1/16", + "induction_constant_A": "n0^(1-z0)", + "k_min": 8, + "m": 50, + "m_to_h_integer_comparison": { + "left": 50, + "right": 65536 + }, + "materialized_winning_word": false, + "n0": { + "H0": 10666731, + "encoding": "2^H0" + }, + "negative_controls": { + "one_less_H0_fails_chosen_exit_bound": true, + "one_less_H1_fails_strict_final_bound": true, + "requested_004_not_supported_by_this_unit": true + }, + "paid": { + "1": 9261, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1465 + }, + "proven_uniform_moment": "P_N/B_N < N^z", + "r_max": 7, + "rank_one_surplus": 225, + "strict_test_margin": { + "lower": "0.06709318391020230634322405194711404", + "upper": "0.06709318391020230634322405194711405" + }, + "t": "997231/1000000", + "targets": { + "8": 3073, + "22": 444, + "50": 222 + }, + "template_definition": "Apply the integer, ordered, denominator-cleared compiler to this exact N. Keep every scalar/route operation and one owned bank per finite unit instance; only final rank-one leaves remain paid.", + "tested_mixed_saving": "2769/1000000", + "theta_integer_comparison": { + "left": 33232930569601, + "right": 35184372088832 + }, + "theta_upper": "15/16", + "uniform_saving": "69/25000", + "z": "24931/25000", + "z0": "1994471/2000000" + } + ], + "new_kappa_claimed_by_this_file": false, + "proof_dependency": "The written native finite-precompilation theorem, including arbitrary-input closure and every exact ordered batch. This receipt checks its quantitative sufficient inequalities, not the theorem as a formal proof.", + "schema": "explicit-compressed-uniform-template-ranks/1", + "scope": "Explicit fixed ranks are specified in compressed form. No near-threshold word, GL cover, B_N, P_N or U_N is materialized. Their exact finite algorithms and replacement constants remain part of the construction.", + "source_sha256": { + "research/kappa-rebuild/architecture/FINITE-PRECOMPILATION.md": "4cb32f6b8e388ecf6ed41b5bbdeeb4539cb40c8424f194974afea58f7e0921e8", + "research/kappa-rebuild/architecture/mixed_target_probe.py": "011c3c9d0cf81103dc84d709a0e36793eb7c5ae3f23c85eac27650ef13afa04a", + "research/kappa-rebuild/bit/MIXED-PRODUCTIVE-BIT.json": "2f69a19f417c0e0d141c180f9b2fae3d358391b9a230f5848ee07c45450e70bf", + "research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz": "facfc6d6ecde151afb7e6a167776f3fdaac1c816e785a21a39f7f0bfd9140f3f", + "research/kappa-rebuild/controller/explicit_template_bounds.py": "4b3fbc216c32821729403d32efa8f75a418079b3bb57eb6204e044411d7d7f6f", + "research/kappa-rebuild/transfer-audit/AUDIT.md": "032488343667d19613cab84df5dad554cfd2a283fb4fc0d4c9b4d165cfa04669", + "research/kappa-rebuild/transfer-audit/finite_dag_audit.py": "f842f0e07ca40a181782a565a381b0ee20c6a4b08b8c2b8a4d5d630381bd1e8e" + }, + "status": "PASS_EXACT_SUPERPOTENTIAL_AND_COMPRESSED_RANK_BOUNDS" +} diff --git a/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.md b/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.md new file mode 100644 index 0000000000..b00acfbe8b --- /dev/null +++ b/research/kappa-rebuild/controller/EXPLICIT-TEMPLATE-BOUNDS.md @@ -0,0 +1,108 @@ +# Explicit finite ranks, without an unperformed search for the first success + +Apache-2.0. Substantial OpenAI assistance. The integer lockstep flow is Jacob +Alan Sussman's Theorem 1G. The native finite precompilation argument and its +independent audit are in `../architecture/FINITE-PRECOMPILATION.md` and +`../transfer-audit/AUDIT.md`. This note strengthens the quantitative choice +of the fixed template rank. It does not claim to materialize that template. + +## 1. Use a strict superpotential rather than an unknown exact root + +Write `L(n)=P_n/B_n` for the exact normalized number of paid rank-one leaves +in the integer compiler, including source complements and every exit batch. +Choose rational `0theta`, and put `h=1-theta_bar>0` and +`delta=z0-t>0`. In our two cases the exact integer inequalities prove +`m^h<2` and `2*r_max<=16`. Since `n>=m`, + +``` +m*E(n)/n < 16*n^(-h). +sum_e x_e*e^z0 <= (16*n^(-h))^delta * sum_s w_s*s^z0. +``` + +The source complements pay for this exit term. Concavity gives + +``` +n^z0-sum_s w_s*(n-s)^z0 >= eta*sum_s w_s*s^z0, +eta=z0*m^(z0-1). +``` + +The exact logarithm intervals check `z0>1/2` and `m^(1-z0)<2`, hence +`eta>1/4`. Set + +``` +H0=ceil((3+4*delta)/(h*delta)), n0=2^H0. +``` + +This exceeds the finite base range. For every `n>=n0`, the exit coefficient +is at most `1/8 25,600,000*N^5, odd, avoiding every fixed native/matcher denominator and rank witness", + "row_reserve": "For the new finite N and role/word constants, log_q S(e,w)=O(w log e); w=ceil(e^(3/1000)).", + "saving_after_cover": { + "decimal": "0.00275", + "rational": "11/4000" + }, + "saving_before_cover": { + "decimal": "0.00276", + "rational": "69/25000" + } + }, + "paid": { + "1": 9261, + "2": 9394, + "3": 553, + "4": 308, + "5": 714, + "6": 77, + "7": 1465 + }, + "precompilation_saving": { + "decimal": "0.00276", + "rational": "69/25000" + }, + "strict_mixed_margin": { + "lower": "0.79440312869001840349907639031552322", + "upper": "0.79440312869001840349907639031552323" + }, + "targets": { + "22": 444, + "50": 222, + "8": 3073 + }, + "uniform_template_definition": "First finite N in the specified exact compiler for which P_N/B_N < N^(1-C). Existence and native closure require the separate written proofs." + }, + "complex": { + "margin_at_004": { + "lower": "-163.47161919550931457258600794984177083", + "upper": "-163.47161919550931457258600794984177082" + }, + "paid": { + "1": 17623, + "2": 5065, + "3": 1738, + "4": 714, + "5": 254 + }, + "precompilation_saving": { + "decimal": "0.002304", + "rational": "36/15625" + }, + "strict_mixed_margin": { + "lower": "0.10548315917752229262732557699459137", + "upper": "0.10548315917752229262732557699459138" + }, + "targets": { + "25": 448, + "57": 224, + "6": 192, + "7": 56, + "8": 93, + "9": 1229 + }, + "uniform_template_definition": "First finite N in the specified exact compiler for which P_N/B_N < N^(1-C). Existence and native closure require the separate written proofs." + } + } +} diff --git a/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz b/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz new file mode 100644 index 0000000000..51a09b3952 Binary files /dev/null and b/research/kappa-rebuild/controller/P3-NATIVE-COVER.json.gz differ diff --git a/research/kappa-rebuild/controller/explicit_template_bounds.py b/research/kappa-rebuild/controller/explicit_template_bounds.py new file mode 100644 index 0000000000..70a110bfb8 --- /dev/null +++ b/research/kappa-rebuild/controller/explicit_template_bounds.py @@ -0,0 +1,132 @@ +#!/usr/bin/env python3 +"""Exact compressed ambient ranks for the finite mixed-job precompiler. + +This certifies inequalities in a written constructive proof. It does not +materialize the enormous finite word or formally verify the proof itself. +Apache-2.0. Substantial OpenAI assistance. The integer lockstep flow is +Jacob Alan Sussman's Theorem 1G; see ../architecture/FINITE-PRECOMPILATION.md. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +sys.path.insert(0, str(HERE.parent/'architecture')) +import mixed_target_probe as E + + +def need(value, message): + if not value: + raise AssertionError(message) + + +def read(path): + raw = Path(path).read_bytes() + return json.loads(gzip.decompress(raw) if str(path).endswith('.gz') else raw) + + +def ceil(value): + value = Q(value) + return -((-value.numerator)//value.denominator) + + +def one_case(name, targets, paid, tested_saving, uniform_saving, theta_bar): + targets = {int(k): int(v) for k,v in targets.items()} + paid = {int(k): int(v) for k,v in paid.items()} + m, kmin, rmax = max(targets), min(targets), max(paid) + need(1 <= rmax < kmin <= m, 'Strict mixed separation') + t, z = 1-tested_saving, 1-uniform_saving + z0 = (t+z)/2 + delta, h = z0-t, 1-theta_bar + need(0 < t < z0 < z < 1 and delta > 0 and h > 0, 'Strict rational exponents') + signed = Counter(targets) + signed.subtract(paid) + margin = E.moment(signed, tested_saving) + need(margin[0] > 0, 'Strict unit superpotential at the chosen rational t') + rank_margin = sum(k*v for k,v in signed.items()) + need(rank_margin > 0, 'Rank-one superpotential gives L(n) <= n') + a,b = theta_bar.numerator,theta_bar.denominator + need(rmax**b < kmin**a, 'Integer proof theta=log(rmax)/log(kmin) < theta_bar') + need(m**h.numerator < 2**h.denominator, 'Integer proof m^h < 2') + need(2*rmax <= 16, 'm E(n)/n < 16 n^-h') + ln_m = E.log_interval(m) + ln_two = E.log_interval(2) + need(z0 > Q(1,2) and (1-z0)*ln_m[1] < ln_two[0], + 'eta=z0*m^(z0-1) > 1/4') + H0 = ceil((3+4*delta)/(h*delta)) + need(H0 >= m.bit_length(), 'n0=2^H0 exceeds the entire finite base range') + need(H0*h*delta >= 3+4*delta, + 'Every n >= n0 has exit coefficient <= 1/8 < eta/2') + ratio = H0*(1-z0)/(z-z0) + H1 = ratio.numerator//ratio.denominator+1 + need(H1 > H0 and H1*(z-z0) > H0*(1-z0), + 'Explicit N=2^H1 has L(N) < N^z') + controls = { + 'one_less_H0_fails_chosen_exit_bound': (H0-1)*h*delta < 3+4*delta, + 'one_less_H1_fails_strict_final_bound': (H1-1)*(z-z0) <= H0*(1-z0), + 'requested_004_not_supported_by_this_unit': E.moment(signed,Q('0.004'))[1] < 0, + } + need(all(controls.values()), 'All deliberate negative controls reject') + return { + 'family':name, 'targets':targets, 'paid':paid, + 'm':m, 'k_min':kmin, 'r_max':rmax, + 'tested_mixed_saving':str(tested_saving), 'strict_test_margin':E.show(margin), + 'uniform_saving':str(uniform_saving), + 't':str(t), 'z0':str(z0), 'z':str(z), 'delta':str(delta), + 'theta_upper':str(theta_bar), 'h':str(h), 'rank_one_surplus':rank_margin, + 'theta_integer_comparison':{'left':rmax**b, 'right':kmin**a}, + 'm_to_h_integer_comparison':{'left':m**h.numerator, 'right':2**h.denominator}, + 'eta_strict_lower':'1/4', + 'n0':{'encoding':'2^H0','H0':H0}, + 'N':{'encoding':'2^H1','H1':H1}, + 'induction_constant_A':'n0^(1-z0)', + 'exit_power_gap':str(H0*h*delta-(3+4*delta)), + 'final_strict_power_gap':str(H1*(z-z0)-H0*(1-z0)), + 'proven_uniform_moment':'P_N/B_N < N^z', + 'exact_integer_test':{'P_and_B_exponent':z.denominator,'N_exponent':z.numerator}, + 'negative_controls':controls, + 'materialized_winning_word':False, + 'template_definition':'Apply the integer, ordered, denominator-cleared compiler to this exact N. Keep every scalar/route operation and one owned bank per finite unit instance; only final rank-one leaves remain paid.' + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out',type=Path,default=HERE/'EXPLICIT-TEMPLATE-BOUNDS.json') + args = parser.parse_args() + cpath = HERE/'P3-NATIVE-COVER.json.gz' + bpath = HERE.parent/'bit/MIXED-PRODUCTIVE-BIT.json' + c = read(cpath)['cover'] + b = next(row for row in read(bpath)['candidates'] if row['pairs']==3 and row['keep_idle_mask']==43) + cases = [ + one_case('complex',c['targets'],c['paid'],Q('0.002305'),Q('0.002304'),Q(9,10)), + one_case('BIT',b['targets'],b['paid'],Q('0.002769'),Q('0.00276'),Q(15,16)), + ] + sources = [Path(__file__),cpath,bpath,HERE.parent/'architecture/mixed_target_probe.py', + HERE.parent/'architecture/FINITE-PRECOMPILATION.md', + HERE.parent/'transfer-audit/finite_dag_audit.py', + HERE.parent/'transfer-audit/AUDIT.md'] + result = { + 'schema':'explicit-compressed-uniform-template-ranks/1', + 'status':'PASS_EXACT_SUPERPOTENTIAL_AND_COMPRESSED_RANK_BOUNDS', + 'cases':cases, + 'source_sha256':{str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources}, + 'proof_dependency':'The written native finite-precompilation theorem, including arbitrary-input closure and every exact ordered batch. This receipt checks its quantitative sufficient inequalities, not the theorem as a formal proof.', + 'scope':'Explicit fixed ranks are specified in compressed form. No near-threshold word, GL cover, B_N, P_N or U_N is materialized. Their exact finite algorithms and replacement constants remain part of the construction.', + 'new_kappa_claimed_by_this_file':False, + } + args.out.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'families':[ + {key:row[key] for key in ('family','uniform_saving','n0','N','final_strict_power_gap')} + for row in cases]})) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/controller/frame_exchange.py b/research/kappa-rebuild/controller/frame_exchange.py new file mode 100644 index 0000000000..0e3213ed01 --- /dev/null +++ b/research/kappa-rebuild/controller/frame_exchange.py @@ -0,0 +1,215 @@ +#!/usr/bin/env python3 +"""Actual owner/phase admission for H wrappers and a same-future-query exchange. + +Apache-2.0; substantial OpenAI assistance. This extends the exact phase +conventions credited in kappa-next-regime/direct-decoder/native_common.py. +The native H block is supplied by the separately certified reconstruction. +This file proves contacts and a reversible dirty-bank primitive, not a closed +global supplier or a rounded recursive induction. +""" +from collections import Counter +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import importlib.util +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +NATIVE = ROOT/'research/kappa-next-regime/direct-decoder/native_common.py' +REBUILD = HERE.parent/'complex/rebuild_native.py' + + +def need(ok, why): + if not ok: + raise ValueError(why) + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + + +def digest(path): + return sha256(Path(path).read_bytes()).hexdigest() + + +def between(n, old, g, new): + """Physical translation-permutation with exact prescribed logical g. + + All F references are commuting C_q products. Their Fourier symbols are + i raised to the stored Z4 quadratic polynomials. An affine permutation + contributes a linear even phase. Equality of these polynomials, including + their zero constant, proves the whole operator, not just polar rank. + """ + actual, wanted = n.polynomial(n.move_frame(old, g)), n.polynomial(new) + need(actual[1] == wanted[1], 'No quadratic phase mismatch') + delta = tuple((a-b) % 4 for a, b in zip(wanted[0], actual[0])) + need(all(x in (0, 2) for x in delta), 'Translation repairs only even linear phase') + shift = sum((x//2) << i for i, x in enumerate(delta)) + need(n.polynomial(n.move_frame(old, g, shift)) == wanted, 'Full Z4 phase identity') + inv = tuple(sum(((g[j] >> i) & 1) << j for j in range(73)) for i in range(73)) + need(n.product(g, inv) == n.product(inv, g) == n.AXES, 'Orthogonal inverse') + inverse_shift = n.act(inv, shift) + need(n.polynomial(n.move_frame(new, inv, inverse_shift)) == n.polynomial(old), + 'Exact inverse phase and coordinate map') + return dict(owner_columns=list(g), physical_shift=shift, + inverse_owner_columns=list(inv), inverse_physical_shift=inverse_shift, + global_phase_Z4=0) + + +def birth(n, before, future): + need(0 <= before < future <= 8, 'Causal future-owner contact') + return n.frame([(a, 1) for a in n.AXES[9:9+8*before]] + + [(a, 1) for a in n.AXES[9+8*(future-1):9+8*future]]) + + +def contact_certificate(n, phases): + records = {} + # H_s is read at preY_s and postX_s. Both logical operators are D_s H D_s. + for stage in range(9): + pre_common = () if stage == 0 else birth(n, stage-1, stage) + post_common = n.completed(stage) + pre, post = [], [] + for p, recipe in enumerate(phases['recipes']): + g = n.source_route(phases, stage, p) + old = () if stage == 0 else n.frame(list(n.completed(stage-1))+[(recipe['q'], -1)]) + pre.append(between(n, old, g, pre_common)) + post.append(between(n, post_common, g, post_common)) + records[str(stage)] = {'preY_common': list(pre_common), 'postX_common': list(post_common), + 'preY_views': pre, 'postX_views': post} + return records + + +def rows_identity(n): + return {i: {(i, n.AXES): Q(1)} for i in range(120)} + + +def exchange(n, geo, phases, rows, future): + """Literal112 Y +8 P word; no owned field, overwrite or clearing.""" + counts = Counter() + for p in range(112): + rows[p] = n.routed(n.source_route(phases, future, p), rows[p]); counts['view'] += 1 + for k, a in enumerate(geo['selected']): + for p, c in geo['L'][k].items(): + n.sparse_add(rows[a], rows[p], c); counts['half_add'] += 1 + for k, a in enumerate(geo['selected']): + b = 112+k + n.sparse_add(rows[a], rows[b], Q(1)) + n.sparse_add(rows[b], rows[a], Q(-1)) + n.sparse_add(rows[a], rows[b], Q(1)) + rows[b] = {key: -c for key, c in rows[b].items()} + counts['unit_add'] += 3; counts['negate'] += 1 + for k, a in enumerate(geo['selected']): + for p, c in geo['L'][k].items(): + n.sparse_add(rows[a], rows[p], -c); counts['half_add'] += 1 + for p in range(112): + rows[p] = n.routed(n.source_route(phases, future, p), rows[p]); counts['view'] += 1 + return counts + + +def physical_witness(n, row): + """A nonzero physical basis-column witness for a nonzero operator row.""" + for var in sorted({v for v, g in row}): + for address in [0]+list(n.AXES): + output = {} + for (v, g), c in row.items(): + if v == var: + at = n.act(g, address) + output[at] = output.get(at, Q(0))+c + output = {k: v for k, v in output.items() if v} + if output: + at = min(output) + return dict(input_field=var, input_address=address, output_address=at, + coefficient=str(output[at])) + raise ValueError('A formal mismatch was not promoted to a physical witness') + + +def run(output): + n, h = load(NATIVE, 'rebuild_native_contacts'), load(REBUILD, 'rebuild_h_for_contacts') + packet, phases = n.load_sources() + geo = h.geometry(packet) + contacts = contact_certificate(n, phases) + original = rows_identity(n) + first = {i: row.copy() for i, row in original.items()} + counts = exchange(n, geo, phases, first, 2) + # Check independent derived formula Y'=(I-EQ)Y+EP, P'=QY. + queries = [] + for k, a in enumerate(geo['selected']): + q = n.routed(n.source_route(phases, 2, a), original[a]) + for p, c in geo['L'][k].items(): + n.sparse_add(q, n.routed(n.source_route(phases, 2, p), original[p]), c) + queries.append(q) + need(first[112+k] == q, 'Actual owner-valued query output') + for p in geo['ordinary']: + need(first[p] == original[p], 'Ordinary Y unchanged') + for k, a in enumerate(geo['selected']): + delta = original[112+k].copy(); n.sparse_add(delta, queries[k], Q(-1)) + wanted = original[a].copy() + n.sparse_add(wanted, n.routed(n.source_route(phases, 2, a), delta), Q(1)) + need(first[a] == wanted, 'Selected right-inverse insertion') + second = {i: row.copy() for i, row in first.items()} + exchange(n, geo, phases, second, 2) + need(second == original, 'All120 arbitrary input fields restored by same query twice') + wrong = {i: row.copy() for i, row in original.items()} + exchange(n, geo, phases, wrong, 1) + exchange(n, geo, phases, wrong, 2) + mismatch = next(i for i in range(112,120) if wrong[i] != original[i]) + diff = wrong[mismatch].copy(); n.sparse_add(diff, original[mismatch], Q(-1)) + negative = {'output_field': mismatch, **physical_witness(n, diff)} + # The two actual cuts are after local0 and after local1. Y is unchanged by + # the intervening backward call, so the second identical logical exchange + # returns every arbitrary P baseline, even when H1 does not commute with H2. + bank_frames = [birth(n, b, 2) for b in (0,1)] + timed = [] + for before, common in zip((0,1), bank_frames): + views = [] + for p, recipe in enumerate(phases['recipes']): + old = n.frame(list(n.completed(before))+[(recipe['q'], -1)]) + views.append(between(n, old, n.source_route(phases, 2, p), common)) + timed.append(dict(after_stage=before, common_frame=list(common), views=views)) + difference = n.frame(list(bank_frames[1])+[(a,-c) for a,c in bank_frames[0]]) + expected = n.frame((a,1) for a in n.AXES[9:17]) + need(n.polynomial(difference) == n.polynomial(expected), 'Bank relative operator is exactly positive rank8') + # A direct tensor product of the eight C_e factors has no hidden phase. + need(all(c == 1 and a in n.AXES[9:17] for a,c in difference), 'Literal rank8 child axes') + result = dict(schema='native-future-query-and-H-wrapper-contacts-v1', + status='PASS_EXACT_CONTACTS_AND_DIRTY_EXCHANGE_NOT_CALLER_CLOSURE', + generator_sha256=digest(__file__), dependency_hashes={str(NATIVE.relative_to(ROOT)):digest(NATIVE), + str(REBUILD.relative_to(ROOT)):digest(REBUILD)}, + immutable_source_pins=n.PINS, + H_wrapper_contacts=contacts, + wrapper_identity='postX(H_s) F_or_B(H_s) preY(H_s) = diag(H_s,H_s) F_or_B(I)', + p1_abstract_endpoint='diag(H2 H1 H0,H2 H1 H0) times signed swap; no commutation assumed', + timed_same_future_query=timed, + exchange_formula='Y_new=Y+E2(P-Q2Y), P_new=Q2Y; Q2E2=I', + one_exchange_counts=dict(counts), + one_exchange_real_scalar_allowance=counts['half_add']*10+counts['unit_add']*8+counts['negate']*2, + exchanged_arbitrary_input_fields=120, + all_arbitrary_Y_P_columns_restored=True, + bank_relative_child_profile={'8':8}, + bank_target_credit='8 independent rank8 transforms only after all main cross-dependence is removed', + wrong_query_physical_negative_control=negative, + no_owned_fields=True, + open_obligations=['head/hole main endpoint closure','main dependence on dirty bank after intervening B', + 'strict child hierarchy if combined with minjob6 recipe','rounded induction and precision', + 'full mixed-job supplier and BIT premise'], no_new_kappa_claimed=True) + output.parent.mkdir(parents=True, exist_ok=True) + raw=json.dumps(result, sort_keys=True, separators=(',',':')).encode() + output.write_bytes(gzip.compress(raw,mtime=0)) + print(json.dumps({k:v for k,v in result.items() if k in ('status','one_exchange_counts', + 'one_exchange_real_scalar_allowance','wrong_query_physical_negative_control')})) + print('phase contacts',9*112*2+2*112,'receipt',str(output),'sha256',digest(output)) + + +if __name__ == '__main__': + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out',type=Path,default=HERE/'FRAME-EXCHANGE.json.gz') + run(parser.parse_args().out) diff --git a/research/kappa-rebuild/controller/global_identity_cover.py b/research/kappa-rebuild/controller/global_identity_cover.py new file mode 100644 index 0000000000..bb6018afd7 --- /dev/null +++ b/research/kappa-rebuild/controller/global_identity_cover.py @@ -0,0 +1,356 @@ +#!/usr/bin/env python3 +"""Bind a complete p2 mixed supplier to native words, owners and exact phases. + +Apache-2.0; substantial OpenAI assistance. Native words and endpoint recipes +are attributed and pinned through the independently reconstructed complex +package. This certificate is a finite ideal construction, not the mixed +recursion or the rounded integer-multiplication transfer theorem. +""" +from collections import Counter, defaultdict, deque +from fractions import Fraction as Q +from hashlib import sha256 +from itertools import product +from pathlib import Path +import argparse +import gzip +import importlib.util +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +COMPLEX = HERE.parent/'complex' +PAIRS = ((93,8,8),(56,7,9),(56,9,7),(19,5,9)) +MIDDLE = ((29,0),(28,5),(18,6),(56,7),(93,8)) +STAGES = ((0,'forward'),(0,'backward'),(0,'forward'),(1,'backward'),(1,'forward')) + + +def need(ok, why): + if not ok: + raise ValueError(why) + + +def read(path): + raw = Path(path).read_bytes() + return json.loads(gzip.decompress(raw) if str(path).endswith('.gz') else raw) + + +def digest(path): + return sha256(Path(path).read_bytes()).hexdigest() + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + sys.modules[name] = module + spec.loader.exec_module(module) + return module + + +def gf2_basis(vectors): + basis = {} + for vector in vectors: + value = vector + while value: + pivot = value.bit_length()-1 + if pivot in basis: + value ^= basis[pivot] + else: + basis[pivot] = value + break + return basis + + +def coordinate_map(axes, dim=41): + """An explicit invertible address map with the desired first columns.""" + columns = list(axes) + need(len(gf2_basis(columns)) == len(columns), 'Independent target axes') + for j in range(dim): + if len(gf2_basis(columns+[1 << j])) > len(columns): + columns.append(1 << j) + need(len(columns) == dim, 'Full invertible coordinate extension') + rows = [sum(((columns[j] >> i) & 1) << j for j in range(dim)) for i in range(dim)] + augmented = [row | (1 << (dim+i)) for i,row in enumerate(rows)] + for j in range(dim): + pivot = next(i for i in range(j,dim) if (augmented[i] >> j) & 1) + augmented[j], augmented[pivot] = augmented[pivot], augmented[j] + for i in range(dim): + if i != j and (augmented[i] >> j) & 1: + augmented[i] ^= augmented[j] + inverse_rows = [row >> dim for row in augmented] + need(all((row & ((1 << dim)-1)) == 1 << i for i,row in enumerate(augmented)), + 'Explicit inverse address matrix') + return dict(columns=columns, inverse_rows=inverse_rows) + + +def local_bindings(): + receipt_path = COMPLEX/'full-identity-certificates/RESULT.json' + data = read(receipt_path) + pins = {str(receipt_path.relative_to(ROOT)):digest(receipt_path)} + helpers = {} + for tag,provider in data['providers'].items(): + need(provider['all_408_helpers_autonomous'] and provider['all_632_persistent_inputs_arbitrary'], + 'Complete native arbitrary-input contract') + need(provider['external_cleanup_fields'] == 0 and provider['owned_C_fields'] == 1, + 'One explicit owned center; no hidden cleanup bank') + for filename,wanted in provider['artifacts'].items(): + path = receipt_path.parent/filename + need(digest(path) == wanted, 'Native artifact '+filename) + pins[str(path.relative_to(ROOT))] = wanted + path = ROOT/provider['helper_boundary']['certificate'] + need(digest(path) == provider['helper_boundary']['certificate_sha256'], 'Actual helper boundaries') + pins[str(path.relative_to(ROOT))] = digest(path) + helpers[tag] = read(path) + return data,pins,helpers + + +def main_contacts(n, F, phases): + """Every complete native entrance and exit, including conjugate B words. + + After the input affine views, all632 local roles share the SAME spectator + axes9..(9+8s-1). Thus the native internal scalar contacts are unchanged. + Different port views cancel on the per-port identity response. + """ + out = [] + for stage,(pair,direction) in enumerate(STAGES): + sign = 1 if direction == 'forward' else -1 + spectator = n.frame((a,1) for a in n.AXES[9:9+8*stage]) + need(all(a >= 512 and a < 1 << 41 for a,c in spectator), 'Spectators outside native nine axes') + for port,recipe in enumerate(phases['recipes']): + q = recipe['q'] + g = n.source_route(phases, stage, port) + need(all(g[i] == 1 << i for i in range(41,73)) and all(a < 1 << 41 for a in g[:41]), + 'The ambient41 construction leaves all higher coordinates untouched') + input_x = n.frame([(q,1)]) if stage == 0 else n.completed(stage-1) + input_y = () if stage == 0 else n.frame(list(n.completed(stage-1))+[(q,-1)]) + local_start_x = n.frame(list(spectator)+[(q,sign)]) + local_start_y = spectator + local_end_x = n.frame(list(spectator)+[(a,sign) for a in n.AXES[:9]]) + local_end_y = n.frame(list(local_end_x)+[(q,-sign)]) + output_x = n.completed(stage) + output_y = n.frame(list(output_x)+[(q,-1)]) + maps = dict(entry_X=F.between(n,input_x,g,local_start_x), + entry_Y=F.between(n,input_y,g,local_start_y), + exit_X=F.between(n,local_end_x,g,output_x), + exit_Y=F.between(n,local_end_y,g,output_y)) + out.append(dict(stage=stage,pair=pair,direction=direction,port=port, + common_spectator_frame=list(spectator),maps=maps)) + return out + + +def make_windows(n, phases): + windows = [] + for kind,pair,stages in (('early',1,[0,1]),('middle',1,[2]),('late',0,[3,4])): + patterns = [([a,b] if kind != 'middle' else [a]) + for spec in (PAIRS if kind != 'middle' else MIDDLE) + for _ in range(spec[0]) + for a,b in ([spec[1:]] if kind != 'middle' else [(spec[1],0)])] + need(len(patterns) == 224, '224 distinct idle owners') + for index,widths in enumerate(patterns): + side,port = ('X' if index < 112 else 'Y'),index % 112 + recipe = phases['recipes'][port] + if kind == 'early': + axes = recipe['idle_children'][0]['orthogonal_axes'] + shift = recipe['idle_children'][0]['translation_after'] + old = n.frame([(recipe['q'],1)]) if side == 'X' else () + target = n.completed(1) + elif kind == 'middle': + axes = list(n.AXES[17:25]); shift = 0 + old = n.completed(1); target = n.completed(2) + else: + axes = list(n.AXES[25:41]); shift = 0 + old = n.completed(2); target = n.completed(4) + if side == 'Y': + if kind != 'early': + old = n.frame(list(old)+[(recipe['q'],-1)]) + target = n.frame(list(target)+[(recipe['q'],-1)]) + need(len(axes) == (8 if kind == 'middle' else 16), 'Exact idle window width') + actual = n.frame(list(old)+[(a,1) for a in axes]+[(shift,2)]) + need(n.polynomial(actual) == n.polynomial(target), 'Whole idle path including absolute phase') + windows.append(dict(kind=kind,owner=f'{side}{pair}:{port}',pair=pair,port=port,side=side, + stages=stages,widths=widths,axes=axes,translation_after=shift)) + return windows + + +def split_axes(axes, maximum=5): + return [axes[i:i+maximum] for i in range(0,len(axes),maximum)] + + +def build_assignment(n, phases, local, helpers, mask): + tags = [('FOUR' if mask[s] else 'THREE')+'-base' for s in range(5)] + windows = make_windows(n, phases) + # FOUR has one fewer8 and one extra7. Replace the FIRST eight-slot in each + # stage: the same physical owner in a two-stage window whenever possible. + for stage in range(5): + if mask[stage]: + candidates = [w for w in windows if stage in w['stages'] and w['widths'][w['stages'].index(stage)] == 8] + need(candidates, 'A hosted width8 exists for FOUR adjustment') + w = candidates[0] + w['widths'][w['stages'].index(stage)] = 7 + pools = [] + for tag in tags: + pool = defaultdict(deque) + for row in helpers[tag]['recipes']: + pool[row['width']].append(row['role']) + pools.append(pool) + bindings = []; residuals = []; paid = Counter(); targets = Counter({41:448}) + for window in windows: + cursor = 0 + for stage,width in zip(window['stages'],window['widths']): + if not width: + continue + need(pools[stage][width], 'Actual available helper role') + role = pools[stage][width].popleft() + axes = window['axes'][cursor:cursor+width]; cursor += width + bindings.append(dict(stage=stage,role=role,owner=window['owner'],axes=axes,width=width, + kind='original_idle',window=window['kind'])) + rest = window['axes'][cursor:] + for axes in split_axes(rest): + residuals.append(dict(owner=window['owner'],after_stage=max(window['stages']),axes=axes, + rank=len(axes),kind='idle_remnant')) + paid[len(axes)] += 1 + fresh = [] + for stage,pool in enumerate(pools): + for width,roles in sorted(pool.items()): + for role in roles: + owner = f'G{stage}:{role}' + target = max(6,width) + axes = list(n.AXES[:width]) + bindings.append(dict(stage=stage,role=role,owner=owner,axes=axes,width=width,kind='fresh')) + fresh.append(dict(owner=owner,target_rank=target,native_width=width,stage=stage)) + targets[target] += 1 + for extra in split_axes(list(n.AXES[width:target])): + residuals.append(dict(owner=owner,after_stage=stage,axes=extra,rank=len(extra),kind='fresh_topup')) + paid[len(extra)] += 1 + for stage,tag in enumerate(tags): + data = local['providers'][tag]['split_replays'][STAGES[stage][1]]['independent_native_replay'] + paid.update({int(r):count for r,count in data['paid_profile'].items()}) + chosen = [b for b in bindings if b['stage'] == stage] + need(len(chosen) == 408 and {b['role'] for b in chosen} == set(range(224,632)), 'Every native role exactly once') + need(len({b['owner'] for b in chosen}) == 408, 'Injective helper physical owners') + active_pair = STAGES[stage][0] + active = {f'{side}{active_pair}:{p}' for side in ('X','Y') for p in range(112)} + need(not active & {b['owner'] for b in chosen}, 'No helper aliases a live data field') + need(targets == Counter({41:448,6:146,9:803}), 'Complete productive output inventory') + need(sum(k*v for k,v in targets.items())-sum(k*v for k,v in paid.items()) == 128, + 'Exact conserved rank margin including every completion') + need(max(paid) < min(targets), 'Strict child hierarchy') + # Explicit wrappers act only around a WHOLE native invocation. They are + # never used as permission to retag an intermediate native scalar contact. + maps = {} + for b in bindings: + key = ','.join(map(str,b['axes'])) + if key not in maps: + maps[key] = coordinate_map(b['axes']) + b['coordinate_map'] = key + b['boundary_recipe'] = dict(provider=tags[b['stage']],role=b['role'], + direction=STAGES[b['stage']][1], + selector='certificate.directions.direction.existing_native_boundary_normal_form') + scalar = sum(local['providers'][tag]['split_replays'][STAGES[s][1]]['independent_native_replay'] + ['conservative_real_scalar_primitive_allowance'] for s,tag in enumerate(tags)) + scalar += 3*224*8+224*2 # Three bridges, conservative terminal negations. + return dict(providers=tags,targets=dict(sorted(targets.items())),paid=dict(sorted(paid.items())), + rank_margin=128,main_fields=448,fresh_fields=len(fresh),productive_fields=448+len(fresh), + owned_center_fields=1,external_cleanup_fields=0,scalar_allowance=scalar, + helper_bindings=bindings,owner_coordinate_maps=maps,windows=windows, + paid_residuals=residuals,fresh_outputs=fresh) + + +def controller_replay(omit_bridge=False): + """Four independent per-port data columns, no forced zeros or equality.""" + rows = [[Q(i == j) for j in range(4)] for i in range(4)] + maximum = Q(1); events = [] + def add(dst,src,c): + nonlocal maximum + rows[dst] = [a+c*b for a,b in zip(rows[dst],rows[src])] + maximum = max(maximum,sum(map(abs,rows[dst]))) + def local(pair,backward): + x,y = 2*pair,2*pair+1 + add(x if backward else y,y if backward else x,Q(-1 if backward else 1)) + events.append(['local',pair,'backward' if backward else 'forward']) + def bridge(src,dst,c): + add(2*dst,2*src,Q(c)); add(2*src+1,2*dst+1,Q(-c)) + events.append(['bridge',src,dst,c]) + local(0,False); local(0,True); bridge(1,0,1); local(0,False) + if not omit_bridge: + bridge(1,0,-1) + local(1,True); local(1,False); bridge(0,1,1) + before = [row[:] for row in rows] + for p in range(2): + x,y = 2*p,2*p+1 + rows[x],rows[y] = rows[y],[-a for a in rows[x]] + identity = [[Q(i == j) for j in range(4)] for i in range(4)] + if not omit_bridge: + need(rows == identity, 'Full controller after signed swap') + need(before == [[Q(0),Q(-1),Q(0),Q(0)],[Q(1),Q(0),Q(0),Q(0)], + [Q(0),Q(0),Q(0),Q(-1)],[Q(0),Q(0),Q(1),Q(0)]], 'Exact preterminal signed swaps') + return dict(schedule=events,preterminal_matrix=[[str(x) for x in row] for row in before], + terminal_matrix=[[str(x) for x in row] for row in rows], + max_logical_prefix_l1=str(maximum),identity=rows == identity) + + +def terminal_phases(n, phases): + records = [] + for port,r in enumerate(phases['recipes']): + q = r['q']; full = n.completed(4) + # Initial logicalX=Cq^-1 rawX, logicalY=rawY. Signed exchange puts + # F41*Cq^-2 on the originalX output and F41 on originalY. + doubled = n.frame(list(full)+[(q,-2),(q,2)]) + need(n.polynomial(doubled) == n.polynomial(full), 'Final translation restores exact full41 transform') + need(n.polynomial(n.frame(list(full)+[(q,-2)])) != n.polynomial(full), + 'Omitting terminal phase repair is actually wrong') + records.append(dict(port=port,original_X_output_translation=q,original_Y_output_translation=0, + both_final_ranks=41,full_phase_equality=True)) + return records + + +def run(): + F = load(HERE/'frame_exchange.py','global_identity_contacts') + n = load(F.NATIVE,'global_identity_native_common') + packet,phases = n.load_sources() + local,pins,helpers = local_bindings() + contacts = main_contacts(n,F,phases) + cases = [] + for mask in product((0,1),repeat=5): + case = build_assignment(n,phases,local,helpers,mask) + cases.append(case) + all_three = cases[0] + need(all_three['paid'] == {1:12565,2:3621,3:1217,4:510,5:169}, 'Independently regenerated complete invoice') + main = controller_replay() + need(not controller_replay(True)['identity'], 'Missing inverse bridge negative control') + phases_final = terminal_phases(n,phases) + for path in (HERE/'frame_exchange.py',F.NATIVE): + pins[str(path.relative_to(ROOT))] = digest(path) + return dict(schema='complete-native-p2-mixed-identity-owner-cover/1', + status='PASS_FINITE_IDEAL_NATIVE_OWNER_PHASE_AND_COMPLETE_INVOICE', + generator_sha256=digest(__file__),sources=pins,immutable_phase_sources=n.PINS, + main_contacts=contacts,contact_count=5*112*4,terminal_phases=phases_final, + controller=main,all_three=all_three, + provider_cases=[{k:v for k,v in case.items() if k in ('providers','targets','paid','rank_margin','scalar_allowance')} + for case in cases], + all32_finite_provider_sequences_have_native_owners=True, + arbitrary_productive_input_fields=1397,owned_center_arbitrary_at_entry_zero_at_return=True, + helper_normalization='Apply the pinned whole-helper pre/post monomials and the listed coordinate inverse/map around each whole native call. The internal native word is unchanged.', + spectator_argument='After main input views each local word has common spectator axes9..9+8s-1. A helper is arbitrary under its independent input permutation; native autonomous return cancels all intermediate helper dependence on main outputs.', + negative_controls={'omitted_inverse_bridge_rejected':True,'omitted_terminal_translation_rejected_all112':True}, + limits=['Finite ideal circuit, not a rounded implementation.', + 'Mixed-size recursion and uniform fixed-tape compilation require separate proof.', + 'All32 finite provider choices are covered; arbitrary layer-dependent provider dispatch is not proved here.', + 'Owned-center acquisition and precision stock must be paid by the outer caller.'],new_kappa=False) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--out',type=Path,default=HERE/'GLOBAL-IDENTITY-COVER.json.gz') + args = ap.parse_args() + result = run() + args.out.write_bytes(gzip.compress(json.dumps(result,sort_keys=True,separators=(',',':')).encode(),mtime=0)) + print(json.dumps(dict(status=result['status'],contacts=result['contact_count'], + targets=result['all_three']['targets'],paid=result['all_three']['paid'], + fields=result['arbitrary_productive_input_fields'],scalar=result['all_three']['scalar_allowance'], + receipt_sha256=digest(args.out)),sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/controller/outer_parameters.py b/research/kappa-rebuild/controller/outer_parameters.py new file mode 100644 index 0000000000..fd4dc2ad7c --- /dev/null +++ b/research/kappa-rebuild/controller/outer_parameters.py @@ -0,0 +1,249 @@ +#!/usr/bin/env python3 +"""Exact conservative outer parameters for the changed recursive family. + +This checks arithmetic and source binding, not informal proof validity. +The source-bound native words and finite-precompilation/cover proofs are +separate admission gates. No selected frontier file is changed. + +Outer formulas: Vlado Vojdanovski PR1673, Chafik Boukhalfa PR1758/1786, +the retained radix-union/Strassen/fine-prime ledger and its credited lineage. +The changed supplier bindings and conservative parameter choice here are +Apache-2.0, with substantial OpenAI assistance. +""" +from collections import Counter +from decimal import Decimal, localcontext +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import argparse +import ast +import gzip +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +sys.path.insert(0,str(HERE.parent/'architecture')) +import mixed_target_probe as E + +OLD = ROOT/'research/kappa-observation-integration/release-work/current/package/outer/check_ledger.py' +COMPLEX = HERE/'P3-NATIVE-COVER.json.gz' +BIT = HERE.parent/'bit/MIXED-PRODUCTIVE-BIT.json' +BASELINE = Q('0.001730970419199163702738') + + +def need(condition, message): + if not condition: + raise AssertionError(message) + + +def read(path): + raw = Path(path).read_bytes() + return json.loads(gzip.decompress(raw) if str(path).endswith('.gz') else raw) + + +def digest(path): + return sha256(Path(path).read_bytes()).hexdigest() + + +def display(value): + value = Q(value) + with localcontext() as context: + context.prec = 70 + return {'rational':str(value),'decimal':str(Decimal(value.numerator)/Decimal(value.denominator))} + + +def encode(value): + if isinstance(value,Q): + return display(value) + if isinstance(value,dict): + return {str(k):encode(v) for k,v in value.items()} + if isinstance(value,(list,tuple)): + return [encode(v) for v in value] + return value + + +def supplier_checks(): + complex_data,bit_data = read(COMPLEX),read(BIT) + cover = complex_data['cover'] + bit = next(case for case in bit_data['candidates'] if case['pairs'] == 3 and case['keep_idle_mask'] == 43) + rows = {} + for name,paid,targets,precompile in ( + ('complex',cover['paid'],cover['targets'],Q('0.002304')), + ('bit',bit['paid'],bit['targets'],Q('0.00276'))): + paid = {int(rank):int(count) for rank,count in paid.items()} + targets = {int(rank):int(count) for rank,count in targets.items()} + need(max(paid) < min(targets), 'Actual strict child/target separation') + signed = Counter(targets); signed.subtract(paid) + margin = E.moment(signed,precompile) + need(margin[0] > 0, 'Exact strict mixed moment at the selected precompilation exponent') + fail = E.moment(signed,Q('0.004')) + need(fail[1] < 0, 'The current complete identity supplier does not support .004') + rows[name] = {'targets':targets,'paid':paid,'precompilation_saving':precompile, + 'strict_mixed_margin':E.show(margin), + 'margin_at_004':E.show(fail), + 'uniform_template_definition':'First finite N in the specified exact compiler for which P_N/B_N < N^(1-C). Existence and native closure require the separate written proofs.'} + bare,covered,atom = Q('0.00276'),Q('0.00275'),Q(3,1000) + old = Q(384599,10**10) + bit_saving = (1-atom)*covered+atom*old + admitted = Q(27,10000) + gap = bare-covered + need(bit_saving == Q(27418653797,10**13), 'Exact BIT atom conversion') + need(atom > bit_saving > admitted and atom < 1-bit_saving, 'All ordinary BIT conversion terms fit') + # For every N>=2, lnN >= ln2 >1/2, so N^gap >1+gap/2. + # q >256*N^5/gap gives 64*N^5/q 25,600,000*N^5, odd, avoiding every fixed native/matcher denominator and rank witness', + 'prime_coefficient':Q(256)/gap,'fallback_moment_upper_bound':fallback_ratio_bound, + 'row_reserve':'For the new finite N and role/word constants, log_q S(e,w)=O(w log e); w=ceil(e^(3/1000)).'} + return rows + + +def ledger(epsilon=Q(99999,100000), beta=Q(99997,100000), + kappa=Q(23,10000), bit_saving=Q(27,10000), layout=Q(1,10**6)): + b,q,a = Q('0.002303'),Q('0.002302'),Q('0.0023025') + r0,stop = Q(1,10**6),Q(1,10**8) + tau,sigma = 1-a,1-b + internal = tau+(1-stop)*max(sigma-tau,Q(0)) + leaf = sigma+stop*(1-sigma) + lam = 1-(q+a)/2 + matrix_exponent = Q(45,16) + c = (3-matrix_exponent)/2 + xi = epsilon*(1+q)-c+layout + key = beta+1-epsilon + margins = { + 'main_transform':epsilon*q, + 'global_coordinate_router':a, + 'fused_gaussian':1-epsilon+q*beta, + 'group_bit_router':1-epsilon+a*beta, + 'group_scans':1-epsilon+beta, + 'boundary_algebraic_matrix':c+xi-epsilon, + 'sparse_union_movement':xi+Q(1,2)-epsilon, + 'polynomial_products':epsilon, + 'microbox_face_corrections':1-(epsilon+beta+48-128+2), + } + checks = { + 'actual_bit_above_effective':bit_saving-a, + 'complex_above_effective':b-a, + 'q_positive':q,'complex_endpoint':b-q,'q_below_effective':a-q, + 'stop_positive':stop,'stop_below_one':1-stop, + 'leaf_saving_above_effective':(1-stop)*b-a, + 'q_below_internal':1-internal-q,'q_below_leaf':1-leaf-q, + 'lambda_above_tau':lam-tau,'lambda_above_sigma':lam-sigma, + 'lambda_above_internal':lam-internal,'lambda_prime_above_lambda':1-q-lam, + 'lambda_prime_above_leaf':1-q-leaf, + 'epsilon_positive':epsilon,'epsilon_below_one':1-epsilon, + 'cap_above_band':xi-Q(1,2),'cap_below_cell':epsilon-xi, + 'group_positive':beta,'group_below_dimension':epsilon-beta, + 'support_router_span':1-beta, + 'group_gauge_power_slack':1-(2*xi-2)-r0-beta, + 'actual_logarithmic_group_guard':1-(2*xi-2)-beta, + 'global_gamma_power_slack':1-epsilon-r0, + 'actual_logarithmic_gamma_guard':1-epsilon, + 'alpha_logarithm_absorption':r0, + 'group_key_fits_record':1-key, + 'membership_arithmetic_fits_record':1-beta-2*(1-epsilon), + 'microbox_halo_volume':128-48-beta, + 'positive_layout_slack':layout, + 'short_record_fallback':epsilon-a, + 'complex_below_one_over_32':Q(1,32)-b, + 'strassen_integer_power_gap':Q(2**45-7**16), + 'theta_positive_polynomial_power':Q(2), + 'cap_below_phase_cell_power':2-xi, + 'halo_covers_generic_inverse_radius':Q(48-43), + 'union_tag_work_per_visit_fits_record':1-beta, + 'union_header_fits_record':1-beta, + 'source_axis_superpolynomial':1-epsilon, + 'gaussian_above_main':margins['fused_gaussian']-margins['main_transform'], + } + checks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + checks['new_complex_uniform_saving_above_b'] = Q('0.002304')-b + checks['improvement_over_user_baseline'] = kappa-BASELINE + checks['new_semantic_guard_with_C1_one'] = 1-epsilon + checks['fine_prime_halo_umbrella'] = 10-(2*2+2+Q(1,2)) + return {'parameters':{'kappa':kappa,'complex_uniform_saving':Q('0.002304'), + 'BIT_saving':bit_saving,'b':b,'q':q,'effective_saving':a, + 'epsilon':epsilon,'group_beta':beta,'cap_xi':xi,'layout_slack':layout, + 'r0':r0,'stop':stop,'lambda':lam,'lambda_prime':1-q, + 'Strassen_exponent':matrix_exponent,'theta_power':2,'halo_power':48,'core_power':128}, + 'margins':margins,'strict_checks':checks} + + +def inherited_crosscheck(ours): + """Execute only the previously reviewed parameter function as a second oracle. + + The old native geometry checks are neither changed nor called. Its m=73 + supplier cannot be used to admit the new family. New native gates are + explicit separate inputs to this package. + """ + source = ast.parse(OLD.read_text()) + function = next(node for node in source.body if isinstance(node,ast.FunctionDef) and node.name == 'ledger') + constants = [node for node in source.body if isinstance(node,ast.Assign) + and any(isinstance(target,ast.Name) and target.id.startswith('KAPPA_') for target in node.targets)] + env = {'Q':Q,'R0':Q(1,10**6),'ZETA':Q(1,10**8),'SLACK':Q(1,10**6), + 'GAP':Q(1,10**6),'STOP':Q(1,10**8),'FLOOR':10**6,'BACKOFF':Q(1,10**6), + 'EPS':Q(99999,100000),'BETA':Q(99997,100000),'THETA_POWER':Q(2), + 'HALO_POWER':48,'CORE_POWER':128,'STRASSEN_TAU':Q(45,16), + 'b':Q('0.002303'),'q':Q('0.002302'),'a':Q('0.0023025'), + 'bit':Q('0.0027'),'online':BASELINE,'old':BASELINE} + module = ast.fix_missing_locations(ast.Module(body=constants+[function],type_ignores=[])) + exec(compile(module,str(OLD),'exec'),env) + original = env['ledger']('new_uniform_supplier_conservative_parameters') + need(original['kappa'] == Q(23,10000), 'Inherited floor/backoff produces the same rational target') + need(original['margins'] == ours['margins'], 'Every inherited outer cost row independently agrees') + common = set(original['strict_checks']).intersection(ours['strict_checks']) + need(all(original['strict_checks'][key] == ours['strict_checks'][key] for key in common), + 'All common strict predicates independently agree') + need(all(value > 0 for value in original['strict_checks'].values()), 'All63 inherited arithmetic checks pass') + return {'matched_cost_rows':len(original['margins']), + 'matched_named_checks':len(common),'all_inherited_checks':len(original['strict_checks']), + 'native_admission_not_inferred_from_parameter_function':True} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out',type=Path,default=HERE/'OUTER-PARAMETERS.json') + args = parser.parse_args() + suppliers = supplier_checks() + chosen = ledger() + need(all(value > 0 for value in chosen['strict_checks'].values()), 'Every declared strict condition') + need(min(chosen['margins'].values()) == chosen['margins']['main_transform'], 'Main transform is limiting') + independent = inherited_crosscheck(chosen) + controls = {} + for label,options in [('unfunded_004',{'kappa':Q(1,250)}), + ('epsilon_one',{'epsilon':Q(1)}), + ('zero_layout_slack',{'layout':Q(0)}), + ('old_BIT_baseline_is_insufficient',{'bit_saving':BASELINE})]: + invalid = ledger(**options) + failed = {key:value for key,value in invalid['strict_checks'].items() if value <= 0} + need(failed, 'Deliberately invalid parameter set is rejected: '+label) + controls[label] = failed + paths = [Path(__file__),COMPLEX,BIT,OLD,HERE.parent/'architecture/mixed_target_probe.py'] + result = {'schema':'changed-family-conservative-outer-parameter-certificate/1', + 'status':'PASS_EXACT_ARITHMETIC_AND_SOURCE_BOUND_MIXED_MOMENTS', + 'suppliers':suppliers,'outer':chosen,'inherited_parameter_crosscheck':independent, + 'controls':controls,'source_sha256':{str(path.relative_to(ROOT)):digest(path) for path in paths}, + 'scope':'Exact rational arithmetic. Native closure and the finite-program existence proof remain distinct gates. This is not a claim that the incumbent m73 admission script accepted a different supplier.', + 'conditional_hypotheses':['The source-bound finite words have the supplied all-column native interpretation.', + 'Finite precompilation and delayed BIT cover have their stated arbitrary-input, ownership, storage and routing closure.', + 'The new uniform templates use replacement semantic precision and row/stock constants.', + 'The inherited fine-prime padding, gauge-free interior, fixed-tape and outer analytic framework hypotheses.'], + 'new_conditional_kappa_claimed_by_this_arithmetic_file':False} + args.out.write_text(json.dumps(encode(result),indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'target':str(Q(23,10000)), + 'strict_checks':len(chosen['strict_checks']), + 'inherited_crosscheck':independent, + 'smallest_cost_margin':display(min(chosen['margins'].values())-Q(23,10000)), + 'source_sha256':digest(args.out)},sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/controller/p3_native_cover.py b/research/kappa-rebuild/controller/p3_native_cover.py new file mode 100644 index 0000000000..8756693610 --- /dev/null +++ b/research/kappa-rebuild/controller/p3_native_cover.py @@ -0,0 +1,274 @@ +#!/usr/bin/env python3 +"""Complete native ownership and phase binding for p3/keep43. + +This binds the exact Laurent controller to all 2,856 original helper roles, +whole-call adapters, arbitrary productive inputs, and the full recursive +invoice. Apache-2.0; substantial OpenAI assistance. Native source credits +and immutable hashes are inherited from the reconstruction receipts. +""" +from collections import Counter, defaultdict, deque +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json +import sys + +sys.dont_write_bytecode = True +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[2] +sys.path.insert(0, str(HERE)) +import global_identity_cover as G + +STAGES = ((0,'forward'),(0,'backward'),(0,'forward'),(1,'backward'), + (1,'forward'),(2,'backward'),(2,'forward')) +FORMAL = HERE.parent/'bit/COMPLEX-CONTROLLER-FACTOR.json' +TARIFF = HERE.parent/'architecture/P3-CONTROLLER-TARIFF-CERTIFICATE.json' +AUDIT = HERE.parent/'transfer-audit/P3-NATIVE-AUDIT.json.gz' +TARGETS = {6:192,7:56,8:93,9:1229,25:448,57:224} +PAID = {1:17623,2:5065,3:1738,4:714,5:254} + + +def need(condition, message): + if not condition: + raise AssertionError(message) + + +def stage_frame(n, stage, q): + if stage == 0: + return n.frame([(a,1) for a in n.AXES[:9]]+[(q,-1)]) + return n.frame((a,1) for a in n.AXES[9+8*(stage-1):9+8*stage]) + + +def contacts(n, F, phases): + result = [] + for stage,(pair,direction) in enumerate(STAGES): + sign = 1 if direction == 'forward' else -1 + spectator = n.frame((a,1) for a in n.AXES[9:9+8*stage]) + for port,recipe in enumerate(phases['recipes']): + q = recipe['q']; route = n.source_route(phases,stage,port) + need(all(route[i] == 1 << i for i in range(57,73)), 'No route touches outside ambient57') + need(all(axis < 1 << 57 for axis in route[:57]), 'All native views lie in ambient57') + input_x = n.frame([(q,1)]) if not stage else n.completed(stage-1) + input_y = () if not stage else n.frame(list(input_x)+[(q,-1)]) + local_x = n.frame(list(spectator)+[(q,sign)]) + local_y = spectator + end_x = n.frame(list(spectator)+[(a,sign) for a in n.AXES[:9]]) + end_y = n.frame(list(end_x)+[(q,-sign)]) + output_x = n.completed(stage) + output_y = n.frame(list(output_x)+[(q,-1)]) + maps = {'entry_X':F.between(n,input_x,route,local_x), + 'entry_Y':F.between(n,input_y,route,local_y), + 'exit_X':F.between(n,end_x,route,output_x), + 'exit_Y':F.between(n,end_y,route,output_y)} + delta = stage_frame(n,stage,q) + # References cancel as operators on arbitrary raw inputs. No + # historical assumption about which idles have run is needed. + need(n.polynomial(n.frame(list(output_x)+[(a,-c) for a,c in input_x])) + == n.polynomial(delta), 'Exact diagonal physical D_s on X') + need(n.polynomial(n.frame(list(output_y)+[(a,-c) for a,c in input_y])) + == n.polynomial(delta), 'Exact diagonal physical D_s on Y') + cross = n.frame(list(output_y if sign == 1 else output_x)+ + [(a,-c) for a,c in (input_x if sign == 1 else input_y)]) + wanted = n.frame(list(delta)+[(q,-1 if sign == 1 else 1)]) + need(n.polynomial(cross) == n.polynomial(wanted), 'Physical cross operator D_s Q^+-1') + result.append({'stage':stage,'pair':pair,'port':port,'direction':direction, + 'common_spectator_frame':list(spectator),'maps':maps}) + return result + + +def make_windows(n, phases, formal): + output = [] + for number,window in enumerate(formal['retained_idle_windows']): + stages = window['stages'] + patterns = [list(spec[1:]) for spec in (G.PAIRS if len(stages) == 2 else G.MIDDLE) + for _ in range(spec[0])] + need(len(patterns) == 224, 'Every retained window has224 distinct possible owners') + for index,widths in enumerate(patterns): + side,port = ('X' if index < 112 else 'Y'),index % 112 + recipe = phases['recipes'][port] + if 0 in stages: + need(stages == [0,1], 'The retained initial window is D0D1') + axes = recipe['idle_children'][0]['orthogonal_axes'] + shift = recipe['idle_children'][0]['translation_after'] + else: + axes = [axis for stage in stages for axis in n.AXES[9+8*(stage-1):9+8*stage]] + shift = 0 + desired = n.frame([term for stage in stages for term in stage_frame(n,stage,recipe['q'])]) + actual = n.frame([(axis,1) for axis in axes]+[(shift,2)]) + need(n.polynomial(actual) == n.polynomial(desired), 'Literal whole idle operator including phase') + need(len(axes) == window['rank'] and sum(widths) <= len(axes), 'Every hosted helper fits') + output.append({'family':number,'index':window['index'],'kind':window['kind'], + 'pair':window['pair'],'stages':stages,'side':side,'port':port, + 'owner':f"{side}{window['pair']}:{port}", 'widths':widths, + 'axes':axes,'translation_after':shift}) + return output + + +def helper_assignment(n, phases, formal, local, helper_recipes): + windows = make_windows(n,phases,formal) + pools = [] + for stage in range(7): + pool = defaultdict(deque) + for recipe in helper_recipes['THREE-base']['recipes']: + pool[recipe['width']].append(recipe['role']) + pools.append(pool) + bindings,residuals,fresh = [],[],[] + paid = Counter(); targets = Counter({25:448,57:224}) + for window in windows: + cursor = 0 + for stage,width in zip(window['stages'],window['widths']): + if not width: + continue + need(pools[stage][width], 'A real local helper of this width exists') + role = pools[stage][width].popleft() + axes = window['axes'][cursor:cursor+width]; cursor += width + bindings.append({'stage':stage,'role':role,'width':width,'owner':window['owner'], + 'axes':axes,'kind':'original_idle','window_index':window['index']}) + for axes in G.split_axes(window['axes'][cursor:]): + residuals.append({'owner':window['owner'],'after_stage':max(window['stages']), + 'axes':axes,'rank':len(axes),'kind':'idle_remnant'}) + paid[len(axes)] += 1 + for stage,pool in enumerate(pools): + for width,roles in sorted(pool.items()): + for role in roles: + owner = f'G{stage}:{role}'; target = max(6,width) + bindings.append({'stage':stage,'role':role,'width':width,'owner':owner, + 'axes':list(n.AXES[:width]),'kind':'fresh'}) + fresh.append({'owner':owner,'stage':stage,'native_width':width,'target_rank':target}) + targets[target] += 1 + for axes in G.split_axes(list(n.AXES[width:target])): + residuals.append({'owner':owner,'after_stage':stage,'axes':axes, + 'rank':len(axes),'kind':'fresh_topup'}) + paid[len(axes)] += 1 + coordinate_maps = {} + for stage,(pair,direction) in enumerate(STAGES): + native = local['providers']['THREE-base']['split_replays'][direction]['independent_native_replay'] + paid.update({int(rank):count for rank,count in native['paid_profile'].items()}) + selected = [row for row in bindings if row['stage'] == stage] + need(len(selected) == 408 and {row['role'] for row in selected} == set(range(224,632)), + 'All408 actual native helper roles are assigned exactly once') + need(len({row['owner'] for row in selected}) == 408, 'Every local helper has a distinct physical owner') + active = {f'{side}{pair}:{port}' for side in ('X','Y') for port in range(112)} + need(not active.intersection(row['owner'] for row in selected), 'No live main role is borrowed') + for row in selected: + key = ','.join(map(str,row['axes'])) + if key not in coordinate_maps: + coordinate_maps[key] = G.coordinate_map(row['axes'],57) + row['coordinate_map'] = key + row['whole_helper_boundary'] = {'provider':'THREE-base','direction':direction,'role':row['role'], + 'selector':'certificate.directions.direction.existing_native_boundary_normal_form'} + need(dict(sorted(paid.items())) == PAID, 'Full actual paid histogram') + need(dict(sorted(targets.items())) == TARGETS, 'All and only productive target fields') + need(len(fresh) == 1570 and len(bindings) == 2856, 'Complete field and helper census') + need(sum(k*v for k,v in targets.items())-sum(k*v for k,v in paid.items()) == 224, 'Rank-unit margin') + need(max(paid) < min(targets), 'Strict recursive separation') + return {'windows':windows,'helper_bindings':bindings,'owner_coordinate_maps':coordinate_maps, + 'paid_residuals':residuals,'fresh_outputs':fresh,'paid':PAID,'targets':TARGETS, + 'rank_margin':224,'productive_fields':2242,'owned_center_fields':1,'external_cleanup_fields':0} + + +def endpoint_normalizers(n, phases, formal): + output = []; histogram = Counter() + for role,term in enumerate(formal['exact_diagonal_monomials']): + q_power,selected = term[0],term[1:] + need(q_power in (-1,1) and set(selected) <= {0,1}, 'Exact Laurent endpoint alphabet') + for port,recipe in enumerate(phases['recipes']): + q = recipe['q'] + physical = n.frame([(q,q_power)]+[entry for stage,yes in enumerate(selected) if yes + for entry in stage_frame(n,stage,q)]) + axes = list(n.AXES[:9]) if selected[0] else [q] + axes += [axis for stage in range(1,7) if selected[stage] + for axis in n.AXES[9+8*(stage-1):9+8*stage]] + translation = q if q_power == -1 else 0 + repaired = n.frame(list(physical)+[(translation,2)]) + canonical_on_axes = n.frame((a,1) for a in axes) + need(n.polynomial(repaired) == n.polynomial(canonical_on_axes), 'Exact endpoint phase including constant') + if translation: + need(n.polynomial(physical) != n.polynomial(canonical_on_axes), 'Missing phase repair really fails') + rank = len(axes) + need(rank == 1+8*sum(selected), 'Earned endpoint rank') + matrix = G.coordinate_map(axes,57) + need(matrix['columns'][:rank] == axes, 'Literal target-rank conjugation') + output.append({'original_input_role':role,'port':port,'target_rank':rank, + 'input_coordinate_columns':matrix['columns'], + 'output_coordinate_inverse_rows':matrix['inverse_rows'], + 'output_translation_before_inverse_map':translation, + 'physical_frame':list(physical),'positive_axes':axes}) + histogram[rank] += 1 + need(histogram == Counter({25:448,57:224}), 'Main endpoint distribution') + return output + + +def scalar_invoice(local, formal): + native = sum(local['providers']['THREE-base']['split_replays'][direction]['independent_native_replay'] + ['conservative_real_scalar_primitive_allowance'] for pair,direction in STAGES) + adapters = formal['right_program']+formal['left_program'] + counts = Counter(row[0] for row in adapters) + need(counts == Counter({'ADD':4,'SWAP':6,'SCALE':2}), 'Literal boundary scalar word') + need(all(str(row[-1]) in ('1','-1') for row in adapters if row[0] in ('ADD','SCALE')), + 'Only unit scalar boundary coefficients') + # Ordinary swap: three additions followed by one negation. Signed swap + # needs three additions. No free whole-role movement is assumed here. + adapter_adds = 4+6*3; adapter_negs = 2+6 + bridges = sum(row[0] == 'ADD' for row in formal['physical_events']) + signed_swaps = sum(row[0] == 'SIGNED_SWAP' for row in formal['physical_events']) + need(bridges == 12 and signed_swaps == 3, 'Literal global bridge and swap census') + total = native+112*((adapter_adds+bridges+3*signed_swaps)*8+adapter_negs*2) + return {'native_real_scalar_allowance':native,'boundary_unit_ADD_per_port':adapter_adds, + 'boundary_negations_per_port':adapter_negs,'bridge_unit_ADD_per_port':bridges, + 'terminal_signed_swap_unit_ADD_per_port':3*signed_swaps, + 'conservative_real_scalar_primitive_allowance':total, + 'coordinate_and_phase_moves':'Explicit separate fixed movement operations; not scalar ADDs and not paid recursive calls.'} + + +def run(): + F = G.load(HERE/'frame_exchange.py','p3_contacts') + n = G.load(F.NATIVE,'p3_native_common') + packet,phases = n.load_sources() + local,pins,helpers = G.local_bindings() + formal_source = G.read(FORMAL); formal = formal_source['selected_p3_keep43'] + tariff = G.read(TARIFF); audit = G.read(AUDIT) + need(tariff['controller'] == formal, 'The native cover binds the exact priced controller') + for path,wanted in tariff['source_sha256'].items(): + need(G.digest(ROOT/path) == wanted, 'Pinned actual tariff source '+path) + for name,wanted in formal_source['source_sha256'].items(): + need(G.digest(FORMAL.parent/name) == wanted, 'Pinned formal word source '+name) + main_contacts = contacts(n,F,phases) + cover = helper_assignment(n,phases,formal,local,helpers) + endings = endpoint_normalizers(n,phases,formal) + scalar = scalar_invoice(local,formal) + need({int(k):int(v) for k,v in tariff['invoice']['paid'].items()} == cover['paid'], 'Independent complete paid invoice agrees') + need({int(k):int(v) for k,v in tariff['invoice']['targets'].items()} == cover['targets'], 'Independent productive credit agrees') + for path in (FORMAL,TARIFF,AUDIT,HERE/'global_identity_cover.py',HERE/'frame_exchange.py',F.NATIVE): + pins[str(path.relative_to(ROOT))] = G.digest(path) + return {'schema':'complete-native-p3-keep43-owner-phase-cover/1', + 'status':'PASS_COMPLETE_FINITE_NATIVE_CONTROLLER_OWNER_PHASE_INVOICE', + 'generator_sha256':G.digest(__file__),'sources':pins,'immutable_phase_sources':n.PINS, + 'main_contacts':main_contacts,'contact_count':3136,'main_endpoint_normalizers':endings, + 'cover':cover,'scalar_invoice':scalar,'root_bracket':tariff['invoice']['root_bracket'], + 'physical_order':'Endpoint input coordinate maps; R; literal serialized controller with whole-call helper adapters; L; endpoint translations; endpoint inverse coordinate maps.', + 'native_local_contract':{'forward':'D_s [[1,0],[Q^-1,1]]','backward':'D_s [[1,-Q],[0,1]]', + 'meaning':'Exact raw-input operators, valid for arbitrary prior histories and after every boundary scalar basis change.'}, + 'arbitrary_productive_input_fields':2242,'all_2856_original_helper_roles_assigned':True, + 'one_owned_center_reused_sequentially':True,'owned_center_arbitrary_at_entry_zero_at_return':True, + 'helpers_normalized_only_around_whole_native_invocations':True, + 'new_kappa':False, + 'remaining_transfer_obligations':['Finite mixed-to-uniform precompilation and ordinary supplier instantiation.', + 'Replacement precision, stock, scalar and movement constants for the resulting finite template.', + 'A compatible BIT supplier and the inherited outer analytic transfer.']} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--out',type=Path,default=HERE/'P3-NATIVE-COVER.json.gz') + args = parser.parse_args() + receipt = run() + args.out.write_bytes(gzip.compress(json.dumps(receipt,sort_keys=True,separators=(',',':')).encode(),mtime=0)) + print(json.dumps({'status':receipt['status'],'targets':receipt['cover']['targets'], + 'paid':receipt['cover']['paid'],'scalar':receipt['scalar_invoice'], + 'root_bracket':receipt['root_bracket'],'sha256':G.digest(args.out)},sort_keys=True)) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/extend_cumulative.py b/research/kappa-rebuild/extend_cumulative.py new file mode 100644 index 0000000000..48285d7330 --- /dev/null +++ b/research/kappa-rebuild/extend_cumulative.py @@ -0,0 +1,191 @@ +#!/usr/bin/env python3 +"""Extend an authenticated cumulative content-addressed overlay without loss. + +The retained byte applier is reused unchanged. Only the new research scope +and the thirteen recovered files may be appended; every prior record and +payload is preserved. Mathematical receipts are separate from this byte +integrity operation. Apache-2.0; substantial OpenAI assistance. +""" +from collections import Counter +from copy import deepcopy +from hashlib import sha256 +from pathlib import Path, PurePosixPath +import argparse +import importlib.util +import json +import os +import shutil +import stat +import tempfile +import zipfile + +HERE = Path(__file__).resolve().parent +APPLIER_SHA256 = '68900a2064f0f0e27c2c34842df235fd4c3a8b8c515e3e4aa628260b645040a8' +SCOPES = ('research/kappa-rebuild', 'research/kappa-accounted-composition') + + +def need(value, message): + if not value: + raise ValueError(message) + + +def metadata(path): + digest = sha256() + count = 0 + with path.open('rb') as source: + for block in iter(lambda:source.read(1024*1024),b''): + digest.update(block) + count += len(block) + return {'bytes':count, 'sha256':digest.hexdigest(), 'mode':path.stat().st_mode&0o777} + + +def zip_info(name): + info = zipfile.ZipInfo(name,(2026,10,21,0,0,0)) + info.create_system = 3 + info.external_attr = (stat.S_IFREG|0o644)<<16 + info.compress_type = zipfile.ZIP_DEFLATED + return info + + +def load_applier(path): + need(sha256(path.read_bytes()).hexdigest() == APPLIER_SHA256, 'Exact retained byte-applier identity') + spec = importlib.util.spec_from_file_location('retained_kappa_overlay',path) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--base',type=Path,required=True) + parser.add_argument('--prior',type=Path,required=True) + parser.add_argument('--applier',type=Path,required=True) + parser.add_argument('--repo',type=Path,required=True) + parser.add_argument('--out',type=Path,required=True) + parser.add_argument('--receipt',type=Path,required=True) + parser.add_argument('--manifest',type=Path) + args = parser.parse_args() + api = load_applier(args.applier) + need(not args.out.exists(), 'Output must be a new file') + need(args.repo.is_dir() and not args.repo.is_symlink(), 'Plain repository directory') + prior, prior_receipt = api.verify_blob_data(args.base,args.prior) + records = {r['path']:deepcopy(r) for r in prior['records']} + for name,record in records.items(): + path = api.checked_target(args.repo,name) + need(path.is_file() and metadata(path) == record['after'], 'Prior cumulative path changed or disappeared: '+name) + added = {} + for scope in SCOPES: + directory = args.repo/scope + need(directory.is_dir(), 'Required cumulative scope is present: '+scope) + for path in sorted(directory.rglob('*')): + need(not path.is_symlink(), 'No symbolic links in appended research') + if not path.is_file() or '__pycache__' in path.parts or path.suffix=='.pyc': + continue + name = path.relative_to(args.repo).as_posix() + api.normalized_member('integer-mult-bounds/'+name) + if name not in records: + added[name] = path + with zipfile.ZipFile(args.base) as base: + base_files,base_safety = api.safe_index(base) + for name,path in added.items(): + before = api.member_hash(base,base_files[name]) if name in base_files else None + after = metadata(path) + kind = 'added' if before is None else 'identical' if before['sha256']==after['sha256'] else 'changed' + records[name] = {'path':name,'before':before,'after':after,'classification':kind, + 'mode_changed':before is not None and before['mode']!=after['mode'], + 'source':'new_supplement','generation':'kappa-rebuild-20261021'} + union = set(base_files)|set(records) + for name in union: + need(all(str(p)=='.' or str(p) not in union for p in PurePosixPath(name).parents), 'No file/directory collision') + updates = [r for r in records.values() if r['classification']!='identical' or r.get('mode_changed')] + representatives = {} + for record in sorted(updates,key=lambda r:r['path']): + representatives.setdefault(record['after']['sha256'],record) + readme = ('''# Complete cumulative kappa research overlay + +This bundle contains every path from the supplied October20 cumulative +overlay, the surviving recovery files, and the complete October21 research +scope. Nothing is deleted. Read research/kappa-rebuild/README.md after +application for the current mathematical result, proof status and replay. +The byte applier itself makes no mathematical claim and remains unchanged. + +Keep the exact attached base ZIP. Extract apply_overlay.py from this bundle, +then run (replace the three paths with your local filenames): + +```sh +python3 apply_overlay.py blob-verify --base BASE.zip --bundle THIS.zip +python3 apply_overlay.py blob-apply --base BASE.zip --bundle THIS.zip --repo CHECKOUT --receipt applied.json +``` + +The applier checks the complete base archive fingerprint and every touched +preimage. It rejects mismatches and symlinks. Per-file writes are atomic; +a repeated application is idempotent. It does not delete or validate +unrelated checkout files. It does not publish, merge, or change a remote. + +An optional complete binary review patch can be generated using: + +```sh +python3 apply_overlay.py review-diff --base BASE.zip --bundle THIS.zip --out cumulative-review.patch.gz +``` + +The earlier reported 332-entry archive discrepancy, including34 reported +dependencies, remains an explicitly unresolved historical recovery flag. +Newly reconstructed files have fresh provenance; they are not presented +as recovered originals with their lost historical identities. +''').encode() + tool = args.applier.read_bytes() + manifest = deepcopy(prior) + manifest.update({ + 'records':[records[name] for name in sorted(records)], + 'update_paths':sorted(r['path'] for r in updates), + 'counts':dict(Counter(r['classification'] for r in records.values())), + 'result_regular_files':len(union),'blob_sha256s':sorted(representatives), + 'supplemental_files':prior['supplemental_files']+len(added), + 'content_deduplication':{'update_paths':len(updates),'unique_blobs':len(representatives), + 'path_bytes':sum(r['after']['bytes'] for r in updates), + 'unique_blob_bytes':sum(r['after']['bytes'] for r in representatives.values())}, + 'control_sha256':{'apply_overlay.py':sha256(tool).hexdigest(),'README.md':sha256(readme).hexdigest()}, + 'prior_overlay':api.archive_fingerprint(args.prior), + 'prior_records_preserved_byte_for_byte':len(prior['records']), + 'current_round_paths':sorted(added),'deletions':[], + 'local_tree_audit':{'scope':'Every prior cumulative path was checked for exact bytes and mode. New files are appended only from the two named research scopes. Runtime caches are excluded.', + 'divergent_files':[],'extra_local_files':[]}, + }) + api.validate_manifest(manifest) + args.out.parent.mkdir(parents=True,exist_ok=True) + temp_path = None + try: + with tempfile.NamedTemporaryFile(prefix='.cumulative-build-',suffix='.zip',dir=args.out.parent,delete=False) as temp: + temp_path = Path(temp.name) + with zipfile.ZipFile(args.prior) as source, zipfile.ZipFile(temp_path,'w',compression=zipfile.ZIP_DEFLATED,compresslevel=6,allowZip64=True) as sink: + for name,data in [('MANIFEST.json',(json.dumps(manifest,indent=2,sort_keys=True)+'\n').encode()), + ('apply_overlay.py',tool),('README.md',readme)]: + sink.writestr(zip_info(name),data) + old_blobs = set(prior['blob_sha256s']) + for digest,record in sorted(representatives.items()): + incoming = source.open('blobs/'+digest) if digest in old_blobs else added[record['path']].open('rb') + copied = sha256(); count = 0 + with incoming, sink.open(zip_info('blobs/'+digest),'w',force_zip64=True) as target: + for block in iter(lambda:incoming.read(1024*1024),b''): + target.write(block); copied.update(block); count += len(block) + need(copied.hexdigest()==digest and count==record['after']['bytes'], 'Source changed while packaging') + os.link(temp_path,args.out) + finally: + if temp_path is not None: + temp_path.unlink(missing_ok=True) + _,verification = api.verify_blob_data(args.base,args.out) + receipt = {'schema':'extended-cumulative-kappa-overlay/1','status':'PASS_ALL_PRIOR_PATHS_PRESERVED_AND_COMPLETE_NEW_OVERLAY_VERIFIED', + 'base':api.archive_fingerprint(args.base),'prior':api.archive_fingerprint(args.prior), + 'bundle':api.archive_fingerprint(args.out),'old_records':len(prior['records']), + 'new_records':len(added),'total_records':len(records),'counts':manifest['counts'], + 'verification':verification,'deletions':[], + 'scientific_claim_not_inferred_from_byte_packaging':True} + args.receipt.parent.mkdir(parents=True,exist_ok=True) + args.receipt.write_text(json.dumps(receipt,indent=2,sort_keys=True)+'\n') + if args.manifest: + args.manifest.write_text(json.dumps(manifest,indent=2,sort_keys=True)+'\n') + print(json.dumps({k:receipt[k] for k in ('status','bundle','old_records','new_records','total_records','counts')})) + + +if __name__=='__main__': + main() diff --git a/research/kappa-rebuild/provenance/BIT-CACHE-RECOVERY.md b/research/kappa-rebuild/provenance/BIT-CACHE-RECOVERY.md new file mode 100644 index 0000000000..2818756abe --- /dev/null +++ b/research/kappa-rebuild/provenance/BIT-CACHE-RECOVERY.md @@ -0,0 +1,228 @@ +# BIT, cache, and relay recovery record + +## Status + +This is a **new recovery record**, created on 2026-10-21 after the execution workspace reset. It is not a replacement for the missing repository, the earlier cumulative checkpoint, or the complete scientific receipts produced before the reset. + +The original BIT and cache sources are unavailable. Their recorded hashes and previously reported outcomes are preserved in [LOST-BIT-CACHE-PINS.json](LOST-BIT-CACHE-PINS.json). A recorded hash cannot recover its file. No approximate reconstruction has been placed under a missing source's original filename. + +The self-contained [RELAY-PROBES.js](RELAY-PROBES.js) and [RELAY-PROBES.json](RELAY-PROBES.json) are runnable recovery artifacts. The relay function was retained in the tool runtime across the filesystem reset. Its wrapper, explanatory checks, noncommuting example, and this record are new. **They establish no new native supplier and no new κ.** + +## Reproduce the surviving probe + +From the recovered workspace root, run: + + node recovery/RELAY-PROBES.js + +This requires Node.js and no external package. It writes recovery/RELAY-PROBES.json and records the source SHA-256 inside that receipt. An optional first argument chooses another output path. + +The receipt is deterministic. The finite menus complete in about one second in the replacement workspace. The principal exact checks concern integer operators and row ranks over the prime field F_65521. The cost screen uses ordinary double precision and is explicitly not an interval-certified admission test. + +## Model and interpretation + +For one of the eight correction channels, write t=L U, a=A, and v=t+a. The actual earlier construction has + +\[ +LK=13I,\quad C=[L,I],\quad B=\begin{bmatrix}-K\\11I\end{bmatrix}, +\quad CB=-2I,\quad H=I+BC,\quad H^2=I. +\] + +The compressed state tracks all t coordinates, all a coordinates, and one arbitrary cache coordinate for each local stage. It never assumes an initially zero cache. + +The default forward functional exchange occurs after the H update. It maps C X to the incoming dirty cache value and saves the old C X in that cache. The default backward exchange occurs before the negative H update and swaps C Y with its dirty cache by changing the selected A coordinates. The script also allows the latter target exchange before a forward update. + +The observed final rows are selected A, all caches, and the ordinary final Y functions. The missing ordinary X corrections are tested against their span. The productive restoration test additionally demands the original independent cache coordinates. + +A modular rank gap is used conservatively. In these negative menus the observation matrix has its full possible row rank; a larger stacked modular rank proves that the required rows are outside the rational observation span. A zero modular gap by itself would not prove rational feasibility. The selected positive below is therefore checked as a complete exact integer matrix and given explicit equations. + +This model grants common scalar charts. It does not establish native owner equivalence, physical frames, legal child phases, bank hosting, or correction costs. + +## Completed bounded menus + +| Menu | Cases | Result | +|---|---:|---| +| Additive relay with coefficient 0 or ±1, p=1 through5 | 15 | Main gap p−1; main plus cache gap p | +| Independent additive coefficients from −2 through2, p=3 | 125 | Gaps2 and3 throughout | +| Every subset of the standard exchanges, p=1 through4 | 680 | Productive restoration gap p throughout | +| Every bank assignment, p=1 or2 | 3,152 | Reusing a bank alone leaves productive gap p | +| Added target exchanges before forward stages, p=1 | 864 | 66 zero-gap scalar cases | +| Same extension, p=2 with the original distinct bank assignment | 256 | No zero-gap productive case | + +The scripts do not silently promote any of these finite menus into a universal lower bound. Earlier exploratory menus mentioned in the research conversation are not reproduced here if their exact generating code did not survive. + +### A pure negative-dirt relay cannot leave its source unchanged + +After a forward exchange, the source function equals d_forward. If a proposed next cache is set to −d_forward while that source function remains unchanged, the outputs obey the exact relation + +\[ +C X_{\rm out}+P_{\rm next}=0. +\] + +That makes the map singular. Additional independent untouched registers do not remove this output relation. A reversible relay must change the source function, retain another independent dirt component, or use a different chronological contract. + +The receipt contains a small exact matrix witness. This does not exclude all catalytic relays. + +## The positive p1 target-entry relay + +The simplest new zero-gap scalar case uses one arbitrary eight-field bank P. Its chronological operations are: + +1. Add H X to Y. +2. Before the backward update, exchange C Y and P, changing only the selected A coordinates. +3. Subtract H Y from X. +4. Before the last forward update, exchange C Y and the same P again. +5. Add H X to Y, then perform the terminal signed swap. + +There are no forward source exchanges in this selected variant. With v=C X_initial and P_initial=d, the exact endpoint is + +\[ +P_{\rm out}=d,\qquad +X_{U,\rm out}=X_U-Kv,\qquad +Y_{U,\rm out}=Y_U-K(v+d),\qquad +C Y_{\rm out}=-v-d. +\] + +Thus the required source function is observable as + +\[ +v=-C Y_{\rm out}-P_{\rm out}. +\] + +In the coordinate order (L X_U, L Y_U, X_A, Y_A, d), the complete matrix is + +\[ +\begin{pmatrix} +-12&0&-13&0&0\\ +-13&1&-13&0&-13\\ +10&1&11&1&-1\\ +12&-1&12&0&12\\ +0&0&0&0&1 +\end{pmatrix}. +\] + +The source checks this matrix exactly. Omitting the second target-entry exchange restores the productive information deficit and is a negative control. + +The physical issue is now specific: the function can be algebraically recovered, but adding K v into the final ordinary X holes is still unpaid. Cache movement and actual owner charts must also be verified. This endpoint is not the desired full outer operator. + +### Exact paired-query admission gate + +Suppose the first exchange uses C1 and selected-coordinate insertion E1, with C1 E1=I. If Y0 is the target before that exchange, then + +\[ +Y_1=(I-E_1C_1)Y_0+E_1d. +\] + +If the later exchange instead reads C2, the returned bank equals + +\[ +P_{\rm out}=C_2(I-E_1C_1)Y_0+C_2E_1d. +\] + +Restoring d for every independent Y0 and d requires + +\[ +C_2(I-E_1C_1)=0,\qquad C_2E_1=I. +\] + +Consequently C2=C2 E1 C1=C1. Actual physical transports must be included before making this comparison. Transporting arbitrary bank contents does not by itself turn one owner function into another. + +The source includes the negative example C1=(1,1), C2=(2,1), E1=(0,1)^T. Both queries have the same right inverse, yet the returned bank is d+Y0_U. This catches the precise mistake of assuming that equal channel count or equal insertion geometry implies equal functions. + +### A more promising changed premise: the same future query at both contacts + +The complex agent identified a useful next gate: earlier causal geometry may permit a future query chart at a current pre-Y contact. That source is unavailable after the reset, so its applicability must be reauthenticated. + +If both exchanges can deliberately use the same future query Q and right inverse E, QE=I, then the bank return does not require the intervening kernels to be equal or commute. + +Set T=I−EQ and let H0, H1, H2 be the actual three kernels, expressed in one correctly transported chart. The full equations are + +\[ +Y_0=Y+H_0X,\qquad +Y_1=TY_0+Ed,\qquad +X_1=X-H_1Y_1. +\] + +The second exchange gives + +\[ +Y_2=Y_1+E(QY_0-QY_1)=Y_0,\qquad P_{\rm out}=QY_1=d, +\] + +because T²=T, TE=0, QT=0, and QE=I. After the last forward update and terminal swap, + +\[ +X_{\rm out}=Y_0+H_2X_1,\qquad Y_{\rm out}=-X_1. +\] + +The recovery script verifies this identity with explicit noncommuting reflection matrices. It does not import the false premise H0=H1=H2. + +This is the strongest next finite native gate: prove that one chosen future Q and E are legal at both contacts, emit their actual frame path, and compare the full remaining correction invoice with the productive-helper budget. A failure of that geometry should stop this particular construction before a large search. + +## A dyadic catalytic reflection identity + +The literature lead was [Folkertsma et al., Fast Cliffords When Your Quantum Memory Is Full](https://arxiv.org/html/2609.40144v1), especially its dirty-workspace cancellation construction. That construction starts from a primitive whose workspace is already restored. An accumulating dirty logger does not satisfy that premise. + +For our dyadic setting, define + +\[ +U(B/2):(x,d)\mapsto(x+\tfrac12Bd,d),\qquad +L(C):(x,d)\mapsto(x,d+Cx). +\] + +The chronological three-factor map R=U(B/2)L(C)U(B/2) satisfies + +\[ +R(x,d)=\left(x+\tfrac12B(Cx+d),\,Cx\right), +\qquad (Cx,d)\mapsto(-d,Cx). +\] + +Therefore R² sends (x,d) to (Hx,−d). A final sign change of the cache gives (Hx,d), restoring arbitrary dirty state. After merging adjacent U factors, this uses two C queries and three B scatters. + +The source verifies a small exact dyadic instance, and rejects omission of the final cache sign and substitution of CB=−I. This identity is useful only if the required queries and scatters are physically available and affordable. It supplies no free native contacts. + +## Exact-bill screen for the productive p1 hosting proposal + +The complete frozen identity supplier was reported to add eight rank-eight calls to each **unpruned** clean-J word. It cannot be attached directly to the newer pruned bill. + +With H_r=r^(1−κ), the reported THREE profiles are: + +| Profile | H1 | H2 | H3 | H4 | H5 | +|---|---:|---:|---:|---:|---:| +| Pruned | 2,453 | 716 | 224 | 74 | 48 | +| Unpruned | 2,487 | 713 | 224 | 102 | 20 | + +Their difference is 34H1−3H2+28H4−28H5, with zero rank-mass difference. + +For p1, 408 extra productive helper fields give width632 and outer rank25. The balanced residual is 129H1+84H2. The resulting numerical screen is: + +| κ | Allowance after raw pruned supplier and residual | Extra complete identity cost | Remainder | +|---|---:|---:|---:| +| .001745 | 148.923223219 | 191.692397443 | −42.769174225 | +| .004 | 52.601345502 | 191.296597415 | −138.695251913 | + +Thus this already-complete local identity fallback fails the relaxed budget at both requested targets. It does not resolve the productive-cache controller. + +The originating profile receipt is missing. These values are a transparent recomputation of profiles supplied in the surviving agent messages, not a replacement for authenticating that receipt. + +## What was completed before the reset, and is now missing + +The JSON pin catalog records the exact known hashes of three BIT documents and five cache source/document/receipt files. These are expected hashes for future recovery, not verification of currently present files. + +The prior BIT scope contained finite native positive constructions. For its strongest 16-output variant, the recorded counts improved from582 helpers and38,415 ADDs to382 helpers and20,439 ADDs, with full finite margin gain83.3295445158 atκ=.004. The full h19 forest screen still failed:24,167,112 helpers and108,002,708 ADDs, with moment1.00631395 atκ=.004. No global BIT supplier or κ increase was established. + +The cache primitive had four provider variants and a full640-input operator audit. The cheaper suffix preserved the raw-J main operator, restored all408 native helpers, and produced P_out=d+C X_initial with8H9 and4,224 additional scalar allowance. It used no additional owned temporary field. The cache itself was not restored and received no productive-output credit. The earlier prefix comparison cost8H1+8H9 and8,560 scalar allowance and used one reusable owned temporary. + +The cache reproduction command, after authenticating and restoring its sources and dependencies, was: + + python3 -B research/kappa-accounted-composition/cache/dirty_log_forward.py + +Its historical generation produced18 artifacts. They are currently unavailable. Their existence and previously reported audit outcomes are not a fresh replay. + +## Recovery order + +1. Restore the attached base and the latest fully packaged cumulative checkpoint into a new checked location. +2. Authenticate restored files against the recorded pins and the package's own complete manifests. Preserve all scopes the user has not said were merged. +3. Recover the newest research sources only from authentic saved bytes. Do not manufacture old hashes, receipts, or missing generation outputs. +4. Re-run this self-contained relay receipt independently. +5. Authenticate the future-query contact geometry, then test the same-Q two-exchange relay with actual owner operators, every arbitrary dirty input, chronological cuts, and the complete bill. +6. Continue to require a whole supplier and outer admission before reporting a new κ. + diff --git a/research/kappa-rebuild/provenance/LOST-BIT-CACHE-PINS.json b/research/kappa-rebuild/provenance/LOST-BIT-CACHE-PINS.json new file mode 100644 index 0000000000..3bc8232e32 --- /dev/null +++ b/research/kappa-rebuild/provenance/LOST-BIT-CACHE-PINS.json @@ -0,0 +1,136 @@ +{ + "schema": "kappa-recovery-lost-bit-cache-pins-v1", + "status": "RECORDED HASHES ONLY; original bytes unavailable after execution workspace reset", + "recorded_provenance": "Hashes and results retained in this agent's pre-reset context. No source byte is claimed restored or reauthenticated by this catalog.", + "absent_files": [ + { + "path": "checkpoint-tree/research/kappa-accounted-composition/bit/README.md", + "sha256": "7fa63351804f367455d87eded9257f7858e6e6730f2dc148c3f1ccd051ea8136" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/bit/PROOF.md", + "sha256": "57b7593935c930ccff777d56c228b8a44c4666f6b96cf73dc9158f475c9179fe" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/bit/LITERATURE.md", + "sha256": "8afadda68d1e62048d8d31a9fc424da8fd85afab1cddb650de5ee79cdd2d1a22" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/cache/dirty_log_forward.py", + "sha256": "5c25d9af21c9d1840bb5a3cbb542333211422d8bce603e16abd9c92993af012e" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/cache/README.md", + "sha256": "87299e83cfc29f3b53f0e0e0ce680fe95618f526b1bb4e9880c91b348672e002" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/cache/REPRODUCE.json", + "sha256": "deda327c5f2ed62c89564f4488a6b0463a72bb793960fd3d2d702f8954db17f9" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/cache/forward-native/RESULT.json", + "sha256": "304ba639ca7627f63a4cfbb8d4b2657b9868b981c78d432976db32af6e73f0e5" + }, + { + "path": "checkpoint-tree/research/kappa-accounted-composition/cache/forward-native/PHASES.json.gz", + "sha256": "c1cdb8113eb45183ad3f76ad2a0b69973bd770da4f6420f618a11106d07cb492" + } + ], + "historical_cache": { + "reproduction_from_repository_root": "python3 -B research/kappa-accounted-composition/cache/dirty_log_forward.py", + "required_source_bytes_currently_available": false, + "provider_variants": [ + "THREE-base", + "THREE-extra", + "FOUR-base", + "FOUR-extra" + ], + "exact_operator_columns": 640, + "dirty_helpers_restored": 408, + "extra_cache_fields": 8, + "endpoint": "Original raw-J X/Y operator; cache P_out=d+C X_initial", + "cache_restored": false, + "productive_output_credit": false, + "strict_suffix": { + "additional_child_histogram": { + "9": 8 + }, + "additional_scalar_allowance": 4224, + "new_owned_persistent_fields": 0, + "owned_temporary_fields": 0, + "full_rank_mass": { + "THREE": 5165, + "FOUR": 5164 + }, + "full_scalar_allowance": { + "THREE": 119606, + "FOUR": 119584 + } + }, + "earlier_prefix_comparison": { + "additional_child_histogram": { + "1": 8, + "9": 8 + }, + "additional_scalar_allowance": 8560, + "owned_temporary_fields": 1, + "full_rank_mass": { + "THREE": 5173, + "FOUR": 5172 + }, + "full_scalar_allowance": { + "THREE": 123942, + "FOUR": 123920 + } + }, + "historical_generated_output_count": 18 + }, + "historical_bit": { + "native_positive_scope": "Finite 16-output blocks; no complete h19 supplier or outer admission", + "rows": [ + { + "variant": "baseline", + "helpers": 582, + "scalar_adds": 38415 + }, + { + "variant": "greedy pools", + "helpers": 441, + "scalar_adds": 38191, + "margin_gain_at_004": 60.96214 + }, + { + "variant": "fixed-dictionary MILP", + "helpers": 434, + "scalar_adds": 38166, + "margin_gain_at_004": 63.31412 + }, + { + "variant": "hierarchical functions", + "helpers": 386, + "scalar_adds": 38277, + "margin_gain_at_004": 81.6968 + }, + { + "variant": "deferred originals", + "helpers": 382, + "scalar_adds": 20439, + "margin_gain_at_004": 83.3295445158, + "margin_gain_at_001745": 36.2678693352 + } + ], + "whole_h19_forest": { + "helpers": 24167112, + "scalar_adds": 108002708, + "incidences": 42325920, + "moment_at_001745": 1.0027421578, + "moment_at_004": 1.00631395, + "native_full_h19_word_emitted": false + }, + "original_source_pins_pre_reset_checked": 123, + "primary_generators_pre_reset": 13, + "result": "Finite positive constructions; full tested supplier invoice failed; no new kappa", + "exact_source_bytes_currently_available": false + } +} + diff --git a/research/kappa-rebuild/provenance/PACKING-INDEPENDENT-AUDIT.md b/research/kappa-rebuild/provenance/PACKING-INDEPENDENT-AUDIT.md new file mode 100644 index 0000000000..7cfd8241ef --- /dev/null +++ b/research/kappa-rebuild/provenance/PACKING-INDEPENDENT-AUDIT.md @@ -0,0 +1,173 @@ +# Independent audit: productive helper packing and paired query swaps + +**Outcome: the conditional packing and recursive invoices pass. Native +admission remains open.** This audit read the three stable source/receipt +pairs without importing their construction routines. It independently +recounted the improved p=1 and p=2 inventories and rank bills, reviewed the +rational interval proof, and checked all twelve reported invoices with +110-digit Decimal evaluation. Every independent value lies inside its +reported outward interval. No predecessor native source was reconstructed. + +## Improved p=2: 213 added productive fields + +There are448 main fields and213 new productive helper fields, for661 +productive fields of ambient rank41. At each of the four outside stages the +original idle bins supply + +`90×9, 44×8, 56×7, 18×6, 16×5`. + +The new fields independently supply + +`123×9, 49×8, 12×5`. + +Their sum is exactly the required per-call inventory +`213×9, 93×8, 56×7, 18×6, 28×5`. At the middle stage, original fields supply +all195 roles of widths5 through8, and new fields supply all213 width9 roles. + +The48 new hosts with outside pattern `(9,9,9,5)` and middle9 use all41 ranks. +The116 with outside pattern `(9,9,9,0)` use36 and leave5. The49 with outside +pattern `(8,8,8,8)` use41. The original two-stage bins leave +`72H1+64H2`; their middle-stage bins leave `56H1+18H2+28H3+29H8`. +Thus the complete remainder is exactly + +\[ +128H_1+82H_2+28H_3+116H_5+29H_8. +\] + +Its rank mass is1188. Five pruned THREE calls have rank mass25465, so + +\[ +661\cdot41-(25465+1188)=448. +\] + +The40 rank-one children of the five H completions are then explicitly +subtracted. With `H_r=r^(1-C)`, the remaining allowances are: + +| C | Remaining allowance after H completions | +|---:|---:| +|0.001745|257.8060663457233916759| +|0.004|65.3169531797331110453| + +The213-host minimum is valid **for the stated middle window**:213 simultaneous +width9 roles cannot fit in any of its rank8 original idle bins, and a field +cannot serve two simultaneous native helper roles. This is not a global +minimum over changed schedules, interfaces, or split roles. The earlier219 +bound held the old original-bin allocation fixed; changing that allocation +does not contradict it. + +## Improved p=1: concentrated residual + +The six independently checked cyclic patterns have counts +`18,1,28,54,168,139`, totaling408 fields per controller. Across three replicas, +each stage has precisely three times the required helper inventory. Their +remainder is + +\[ +54H_1+28H_2+H_7+18H_{10}, +\] + +of rank mass297. All residual children are strictly smaller than the ambient +rank25. The rank identity is `632×25−3×5093−297=224`. After the24 rank-one H +completion children, the allowances are125.5333563352732444235 at C=0.001745 +and29.9957695582853966376 at C=0.004. This pattern is an exact feasible +construction in the width model; the MILP discovery is not promoted to a +rational optimality proof. + +## Interval verification + +The source uses the positive series +`log(x)=2 sum z^(2j+1)/(2j+1)`, with `z=(x−1)/(x+1)`. Its displayed geometric +tail bounds the omitted terms. Range reduction keeps z at most1/3. The +positive exponential series uses an upper geometric ratio for its remaining +terms; the largest exponent used here is below1. Reciprocal interval +directions, subtraction of positive cost coefficients, and45-place outward +rounding are correct. Independent Decimal evaluation is an additional +numerical backstop, not the source of the exact interval claim. + +## Why the whole-block helper lemma is sufficient, and what it assumes + +If a fully checked physical word is +`diag(DataOp,T1,...,T408)`, multiplying it on the input and output by block +diagonal operators acting only on individual helper registers leaves +`DataOp` and every zero data/helper cross block unchanged. Therefore free +monomial normal forms of the complete `T_i` can replace canonical helper +entrances with ordinary productive Fourier jobs without changing internal +ADDs. The371-rank synchronous-entrance objection does not apply to this +whole-block construction. + +This conclusion requires the following actual native facts: + +- Arbitrary-input restoration must make every physical helper block diagonal, + including zero helper-to-helper cross terms. +- Each actual relative transform must have the claimed Fourier rank, with + **allowed free monomial** input/output factors and correct absolute phase. + An arbitrary Clifford factorization is insufficient. Backward conjugate + conventions must be checked explicitly. +- Successive normalized jobs on one field must act on disjoint unfinished + axes. Original host windows must correspond to actual named registers + disjoint from that call's active main ports. The packing checks nominal + capacities and stage assignments; it does not emit this physical owner + schedule. +- Every credited productive field must finally compute a complete rank25 or + rank41 Fourier transform of its own arbitrary original input. A cache + containing another source function does not automatically earn that credit. +- The main-data controller still needs complete endpoint closure. Its credit + is conditional in these invoices. The inherited H owned C0 is an additional + reusable, uncredited scratch resource; no full Fourier job is credited to + it. Any outer simultaneous-storage requirement must allow that resource. +- Boundary scalar work, coefficient growth and precision obligations must be + integrated before an asymptotic admission. The reported leading allowance + does not itself perform those checks. + +No hidden additional recursive-rank debit was found **under these stated +premises**. The complete historical identity supplier remains over budget in +the reported favorable screens, so helper packing alone does not close the +research objective. + +## Same-future-query paired swaps + +The paired-swap identity is valid over a noncommuting operator ring. Let +`Q E=I`, `T=I−E Q`, and + +\[ +S(Y,P)=(TY+EP,QY). +\] + +Then `T²=T`, `TE=0`, `QT=0`, so `S²=I`. An intervening backward shear that +changes X but preserves Y and P does not prevent the second exchange from +restoring the bank. For arbitrary H0,H1,H2 the full equations are + +\[ +Y_0=Y+H_0X,\quad Y_1=TY_0+Ed,\quad X_1=X-H_1Y_1, +\] + +followed by `Y2=Y0, P=d` and terminal outputs +`Xout=Y0+H2 X1, Yout=−X1`. This is not an identity main-data endpoint. + +Choosing the same future Q at both pre-Y contacts is a useful changed premise, +but cheap evaluation of Q **does not imply** a cheap right inverse E. A right +inverse can require non-dyadic coefficients, unavailable frame changes, or +expensive scattering. The two physical bank transports must also intertwine +the chosen Q and E; equality of untransported symbolic labels is insufficient. + +There is a concrete candidate for the frozen selected-port query. If the +recovered physical formula is `Q_t=(R_A^T R/4) D_t` and +`R_A^T R_A=4I`, its selected-column restriction is `diag(D_t,a)`. Then +`E_t=P_A diag(D_t,a^-1)`, where `P_A` injects into the eight selected ports, +is an explicit right inverse. It is a cheap candidate only if those inverse +owner maps are allowed monomials preserving both contact frames, with the +required phases. This is the next finite native test, not an admitted free +scatter. + +## Audited source pins + +| File | SHA256 | +|---|---| +|productive_helper_forest.py|fc8ce7bd7ec4803d2cff816af54feee9ebb5198afe6e58bd33f8f85fde1014f4| +|productive_helper_refinement.py|d70e2feb935f93e9c9ce30340ce099dbefdc3ffe9ef54b3b338333efe815df71| +|productive_helper_p1_refinement.py|2982ef4fab8e7844a460579640ad13bf9eb0381b0e7b66fb805c21f73cea1b8d| +|PRODUCTIVE-HELPER-FOREST.json|63a4197062a6ea4a96555b4136210efc716317c1251d88c40a47a22b9926b89d| +|PRODUCTIVE-HELPER-REFINEMENT.json|8ae18b7b972d216a31dba54514925ee244824323eba452049b02ebce746b53d1| +|PRODUCTIVE-HELPER-P1-REFINEMENT.json|3437b3179af26bdde44278bfeb07247735eed3c6c7b8e2a2275057caf7ad4dbc| + +No new kappa is established by this audit. diff --git a/research/kappa-rebuild/provenance/PRIOR-SOURCE-PINS.json b/research/kappa-rebuild/provenance/PRIOR-SOURCE-PINS.json new file mode 100644 index 0000000000..2ce01c7882 --- /dev/null +++ b/research/kappa-rebuild/provenance/PRIOR-SOURCE-PINS.json @@ -0,0 +1,71 @@ +{ + "schema": "kappa-research-recovery-source-pins/1", + "recorded_date": "2026-10-21", + "status": "Historical identities retained from the completed pre-reset checks. These bytes are NOT present in this recovery package.", + "incident": { + "round_start_utc": "2026-10-21 00:13:31", + "environment_offline_window_utc": "approximately 02:25 to 02:43", + "observed_failure": "409 Conflict: environment_offline / Environment is not connected", + "reconnected_workspace": "The workspace returned with the attached base, prior cumulative archives, and almost all current-round sources absent. Three delayed files initially survived; additional self-contained recovery sources were then written.", + "full_current_round_replay_completed": false, + "full_current_round_cumulative_package_completed": false + }, + "reattach_to_restore_predecessor": [ + { + "filename": "kappa-evaluation-compact(7).zip", + "size_bytes": 413152788, + "sha256": "be617301d74b0e586055c9ff702582a4c22c796008db8362781230ed9b453e11", + "previous_regular_file_count": 78395, + "purpose": "Exact user-supplied base repository. A current public clone does not replace this artifact." + }, + { + "filename": "kappa-function-composition-checkpoint-20261020.zip", + "size_bytes": 155846068, + "sha256": "0ecb9ad58b6fb7f9bc97d47a3c692d3c6fbfb79dec087d6da91debd7973867ba", + "purpose": "Last fully packaged cumulative checkpoint; restores predecessor work, not the lost sources first created on October 21." + } + ], + "prior_inner_overlay": { + "filename": "kappa-function-composition-cumulative-overlay.zip", + "size_bytes": 128696496, + "sha256": "253bc0d5833fdd2d143879eb6c505092417ca5ec34373ce3b31d08b148333157", + "path_mappings": 3214, + "supplemental_mappings": 759, + "writes": 3208, + "base_identical_mappings": 6, + "content_blobs": 2267, + "historical_unavailable_entries": 332, + "historical_unavailable_dependencies": 34, + "selected_binary_omissions": 130, + "qualification": "Counts were authenticated before this reset. Historical unavailable entries and binary omissions must remain explicit; do not fabricate their recovery." + }, + "prior_checkpoint_receipts": { + "INTEGRITY_sha256": "aa3234229dcd09054354d7f85e68c9f8fb0a361c4d0626c281b7070202871180", + "VALIDATION_sha256": "ccf8ce9a5e4e8e9ecd90f4f2b5e3200d573e8299cc4a31396cd4fd5a79952b2c" + }, + "public_baseline_observed": { + "checked_utc": "2026-10-21 01:26:40", + "main_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "conditional_complex_endpoint_PR4085": "0.001730970419199163702738", + "separate_stronger_BIT_checkpoint": "0.001738137188329345368427", + "active_outer_BIT_endpoint": "0.001733427039589844234749", + "outer_formula": "kappa = C - 10^-46 - 10^-70; BIT premise > C - 1.5*10^-104", + "new_kappa_admitted_this_round": false, + "public_writes_performed": false + }, + "lost_current_round_pins": { + "complex_H112_receipt": "d14edeacd37e526f8723ebdfb22f6b8d470ebf1daa7ef7c87aacfe7bae00484c", + "complex_H112_KERNEL_gzip": "12c17f4e14a58a2b202ce21c55458366cbf9f9377b5255ececb087c9c87d5fef", + "dirty_log_forward.py": "5c25d9af21c9d1840bb5a3cbb542333211422d8bce603e16abd9c92993af012e", + "dirty_log_forward_RESULT": "304ba639ca7627f63a4cfbb8d4b2657b9868b981c78d432976db32af6e73f0e5", + "dirty_log_forward_PHASES_gzip": "c1cdb8113eb45183ad3f76ad2a0b69973bd770da4f6420f618a11106d07cb492", + "dirty_log_forward_README": "87299e83cfc29f3b53f0e0e0ce680fe95618f526b1bb4e9880c91b348672e002", + "dirty_log_forward_REPRODUCE": "deda327c5f2ed62c89564f4488a6b0463a72bb793960fd3d2d702f8954db17f9", + "BIT_README": "7fa63351804f367455d87eded9257f7858e6e6730f2dc148c3f1ccd051ea8136", + "BIT_PROOF": "57b7593935c930ccff777d56c228b8a44c4666f6b96cf73dc9158f475c9179fe", + "BIT_LITERATURE": "8afadda68d1e62048d8d31a9fc424da8fd85afab1cddb650de5ee79cdd2d1a22", + "global_BIT_forest_source": "cceb35b6cfa0ff65b24a7d8a0a25e5517e5fc011542b5cb3a1283f51cdf446c7", + "global_BIT_forest_RESULT": "b070d4c2c138e0f6be654f0590f0443a038645248c00b82cbf891ace3d18c6e5" + }, + "recovery_rule": "Only claim restoration of an original frozen file if its bytes match its recorded hash. New algebraic reconstructions must use distinct filenames and state their new provenance." +} diff --git a/research/kappa-rebuild/provenance/RECOVERY-VALIDATION.json b/research/kappa-rebuild/provenance/RECOVERY-VALIDATION.json new file mode 100644 index 0000000000..180fc34ea8 --- /dev/null +++ b/research/kappa-rebuild/provenance/RECOVERY-VALIDATION.json @@ -0,0 +1,125 @@ +{ + "alias_freeze_verified": true, + "complete_native_supplier_replayed": false, + "independent_invoice_checks": [ + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p1" + }, + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p1_every_host_has_width9" + }, + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p2" + }, + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p2_219_hosts" + }, + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p2_changed_original_assignment_213_hosts" + }, + { + "checks": [ + { + "C": "0.001745", + "independent_decimal_inside_rational_interval": true + }, + { + "C": "0.004", + "independent_decimal_inside_rational_interval": true + } + ], + "name": "p1_concentrated_residual" + } + ], + "integrity_inventory_files_checked": null, + "native_alias_checked": "syntax only; not imported or executed", + "new_kappa_claimed": false, + "outer_theorem_replayed": false, + "prior_cumulative_overlay_replayed": false, + "qualification": "The old native source profile is a stated input. A positive conditional allowance does not certify main-controller closure or helper monomial adapters.", + "schema": "kappa-recovery-validation/1", + "source_receipt_pairs": 5, + "stages": [ + { + "receipt": "checkpoint-tree/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-FOREST.json", + "sha256": "63a4197062a6ea4a96555b4136210efc716317c1251d88c40a47a22b9926b89d", + "stage": "productive_helper_forest", + "status": "PASS_BYTE_IDENTICAL" + }, + { + "receipt": "checkpoint-tree/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-REFINEMENT.json", + "sha256": "8ae18b7b972d216a31dba54514925ee244824323eba452049b02ebce746b53d1", + "stage": "productive_helper_213", + "status": "PASS_BYTE_IDENTICAL" + }, + { + "receipt": "checkpoint-tree/research/kappa-accounted-composition/recursion/PRODUCTIVE-HELPER-P1-REFINEMENT.json", + "sha256": "3437b3179af26bdde44278bfeb07247735eed3c6c7b8e2a2275057caf7ad4dbc", + "stage": "productive_helper_p1", + "status": "PASS_BYTE_IDENTICAL" + }, + { + "receipt": "checkpoint-tree/research/kappa-accounted-composition/alias/CONSTANT-CHART-AUDIT.json", + "sha256": "18b78c0877364e75bd487bec9a2f9c31c08db3401deabc026fa5bc64e6be8425", + "stage": "alias_constant_chart", + "status": "PASS_BYTE_IDENTICAL" + }, + { + "receipt": "recovery/RELAY-PROBES.json", + "sha256": "cf970b524e074c032cf57e3d5aa0bdd6b7017159a4ef59bb5d583f3083a17df6", + "stage": "relay_menus_and_identities", + "status": "PASS_BYTE_IDENTICAL" + } + ], + "status": "PASS_RECOVERY_SCOPE_ONLY" +} diff --git a/research/kappa-rebuild/provenance/RELAY-PROBES.js b/research/kappa-rebuild/provenance/RELAY-PROBES.js new file mode 100644 index 0000000000..e87a46b4f3 --- /dev/null +++ b/research/kappa-rebuild/provenance/RELAY-PROBES.js @@ -0,0 +1,330 @@ +#!/usr/bin/env node +"use strict"; +/* +Recovery-only scalar relay probes, 2026-10-21. +The original repository and frozen research sources disappeared during an +execution-environment reset. This file is NEW recovery work. It is NOT a +byte-identical reconstruction of any missing scientific source. + +Model: one constant chart per eight-dimensional channel, LK=13 I, +C=[L,I], B=[-K;11 I], CB=-2 I, H=I+BC. +The scalar coordinates are t=L U, a=A, with Cx=t+a. +No native frames, owner maps, child phases, dirty-helper hosting, or outer +admission are established here. Exact modular row ranks are screening +witnesses; the positive p=1 endpoint is additionally checked as an exact +integer matrix. Numerical H_r cost screens are ordinary double precision. + +The core function probe below was retained in functions.store across the +filesystem reset. Explicit safe-integer and observation-rank guards were then +added. Its companion probeMatrix changes only the returned data. +*/ +const assert=require("node:assert/strict"); +const fs=require("node:fs"); +const path=require("node:path"); +const crypto=require("node:crypto"); +function safeMul(a,b){const v=a*b;assert(Number.isSafeInteger(v),"unsafe integer product");return v;} +function safeSub(a,b){const v=a-b;assert(Number.isSafeInteger(v),"unsafe integer difference");return v;} + +function probe(p,lam=0,ids=[],premask=0,swapmask=-1){ +const n=2*p,N=3*n+1,E=Array.from({length:N},(_,i)=>Array.from({length:N},(_,j)=>+(i===j))), M=E.map(r=>r.slice()); +const plus=(a,b,k=1)=>a.map((v,j)=>v+k*b[j]), row=(i,j,k)=>{M[i]=plus(M[i],M[j],k)}; +const ev=[]; +let s=0;ev.push(['F',0,s++],['B',0,s++]); +for(let j=1;j=0;j--)ev.push(['bridge',j,j+1,1]); +for(const q of ev){ +if(q[0]==='bridge'){let x=2*q[1],y=2*q[2],k=q[3];for(const off of [0,n]){row(off+y,off+x,k);row(off+x+1,off+y+1,-k);}continue;} +const [type,i,st]=q,x=2*i,y=x+1,di=2*n+(ids[st]??st); +if(type==='F'){ +if((premask>>(st/2))&1){const oldq=plus(M[y],M[n+y]);M[n+y]=plus(M[di],M[y],-1);M[di]=oldq;} +let t=M[x].slice(),a=M[n+x].slice(),v=plus(t,a),d=M[di].slice(); +M[y]=plus(plus(M[y],t),v,-13);M[n+y]=plus(plus(M[n+y],t,-1),v,12); +if((swapmask>>st)&1){M[x]=plus(plus(t,d,13),v,-13);M[n+x]=plus(d,M[x],-1);M[di]=v;} +if(lam!==0&&st<2*p){M[di+1]=plus(M[di+1],d,-(Array.isArray(lam)?lam[st/2]:lam));} +}else{ +let t=M[y].slice(),v=plus(t,M[n+y]),d=M[di].slice();if((swapmask>>st)&1){M[n+y]=plus(d,t,-1);M[di]=v;}else{d=v;} +M[x]=plus(plus(M[x],t,-1),d,13);M[n+x]=plus(plus(M[n+x],t),d,-12); +} +} +for(let i=0;i-v);}} +assert(M.every(row=>row.every(Number.isSafeInteger)),"non-safe whole operator entry"); +const obs=M.slice(n,2*n).concat(M.slice(2*n),Array.from({length:p},(_,i)=>M[2*i+1])); +const demanded=Array.from({length:p},(_,i)=>plus(M[2*i],E[2*i],-1)); +const cacheDem=Array.from({length:n+1},(_,j)=>E[2*n+j]); +const mod=65521,norm=x=>{assert(Number.isSafeInteger(x),"non-safe modular input");return (x%mod+mod)%mod;}; +function rank(A){A=A.map(r=>r.map(norm));let k=0; +for(let j=0;jArray.from({length:N},(_,j)=>+(i===j))), M=E.map(r=>r.slice()); +const plus=(a,b,k=1)=>a.map((v,j)=>v+k*b[j]), row=(i,j,k)=>{M[i]=plus(M[i],M[j],k)}; +const ev=[]; +let s=0;ev.push(['F',0,s++],['B',0,s++]); +for(let j=1;j=0;j--)ev.push(['bridge',j,j+1,1]); +for(const q of ev){ +if(q[0]==='bridge'){let x=2*q[1],y=2*q[2],k=q[3];for(const off of [0,n]){row(off+y,off+x,k);row(off+x+1,off+y+1,-k);}continue;} +const [type,i,st]=q,x=2*i,y=x+1,di=2*n+(ids[st]??st); +if(type==='F'){ +if((premask>>(st/2))&1){const oldq=plus(M[y],M[n+y]);M[n+y]=plus(M[di],M[y],-1);M[di]=oldq;} +let t=M[x].slice(),a=M[n+x].slice(),v=plus(t,a),d=M[di].slice(); +M[y]=plus(plus(M[y],t),v,-13);M[n+y]=plus(plus(M[n+y],t,-1),v,12); +if((swapmask>>st)&1){M[x]=plus(plus(t,d,13),v,-13);M[n+x]=plus(d,M[x],-1);M[di]=v;} +if(lam!==0&&st<2*p){M[di+1]=plus(M[di+1],d,-(Array.isArray(lam)?lam[st/2]:lam));} +}else{ +let t=M[y].slice(),v=plus(t,M[n+y]),d=M[di].slice();if((swapmask>>st)&1){M[n+y]=plus(d,t,-1);M[di]=v;}else{d=v;} +M[x]=plus(plus(M[x],t,-1),d,13);M[n+x]=plus(plus(M[n+x],t),d,-12); +} +} +for(let i=0;i-v);}} +assert(M.every(row=>row.every(Number.isSafeInteger)),"non-safe whole operator entry"); +const obs=M.slice(n,2*n).concat(M.slice(2*n),Array.from({length:p},(_,i)=>M[2*i+1])); +const demanded=Array.from({length:p},(_,i)=>plus(M[2*i],E[2*i],-1)); +const cacheDem=Array.from({length:n+1},(_,j)=>E[2*n+j]); +const mod=65521,norm=x=>{assert(Number.isSafeInteger(x),"non-safe modular input");return (x%mod+mod)%mod;}; +function rank(A){A=A.map(r=>r.map(norm));let k=0; +for(let j=0;jArray(b[0].length).fill(0)); + for(let i=0;i (-d,Cx)", + general_closed_contract:"(x,d) -> ((I+BC)x,d), provided CB=-2I", + primitive_counts_after_adjacent_U_merge:{C_queries:2,B_scatters:3,cache_sign_changes:1}, + negative_controls:["omit final cache negation","replace CB=-2 by CB=-1"], + native_contact_cost_established:false}; +} +function positiveEndpoint(){ + const r=probeMatrix(1,0,[0,0,0],2,2),indices=[0,1,2,3,4]; + const actual=indices.map(i=>indices.map(j=>r.M[i][j]||0)); + const expected=[ + [-12,0,-13,0,0], + [-13,1,-13,0,-13], + [10,1,11,1,-1], + [12,-1,12,0,12], + [0,0,0,0,1] + ]; + assert.deepEqual(actual,expected); + assertExpected(probe(1,0,[0,0,0],2,2),0,0); + assert.equal(probe(1,0,[0,0,0],0,2).mainAndCacheGap,1); + return { + p:1,used_banks:1,coordinate_order:["L X_U","L Y_U","X_A","Y_A","d"], + itinerary:[ + "F0: add H X to Y", + "B0-pre: exchange C Y and P using Y_A only", + "B0: subtract H Y from X", + "F0-last-pre: exchange C Y and the same P using Y_A only", + "F0-last: add H X to Y", + "terminal signed swap: X<-Y, Y<-(-old X)" + ], + exact_integer_matrix:actual, + endpoint:{ + P_out:"d", + X_U_out:"X_U - K v", + Y_U_out:"Y_U - K(v+d)", + C_Y_out:"-v-d", + v:"C X_initial", + recover_v:"-(C Y_out)-P_out" + }, + proof_status:"exact integer scalar-chart operator; no native proof", + unpaid:["bank movement between the two entry frames","native owner-map equivalence for C", + "scatter K v into final X holes","all main/selected-output endpoint corrections", + "full productive-bank frame journey and recursive invoice"], + negative_control:"omit second target-entry exchange: productive observation deficit returns to1" + }; +} +function invoiceScreens(){ + const pruned={1:2453,2:716,3:224,4:74,5:48},old={1:2487,2:713,3:224,4:102,5:20}; + const rows=[]; + for(const kappa of [0.001745,0.004]){ + const H=r=>r?Math.pow(r,1-kappa):0; + const sum=h=>Object.entries(h).reduce((s,[r,n])=>s+n*H(+r),0); + const local=sum(pruned),delta=sum(old)-local,residual=129*H(1)+84*H(2); + const raw=632*H(25)-3*local-residual; + rows.push({kappa,H_definition:"r^(1-kappa)",p:1,productive_extra_fields:408,width:632, + outer_rank:25,residual_profile:{1:129,2:84},raw_allowance:raw, + local_unpruned_minus_pruned:delta, + complete_identity_surcharge:3*(8*H(8)+delta), + after_complete_identity:raw-3*(8*H(8)+delta)}); + } + return {status:"double-precision screen, not interval-certified admission",provider:"THREE", + pruned_histogram:pruned,unpruned_histogram:old,rows, + provenance:"histograms supplied by C from frozen receipt before reset; old receipt bytes currently unavailable"}; +} +function singularRelay(){ + const bad=[[0,1,0],[0,-1,0],[0,0,1]]; + assert.deepEqual(bad[0].map((v,j)=>v+bad[1][j]),[0,0,0]); + return {input:["v","d","b"],bad_output:["d","-d","b"],matrix:bad, + dependent_output_relation:"output0+output1=0", + conclusion:"Pure next dirt=-d while retaining source query=d is singular; extra independent untouched registers do not remove this output relation.", + scope:"Does not rule out changing the source query, retaining extra dirt, or another chronological controller."}; +} +function ownerGate(){ + const C1=[[1,1]],C2=[[2,1]],E=[[0],[1]],I=[[1,0],[0,1]]; + const EC=matrixProduct(E,C1),T=I.map((row,i)=>row.map((v,j)=>v-EC[i][j])); + const remaining=matrixProduct(C2,T),dCoefficient=matrixProduct(C2,E); + assert.deepEqual(remaining,[[1,0]]);assert.deepEqual(dCoefficient,[[1]]); + return {C1,C2,E,remaining_old_Y_coefficient:remaining,d_coefficient:dCoefficient, + mismatch_control:"P_out=Y0_U+d, so the dirty bank is not restored", + exact_condition:"C2(I-E1 C1)=0 and C2 E1=I, with C1 E1=I, force C2=C1", + native_condition:"Use actual common-chart conjugations and cache/target transports before comparing C1 and C2", + native_owner_gate_evaluated:false}; +} +function commonFutureQuery(){ + // This checks a noncommuting matrix instance of the operator identity in + // README. It does not certify availability of these queries in native frames. + const H0=[[-1,0],[0,1]],H1=[[0,-1],[-1,0]],H2=[[1,0],[-4,-1]]; + const Q=[2,1],E=[0,1],N=5; + const id=Array.from({length:N},(_,i)=>Array.from({length:N},(_,j)=>+(i===j))); + const M=id.map(row=>row.slice()); + const plus=(a,b,k=1)=>a.map((v,j)=>v+k*b[j]); + function addBlock(dst,src,H,sign){ + const old=M.map(row=>row.slice()); + for(let i=0;i<2;i++)for(let j=0;j<2;j++)M[dst+i]=plus(M[dst+i],old[src+j],sign*H[i][j]); + } + function swapQuery(){ + const oldP=M[4].slice(),value=plus(M[2].map(v=>Q[0]*v),M[3],Q[1]); + for(let i=0;i<2;i++)M[2+i]=plus(M[2+i],plus(oldP,value,-1),E[i]); + M[4]=value; + } + addBlock(2,0,H0,1);const Y0=[M[2].slice(),M[3].slice()]; + swapQuery();addBlock(0,2,H1,-1);const X1=[M[0].slice(),M[1].slice()]; + swapQuery();assert.deepEqual([M[2],M[3]],Y0);assert.deepEqual(M[4],id[4]); + addBlock(2,0,H2,1); + for(let i=0;i<2;i++){const old=M[i];M[i]=M[2+i];M[2+i]=old.map(v=>-v||0);} + const expectedX=Y0.map(row=>row.slice()); + for(let i=0;i<2;i++)for(let j=0;j<2;j++)expectedX[i]=plus(expectedX[i],X1[j],H2[i][j]); + assert.deepEqual(M.slice(0,2),expectedX);assert.deepEqual(M.slice(2,4),X1.map(row=>row.map(v=>-v||0))); + assert.notDeepEqual(matrixProduct(H1,H2),matrixProduct(H2,H1)); + return {H0,H1,H2,Q,E,noncommuting:true,exact_final_matrix:M, + scalar_identity_assumed:false,bank_restored:true, + general_equations:{T:"I-EQ",Y0:"Y+H0 X",Y1:"T Y0+E d",X1:"X-H1 Y1", + second_swap:"Y2=Y0 and P=d",terminal:"X_out=Y0+H2 X1; Y_out=-X1"}, + native_gate:"The same future query Q and its right inverse E must be legal at both pre-Y contacts after actual owner transports", + still_unpaid:["native bank path","main-output correction","complete recursive invoice"]}; +} +function main(){ + const report={ + schema:"kappa-recovery-relay-probes-v1", + status:"PASS scalar recovery probes; native supplier and new kappa NOT established", + recovery_date:"2026-10-21", + provenance:{ + original_scientific_sources_available:false, + reconstruction_of_missing_files:false, + model_source:"function source retained in functions.store across workspace reset", + modular_prime:65521, + integer_arithmetic:"Small exact integer matrices; elimination products below 2^53", + rank_interpretation:"Ranks over F_65521. A full-row-rank observation matrix and a larger stacked rank give a rational obstruction. A modular zero gap alone is not a rational certificate; selected positive endpoint is separately checked exactly." + }, + signed_additive_relays:signedAdditiveMenu(), + optional_swaps:optionalSwapMenu(), + reused_bank_assignments:reuseMenu(), + early_target_exchange:earlyTargetMenu(), + selected_positive:positiveEndpoint(), + quarter_turn:quarterTurn(), + singular_pure_relay:singularRelay(), + owner_mismatch_gate:ownerGate(), + same_future_query_identity:commonFutureQuery(), + invoice_screens:invoiceScreens(), + source_sha256:crypto.createHash("sha256").update(fs.readFileSync(__filename)).digest("hex") + }; + const output=process.argv[2]||path.join(__dirname,"RELAY-PROBES.json"); + fs.writeFileSync(output,JSON.stringify(report,null,2)+"\n"); + process.stdout.write(JSON.stringify({status:report.status,output, + source_sha256:report.source_sha256, + complete_native_supplier:false, + recovery_positive_p1_count:report.early_target_exchange[0].successes.length})+"\n"); +} +if(require.main===module)main(); +module.exports={probe,probeMatrix,matrixProduct}; + diff --git a/research/kappa-rebuild/provenance/RELAY-PROBES.json b/research/kappa-rebuild/provenance/RELAY-PROBES.json new file mode 100644 index 0000000000..4b07518760 --- /dev/null +++ b/research/kappa-rebuild/provenance/RELAY-PROBES.json @@ -0,0 +1,1204 @@ +{ + "schema": "kappa-recovery-relay-probes-v1", + "status": "PASS scalar recovery probes; native supplier and new kappa NOT established", + "recovery_date": "2026-10-21", + "provenance": { + "original_scientific_sources_available": false, + "reconstruction_of_missing_files": false, + "model_source": "function source retained in functions.store across workspace reset", + "modular_prime": 65521, + "integer_arithmetic": "Small exact integer matrices; elimination products below 2^53", + "rank_interpretation": "Ranks over F_65521. A full-row-rank observation matrix and a larger stacked rank give a rational obstruction. A modular zero gap alone is not a rational certificate; selected positive endpoint is separately checked exactly." + }, + "signed_additive_relays": { + "uniform_count": 15, + "rows": [ + { + "p": 1, + "lam": 0, + "obsRank": 6, + "mainDemandGap": 0, + "mainAndCacheGap": 1, + "integralMax": 13 + }, + { + "p": 1, + "lam": 1, + "obsRank": 6, + "mainDemandGap": 0, + "mainAndCacheGap": 1, + "integralMax": 13 + }, + { + "p": 1, + "lam": -1, + "obsRank": 6, + "mainDemandGap": 0, + "mainAndCacheGap": 1, + "integralMax": 13 + }, + { + "p": 2, + "lam": 0, + "obsRank": 11, + "mainDemandGap": 1, + "mainAndCacheGap": 2, + "integralMax": 13 + }, + { + "p": 2, + "lam": 1, + "obsRank": 11, + "mainDemandGap": 1, + "mainAndCacheGap": 2, + "integralMax": 13 + }, + { + "p": 2, + "lam": -1, + "obsRank": 11, + "mainDemandGap": 1, + "mainAndCacheGap": 2, + "integralMax": 13 + }, + { + "p": 3, + "lam": 0, + "obsRank": 16, + "mainDemandGap": 2, + "mainAndCacheGap": 3, + "integralMax": 13 + }, + { + "p": 3, + "lam": 1, + "obsRank": 16, + "mainDemandGap": 2, + "mainAndCacheGap": 3, + "integralMax": 13 + }, + { + "p": 3, + "lam": -1, + "obsRank": 16, + "mainDemandGap": 2, + "mainAndCacheGap": 3, + "integralMax": 13 + }, + { + "p": 4, + "lam": 0, + "obsRank": 21, + "mainDemandGap": 3, + "mainAndCacheGap": 4, + "integralMax": 13 + }, + { + "p": 4, + "lam": 1, + "obsRank": 21, + "mainDemandGap": 3, + "mainAndCacheGap": 4, + "integralMax": 13 + }, + { + "p": 4, + "lam": -1, + "obsRank": 21, + "mainDemandGap": 3, + "mainAndCacheGap": 4, + "integralMax": 13 + }, + { + "p": 5, + "lam": 0, + "obsRank": 26, + "mainDemandGap": 4, + "mainAndCacheGap": 5, + "integralMax": 13 + }, + { + "p": 5, + "lam": 1, + "obsRank": 26, + "mainDemandGap": 4, + "mainAndCacheGap": 5, + "integralMax": 13 + }, + { + "p": 5, + "lam": -1, + "obsRank": 26, + "mainDemandGap": 4, + "mainAndCacheGap": 5, + "integralMax": 13 + } + ], + "independent_p3_count": 125, + "independent_p3_histogram": { + "2,3": 125 + } + }, + "optional_swaps": { + "count": 680, + "rows": [ + { + "p": 1, + "count": 8, + "histogram": { + "1,1": 4, + "0,1": 4 + } + }, + { + "p": 2, + "count": 32, + "histogram": { + "2,2": 16, + "1,2": 16 + } + }, + { + "p": 3, + "count": 128, + "histogram": { + "3,3": 64, + "2,3": 64 + } + }, + { + "p": 4, + "count": 512, + "histogram": { + "4,4": 256, + "3,4": 256 + } + } + ] + }, + "reused_bank_assignments": { + "count": 3152, + "rows": [ + { + "p": 1, + "count": 27, + "histogram": { + "1,1": 9, + "0,1": 18 + } + }, + { + "p": 2, + "count": 3125, + "histogram": { + "2,2": 625, + "1,2": 2500 + } + } + ] + }, + "early_target_exchange": [ + { + "p": 1, + "count": 864, + "histogram": { + "1,1": 312, + "0,1": 486, + "0,0": 66 + }, + "successes": [ + { + "ids": [ + 0, + 0, + 0 + ], + "preMask": 2, + "swapMask": 2 + }, + { + "ids": [ + 0, + 0, + 0 + ], + "preMask": 2, + "swapMask": 3 + }, + { + "ids": [ + 0, + 0, + 0 + ], + "preMask": 2, + "swapMask": 6 + }, + { + "ids": [ + 0, + 0, + 0 + ], + 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"preMask": 2, + "swapMask": 7 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 2, + "swapMask": 2 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 2, + "swapMask": 3 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 2, + "swapMask": 6 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 2, + "swapMask": 7 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 3, + "swapMask": 0 + }, + { + "ids": [ + 2, + 2, + 2 + ], + "preMask": 3, + "swapMask": 4 + } + ] + }, + { + "p": 2, + "count": 256, + "histogram": { + "2,2": 32, + "1,2": 128, + "0,2": 96 + }, + "successes": [] + } + ], + "selected_positive": { + "p": 1, + "used_banks": 1, + "coordinate_order": [ + "L X_U", + "L Y_U", + "X_A", + "Y_A", + "d" + ], + "itinerary": [ + "F0: add H X to Y", + "B0-pre: exchange C Y and P using Y_A only", + "B0: subtract H Y from X", + "F0-last-pre: exchange C Y and the same P using Y_A only", + "F0-last: add H X to Y", + "terminal signed swap: X<-Y, Y<-(-old X)" + ], + "exact_integer_matrix": [ + [ + -12, + 0, + -13, + 0, + 0 + ], + [ + -13, + 1, + -13, + 0, + -13 + ], + [ + 10, + 1, + 11, + 1, + -1 + ], + [ + 12, + -1, + 12, + 0, + 12 + ], + [ + 0, + 0, + 0, + 0, + 1 + ] + ], + "endpoint": { + "P_out": "d", + "X_U_out": "X_U - K v", + "Y_U_out": "Y_U - K(v+d)", + "C_Y_out": "-v-d", + "v": "C X_initial", + "recover_v": "-(C Y_out)-P_out" + }, + "proof_status": "exact integer scalar-chart operator; no native proof", + "unpaid": [ + "bank movement between the two entry frames", + "native owner-map equivalence for C", + "scatter K v into final X holes", + "all main/selected-output endpoint corrections", + "full productive-bank frame journey and recursive invoice" + ], + "negative_control": "omit second target-entry exchange: productive observation deficit returns to1" + }, + "quarter_turn": { + "B": [ + [ + -1 + ], + [ + 1 + ] + ], + "C": [ + [ + 1, + -1 + ] + ], + "CB": -2, + "R": [ + [ + 0.5, + 0.5, + -0.5 + ], + [ + 0.5, + 0.5, + 0.5 + ], + [ + 1, + -1, + 0 + ] + ], + "R_squared": [ + [ + 0, + 1, + 0 + ], + [ + 1, + 0, + 0 + ], + [ + 0, + 0, + -1 + ] + ], + "closed": [ + [ + 0, + 1, + 0 + ], + [ + 1, + 0, + 0 + ], + [ + 0, + 0, + 1 + ] + ], + "expected": [ + [ + 0, + 1, + 0 + ], + [ + 1, + 0, + 0 + ], + [ + 0, + 0, + 1 + ] + ], + "query_map": "(Cx,d) -> (-d,Cx)", + "general_closed_contract": "(x,d) -> ((I+BC)x,d), provided CB=-2I", + "primitive_counts_after_adjacent_U_merge": { + "C_queries": 2, + "B_scatters": 3, + "cache_sign_changes": 1 + }, + "negative_controls": [ + "omit final cache negation", + "replace CB=-2 by CB=-1" + ], + "native_contact_cost_established": false + }, + "singular_pure_relay": { + "input": [ + "v", + "d", + "b" + ], + "bad_output": [ + "d", + "-d", + "b" + ], + "matrix": [ + [ + 0, + 1, + 0 + ], + [ + 0, + -1, + 0 + ], + [ + 0, + 0, + 1 + ] + ], + "dependent_output_relation": "output0+output1=0", + "conclusion": "Pure next dirt=-d while retaining source query=d is singular; extra independent untouched registers do not remove this output relation.", + "scope": "Does not rule out changing the source query, retaining extra dirt, or another chronological controller." + }, + "owner_mismatch_gate": { + "C1": [ + [ + 1, + 1 + ] + ], + "C2": [ + [ + 2, + 1 + ] + ], + "E": [ + [ + 0 + ], + [ + 1 + ] + ], + "remaining_old_Y_coefficient": [ + [ + 1, + 0 + ] + ], + "d_coefficient": [ + [ + 1 + ] + ], + "mismatch_control": "P_out=Y0_U+d, so the dirty bank is not restored", + "exact_condition": "C2(I-E1 C1)=0 and C2 E1=I, with C1 E1=I, force C2=C1", + "native_condition": "Use actual common-chart conjugations and cache/target transports before comparing C1 and C2", + "native_owner_gate_evaluated": false + }, + "same_future_query_identity": { + "H0": [ + [ + -1, + 0 + ], + [ + 0, + 1 + ] + ], + "H1": [ + [ + 0, + -1 + ], + [ + -1, + 0 + ] + ], + "H2": [ + [ + 1, + 0 + ], + [ + -4, + -1 + ] + ], + "Q": [ + 2, + 1 + ], + "E": [ + 0, + 1 + ], + "noncommuting": true, + "exact_final_matrix": [ + [ + 2, + 0, + -1, + 0, + 1 + ], + [ + -11, + 0, + 7, + 1, + -4 + ], + [ + -3, + 0, + 2, + 0, + -1 + ], + [ + 1, + -1, + -1, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 1 + ] + ], + "scalar_identity_assumed": false, + "bank_restored": true, + "general_equations": { + "T": "I-EQ", + "Y0": "Y+H0 X", + "Y1": "T Y0+E d", + "X1": "X-H1 Y1", + "second_swap": "Y2=Y0 and P=d", + "terminal": "X_out=Y0+H2 X1; Y_out=-X1" + }, + "native_gate": "The same future query Q and its right inverse E must be legal at both pre-Y contacts after actual owner transports", + "still_unpaid": [ + "native bank path", + "main-output correction", + "complete recursive invoice" + ] + }, + "invoice_screens": { + "status": "double-precision screen, not interval-certified admission", + "provider": "THREE", + "pruned_histogram": { + "1": 2453, + "2": 716, + "3": 224, + "4": 74, + "5": 48 + }, + "unpruned_histogram": { + "1": 2487, + "2": 713, + "3": 224, + "4": 102, + "5": 20 + }, + "rows": [ + { + "kappa": 0.001745, + "H_definition": "r^(1-kappa)", + "p": 1, + "productive_extra_fields": 408, + "width": 632, + "outer_rank": 25, + "residual_profile": { + "1": 129, + "2": 84 + }, + "raw_allowance": 148.92322321859046, + "local_unpruned_minus_pruned": 0.1292770134168677, + "complete_identity_surcharge": 191.69239744322454, + "after_complete_identity": -42.76917422463407 + }, + { + "kappa": 0.004, + "H_definition": "r^(1-kappa)", + "p": 1, + "productive_extra_fields": 408, + "width": 632, + "outer_rank": 25, + "residual_profile": { + "1": 129, + "2": 84 + }, + "raw_allowance": 52.601345501993364, + "local_unpruned_minus_pruned": 0.2956617045765597, + "complete_identity_surcharge": 191.29659741548195, + "after_complete_identity": -138.6952519134886 + } + ], + "provenance": "histograms supplied by C from frozen receipt before reset; old receipt bytes currently unavailable" + }, + "source_sha256": "d40c208b691ff225811730887cc0b74ce42520e75c58e1195a3e4554ec9819cd" +} diff --git a/research/kappa-rebuild/provenance/RESEARCH-HANDOFF.md b/research/kappa-rebuild/provenance/RESEARCH-HANDOFF.md new file mode 100644 index 0000000000..5d4f6095fa --- /dev/null +++ b/research/kappa-rebuild/provenance/RESEARCH-HANDOFF.md @@ -0,0 +1,188 @@ +# Accounting-aware kappa research: October 21 recovery handoff + +## Outcome and integrity status + +The objective remains **raising kappa by lowering the complete recursive bill**. No new kappa is admitted by this round. The work did produce a cheaper local complex interface, finite BIT improvements with full accounting, a new productive-helper recursive family, and a promising reversible cache exchange. The outstanding task is to close the physical caller and its endpoint contract within the measured allowance while also meeting the independent BIT premise. + +The execution environment went offline approximately 02:25–02:43 UTC on October 21, after the round began at 00:13:31 UTC. On reconnection the workspace had been reset. The attached base repository, prior cumulative checkpoint, and almost all current-round sources were absent. This was an execution-environment failure, not a repository deletion or an approval rejection. The planned complete fresh replay and cumulative package had not finished. + +This archive is consequently a **recovery checkpoint**, not a replacement for the earlier cumulative overlay. It contains surviving or newly written self-contained probes, their fresh receipts, explicit unverified native work, and the research record below. Historical passes are clearly separated from checks reproducible from this archive. No missing file has been approximated under its original frozen name. + +The source identities needed for recovery are in `PRIOR-SOURCE-PINS.json`. Reattach the exact base `kappa-evaluation-compact(7).zip` and the last fully packaged `kappa-function-composition-checkpoint-20261020.zip` to restore the predecessor. Those two files do not automatically recover code first written during this round. Such missing work must either be recovered byte-for-byte against its pins or independently reconstructed and reverified with new provenance. + +## What the accounting means + +Write + +\[ +H_r(C)=r^{1-C},\qquad H_0(C)=0. +\] + +The proposed supplier must earn credit for complete productive jobs and pay every recursive child, boundary adapter that is not free, state transport, and final cleanup. A positive remainder in a partial construction is an **allowance for the omitted obligations**, not a kappa theorem. Scalar-operation allowances must also fit the model; a zero recursive-child cost does not mean zero physical work. + +The public baseline observed at 01:26:40 UTC was main commit `12fa8c5c844be21bfefb38a19503fbd867a11244`. Merged [PR 4085](https://github.com/CrocSwap/integer-mult-bounds/pull/4085) gives a conditional complex endpoint `0.001730970419199163702738`; the outer statement subtracts `10^-46 + 10^-70`. The separately stronger BIT checkpoint was `0.001738137188329345368427`, while the active outer BIT endpoint was `0.001733427039589844234749`. These values refer to distinct suppliers and must not be silently combined. + +Even reaching `C=0.001745` on the complex side requires improvement beyond the stronger BIT checkpoint by more than `0.000006862811670654631573`. Reaching `C=0.004` needs a much larger BIT advance. The outer formula is not `C/(1+C)` and does not introduce a factor of five. + +## Direction 1: replace the local response with a composition-friendly involution + +### Algebraic construction + +Split the 112 local ports into 104 ordinary ports U and eight selected ports A. The authenticated pre-reset geometry had operators K and L with `LK=13I`. Its pruned response was + +\[ +J=\begin{pmatrix}I-KL&-K\\-L&0\end{pmatrix}. +\] + +Completing the selected response gives + +\[ +H=\begin{pmatrix}I-KL&-K\\11L&12I\end{pmatrix} + =I+BC, +\qquad B=\binom{-K}{11I},\quad C=(L\ I). +\] + +Since `CB=-2I`, `H^2=I`. This is a useful interface change: a second application undoes the first, and the discrepancy from the identity is carried by eight state functions. It is an actual changed data operation; callers must be synthesized for H rather than pretending it still performs the old identity response. + +### Verified before the reset; sources presently unavailable + +The native H completion passed all four provider variants in both directions, retaining all 632 arbitrary persistent input columns. It restored all 408 helpers, used one owned C0 field that was cleared and reused, and required no new persistent width. The increment was **8 rank-one children and 4,336 real scalar allowance**. The independent matrix audit included a negative control with the expected rank-eight defect when the completion was removed. + +The resulting scalar totals were 119,718 for THREE and 119,696 for FOUR; paid rank masses were 5,101 and 5,100. These are historical native passes, not receipts reproducible from the present recovery archive. The surviving abstract identities do not replace the lost phase and owner checks. + +A zero-child same-completed-prefix state shear was also verified: two eight-state banks can implement `d += C_s B_t u` using 1,672 ADDs and 20,112 real scalar allowance while restoring the sources. Its acquisition, host paths, and return are not paid by that local statement. Actual cross-owner operators were checked as physical operators; treating them as scalar multiples or one monomial is invalid. + +### What failed and how it changes the search + +Replacing all physical frames by a common chart gave the right small scalar histogram but the wrong physical residual rank. The tested complementary two-flag repair requires at least 328 crossings, hence 656 extra rank units; that exceeds the relevant budget. This is a scoped failure of that repair menu, not a theorem excluding all frame changes. + +The former identity supplier cannot be obtained by placing its old correction assembly on the pruned profile for free. The old query contract needs data that the pruning removed. Forty-six tested assembly/inverse deletions all failed, with an exact sensitivity rank of 46 for that menu. The old full-identity THREE profile includes `2487H1+713H2+224H3+102H4+20H5+8H8`; the new pruned profile is `2453H1+716H2+224H3+74H4+48H5` before H completion. + +## Direction 2: productive helpers change the recursive family + +The old family placed every borrowed helper inside an idle segment of an existing main-data register. Even a fractional idealization of its historical repaired-identity packing missed `C=0.001745`; the best tested p=4 allowance was about `-1.5625611533`. Searching that packing family more finely cannot supply the missing gain under the same constraints. + +The new proposal adds registers whose job is a complete transform of their own arbitrary input. They host existing cleaned helper blocks on successive unfinished axes and finish the remaining axes afterward. Their full output credit is legitimate only once the whole transform and all boundary interfaces are proved. A cache ending with `d+v` is not such a productive output. + +### The boundary lemma that makes the proposal plausible + +An exactly cleaned native word has physical operator + +\[ +\operatorname{diag}(D,T_1,\ldots,T_{408}), +\qquad T_i=F_{i,\mathrm{end}}F_{i,\mathrm{start}}^{-1}. +\] + +Suppose each actual `T_i` has a certified allowed monomial normal form `T_i=L_i F_{r_i} R_i`. Applying `R_i^-1` before and `L_i^-1` after the **whole helper block** changes that block to `F_{r_i}` while preserving D and all zero cross terms. Internal ADDs and contacts stay intact. This avoids a spurious requirement to separately prepare all canonical helper entrance frames, which would otherwise add 371 rank units in the one-pair proposal. + +The lemma is elementary block algebra. Its application still needs each actual relative transform, both allowed monomials, and the absolute phase checked, including the backward conjugate convention. The missing native `clifford_exact.lower_child` machinery should provide these checks after source recovery. Independent retagging of internal helper frames would not be a substitute. + +### Reproducible packings + +`productive_helper_forest.py` expands integer host assignments and verifies every stage inventory, chronological eligibility, residual rank, strict child rank, and complete conditional tariff. It uses rational logarithm and exponential bounds and rounds the final intervals outward. The exact current cases and numbers are in `PRODUCTIVE-HELPER-FOREST.json` and the accompanying note; those files are the authoritative final packing record. + +The THREE helper inventory per call is 28 of width 5, 18 of width 6, 56 of width 7, 93 of width 8, and 213 of width 9, totaling 408 helpers and rank mass 3301. The tariff certificate takes this authenticated pre-reset profile as an input; it does not regenerate the lost native word. + +The one-pair family uses three calls, ambient rank 25, 224 main fields and 408 productive helper fields. A first improved cyclic packing had remainder `54H1+9H2+75H3`. A subsequent exact integer packing concentrates the residual into `54H1+28H2+H7+18H10`. After all three H completions, the latter leaves `125.5333563352732444235` at C=.001745 and `29.9957695582853966376` at C=.004. A numerical MILP suggested the pattern; the delivered certificate independently checks its integer assignments and interval invoice without using an optimizer. No exact global optimality claim is made. + +The two-pair family uses five calls and ambient rank 41. Starting from 224 added fields, an exact 219-field construction was found. Changing the original idle allocation then permits 213 added fields: replace twelve pairs of `(9,5),(5,9)` by `(8,8),(8,8)` in each two-stage idle window. This frees width5 roles for the global hosts, allowing outside patterns `(9,9,9,5)` to fill their remaining capacity exactly. Use 48 such hosts, 116 with `(9,9,9,0)`, and 49 with `(8,8,8,8)`; every host has a middle width 9 role. Distribute the 5 and 0 positions equally among the four outside stages. This attains the 213-field lower bound imposed by the middle window in this unstaggered family. It is not a global lower bound over different schedules or helper interfaces. + +For this 213-field construction, `W=661` and the residual profile is `128H1+82H2+28H3+29H8+116H5`. After all 40 H1 completion calls it leaves `257.8060663457233916759` at C=.001745 and `65.3169531797331110453` at C=.004. Replacing the changed kernel by the historical full identity still fails, with remainders about `-21.6812627263` and `-213.5107091794` respectively. Changed caller closure is essential. + +| Candidate | Main fields | Added productive fields | Ambient rank | Allowance at .001745 | Allowance at .004 | +|---|---:|---:|---:|---:|---:| +|One pair, concentrated residual|224|408|25|125.5333563353|29.9957695583| +|Two pairs, original224-host construction|448|224|41|256.6925912501|62.7790886213| +|Two pairs, revised213-host construction|448|213|41|257.8060663457|65.3169531797| + +All three rows charge the local H completions and all displayed helper residuals. They do not yet charge an emitted global controller closure. The inherited owned C0 scratch field remains uncredited; the final theorem must permit its simultaneously live capacity or explicitly supply it. It does not receive full productive-job credit. + +## Direction 3: causal coding and reversible dirty state + +### Native logger verified before reset + +A suffix after a forward pruned word passed all 640 arbitrary input columns and all four forward providers. It realizes + +\[ +X'=X,\quad Y'=Y+JX,\quad P'=d+(L\ I)X, +\] + +while restoring all 408 old helpers. Its suffix costs **8H9, 4,224 scalar allowance, no H1 child, and no owned field**. A previous implementation of the same full operator needed8H1+8H9 and 8,560 scalar allowance. The new native suffix is a real local improvement, but P still contains source information; neither restored-cache nor full productive-output credit follows. The source and its recorded hashes are presently unavailable after the reset. + +### Surviving exact alias audit + +`alias/constant_chart_audit.py` independently constructs the complete selected-register controller using literal shears and both bridge directions, with all original and cache inputs arbitrary. Exact rational elimination proves whole-operator invertibility for p=1,...,5. It also finds that the specified final selected fields, forward caches, and completed ordinary Y fields leave p-1 ordinary-error functions per active channel unobservable. The JSON includes integer null witnesses, so this is more than a numerical rank impression. + +The native `alias_primitives.py` is explicitly WIP. It could not import its frozen dependencies after the reset and has no native replay receipt. Its proposed forward and backward wrapper costs are predictions to test, not admitted additions to the complete supplier. + +### A useful positive from changing the contact location + +The relay probes tested optional exchanges, small signed additive relays, and repeated cache-bank assignments. Those finite-field menus did not close the larger controllers. Then the exchange was moved to the target's entrance, where the required future query may be accessible before its frame changes. + +For p=1, a constant-chart exact integer example restores one arbitrary cache bank and exposes the missing correction. There were 66 modular closing variants in the 864-case entrance-exchange menu, but that count is only a finite-field screening result. The selected endpoint is independently checked as an integer matrix. It is not a physical supplier. + +The stronger operator identity does not require equal or commuting local H operators. Let `QE=I`, `T=I-EQ`, and begin with `Y0=Y+H0 X`. Exchange `QY0` with arbitrary P=d using insertion E. This produces `Y1=T Y0+Ed`. Apply the backward word only to X: `X1=X-H1 Y1`. Repeat the **same query exchange** on Y. Then exactly `Y2=Y0` and `P=d`. The final forward word and signed swap give + +\[ +X_{out}=Y_0+H_2X_1,\qquad Y_{out}=-X_1,\qquad P_{out}=d. +\] + +The fresh relay receipt checks a noncommuting matrix instance. The result keeps arbitrary borrowed data without assuming all owner charts coincide. Native access to both Q and its right inverse E at both contacts, their actual transports, the cache path, and correction of the main outputs remain required. Cheap query access alone does not imply a cheap insertion or decoder. + +Using different queries silently is invalid. If the second query is C2 and the first is C1 with `C1 E1=I`, the bank returns + +\[ +P_{out}=C_2(I-E_1C_1)Y_0+C_2E_1d. +\] + +Requiring `P_out=d` for every input forces `C2=C1` after actual transport into a common chart. The proposed remedy is the same future query at both pre-Y contacts, not an assumption that different native queries happen to match. + +### Catalytic quarter-turn identity + +When `CB=-2I`, the three shears `U(B/2) L(C) U(B/2)` send `(Cx,d)` to `(-d,Cx)`. Squaring this word and negating d implements `(x,d)->((I+BC)x,d)`. After adjacent upper shears merge, it needs two C queries and three B scatters. This is an exact useful identity over dyadic coefficients; its timed native contacts and complete recursive invoice are still open. The source includes negative controls for an omitted final sign and a wrong CB value. + +## BIT: worthwhile finite progress, independent asymptotic barrier + +The strongest pre-reset deferred-source finite words reduced both helpers and additions: + +| Word | Original helpers | New helpers | Original ADDs | New ADDs | Full normalized saving at .001745 | At .004 | +|---|---:|---:|---:|---:|---:|---:| +|15 outputs|265|176|34,829|19,522|16.1390022847|37.0807501005| +|16 changed-core outputs|582|382|38,415|20,439|36.2678693352|83.3295445158| + +The checks covered signs, arbitrary helper values, input columns, and the final boundary. Intermediate original registers deliberately retain function state; demanding they be restored at every cut would erase the saving. Independent checks passed before reset. Their implementation bytes are currently missing. + +These improvements did not yet give the necessary infinite recursive family. An exact h=19 global forest with 42,325,920 incidences reduced helpers from 29,078,731 to 24,167,112 and used 108,002,708 ADDs, but its covered moments were `1.0027421578257865` at .001745 and `1.0063139501663587` at .004, where admission requires less than 1. The h=19 result was an exact combinatorial/rank/invoice calculation, not a replay of 100 million native instructions; a smaller h=9 expansion was replayed. + +The finite result should be preserved as an optimization primitive. It must not be extrapolated to an exponent. Further fixed-coordinate center compression also encountered exact scoped obstructions. The next BIT work needs a changed global state boundary or center family, not a claim that the existing finite reduction by itself funds .004. + +## Literature: what transferred, and what did not + +The productive results came from combining exact operator algebra, causal realization, reversible state, and full recursive accounting. The papers below motivate techniques; none proves this repository's target exponent. + +| Source | Applicable idea | Specific use or qualification | +|---|---|---| +|[Daubechies–Sweldens, factoring wavelet transforms into lifting steps (1998)](https://9p.io/who/wim/papers/factor/factor.pdf)|Factor a reconstruction operator into elementary reversible steps.|Motivates changing the local response and factoring the eight-state correction. Native timing and allowed coefficients must still be certified.| +|[Brislawn, causal lifting via Diophantine equations (2024 version)](https://arxiv.org/abs/1902.09040)|Choose causal factors with degree/divisibility constraints.|A close match to asking which correction is available at a legal contact; an unconstrained low-rank factorization is insufficient.| +|[Connelly–Zeger, linear network coding over rings, Part II](https://arxiv.org/pdf/1608.01737)|The coefficient ring matters to linear coding.|Useful discipline for keeping dyadic/legal operators; field-only rank improvements are screened separately.| +|[Chistikov et al., rectifier networks (2017)](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23)|Share linear information through intermediate nodes.|Motivates the BIT sharing/forest probes; the tested full forest still misses the moment inequality.| +|[Pernet–Storjohann, quasiseparable matrix generators](https://arxiv.org/abs/1701.00396)|Represent exact matrices through low-rank off-diagonal structure.|Motivates carrying only the necessary state across cuts. The measured observer gaps show where the proposed state is insufficient.| +|[Kostrykin–Schrader, generalized star products for scattering matrices](https://arxiv.org/abs/math-ph/0008022)|Compose systems through retained interface variables.|A useful structural analogy for eliminating internal ports. It is not permission to use an illegal inverse or ignore a feedback path's recursive cost.| +|[Buhrman et al., catalytic space (2014)](https://eccc.weizmann.ac.il/report/2014/053/)|Borrow arbitrary memory while restoring it exactly.|Matches the arbitrary-input restoration requirement and led to explicit reversible dirty-bank probes.| +|[Folkertsma et al., Fast Cliffords When Your Quantum Memory Is Full (2026)](https://arxiv.org/html/2609.40144v1)|Catalytic circuit synthesis and a cycling cancellation construction.|The cycling lemma requires its workspace to be restored by the constituent block. Our dirty logger changes its workspace, so directly applying that lemma would be invalid. The native arithmetic and cost model also differ.| +|[Urschel, block unitriangular generation of special linear groups (2023)](https://arxiv.org/abs/2305.00485)|Small numbers of block shears for general linear transformations.|A lead for controller synthesis; field-level factor counts do not certify dyadic coefficients, frame legality, or the recursive invoice.| + +A 2026 operator-catalytic lead at [arXiv:2609.18692](https://arxiv.org/abs/2609.18692) was identified from its abstract. Full text was unavailable during this round; no uninspected theorem from that paper is used here. + +The useful cross-field connection is concrete: a scattering-style interface exposes eight state functions; causal lifting constrains when they can move; catalytic synthesis restores arbitrary storage; and productive helper packing pays for the hosts by assigning them complete transform jobs. Physical analogies that do not produce an operator identity and a complete bill were not promoted into results. + +## Reevaluation and decisive next steps + +1. **Recover and authenticate sources first.** Restore the attached base and the last cumulative overlay against their hashes. Keep all unmerged work and the historical missing-entry list. Do not overwrite either baseline with this recovery archive. +2. **Certify the productive helper interface.** For all 408 actual helper relative transforms in each relevant direction/provider, emit the two monomial adapters, child rank, and absolute phase. Verify the complete data/helper block diagonal operator. This is the admission gate for the new accounting family. +3. **Prioritize the one-pair main controller.** It has the simplest state boundary. Test the same future query and right inverse at both pre-Y exchanges using actual owner transports, then synthesize the remaining main endpoint correction. Charge initial heads, final normalization, cache movement, and all cleanup in one ledger. A lone 112H1 normalization could fit the .001745 allowance in the recorded one-pair packing, while two such normalizations could not; at .004 even one is too expensive. The contract cannot simply waive the initial head. +4. **Use the two-pair 213-host construction if the extra allowance outweighs its added closure state.** Its packing reaches the unavoidable middle-window host count within this family. The extra state is not free; use the exact observer audit to specify what the decoder must retain. +5. **Advance BIT independently with the successful deferred state principle.** Integrate the finite primitive after source recovery, but judge every extension by the covered moment. Search altered center/state interfaces rather than keep optimizing a fixed forest whose complete moment exceeds 1. +6. **Only then run the full supplier and outer replay.** A valid new kappa requires both suppliers, legal recursive ranks, output contracts, all scalar allowances, full phase checks, and a cumulative artifact that freshly reproduces from the exact base. + +The current stop is caused by missing execution inputs after an environment reset, not by a need for more authorization. The mathematical priority is now a specific boundary closure on a newly funded recursive family. Neither the remaining budget nor the restored scalar cache is, on its own, a claim that the target has been achieved. diff --git a/research/kappa-rebuild/provenance/RESTORED-OVERLAY.json b/research/kappa-rebuild/provenance/RESTORED-OVERLAY.json new file mode 100644 index 0000000000..c50bcaa105 --- /dev/null +++ b/research/kappa-rebuild/provenance/RESTORED-OVERLAY.json @@ -0,0 +1,33 @@ +{ + "already_exact_files": 6, + "atomicity": "Each individual file replacement is atomic; a process crash is not an all-files transaction. Re-running is idempotent.", + "deletions": [], + "inherited_scope_flags": { + "later_reported_archive_gap": { + "available_cumulative_files": 2455, + "exact_missing_path_manifest_available": false, + "later_reported_archive": { + "bytes": 184572857, + "dependency_receipt_sha256": "f5f60f8d8030e9e2b01ef9e9b0efc46309db42d5d5915ab130c6baa429919cf5", + "entries_including_manifest": 2787, + "filename": "kappa-literature-breakout-cumulative.zip", + "manifest_sha256": "b52704f372dd31d364338251a338a003abda7c900c483d717dc3f07a26e861f6", + "sha256": "21a0d692c5c48899165e4c1a53d410e083afa177dbdf4711665dcc77983f9629" + }, + "literature_prefix_files_in_available_cumulative": 0, + "literature_prefix_files_in_compact": 0, + "meaning": "Later reported archive had 2787 entries versus 2455 available files. The difference includes 34 reported repository dependencies; it is not a claim of 332 files under the literature prefix. No missing path list is invented, and this comparison does not claim recovery of all later work.", + "new_admitted_kappa": false, + "reported_additional_archive_entries_not_recovered": 332, + "reported_supplemental_unchanged_repository_dependencies": 34, + "source": "recovered/continuation.md, Cumulative package recovery record", + "status": "RETAINED_REPORTED_COUNT_NOT_PATH_VERIFIED" + } + }, + "new_admitted_kappa": false, + "precondition_scope": "Every touched path is checked against its exact before or after digest and mode. No claim is made about unrelated existing checkout files.", + "schema": "applied-content-addressed-overlay/1", + "status": "PASS_ATOMIC_FILE_OVERLAY", + "verified_path_mappings": 3214, + "written_files": 3208 +} diff --git a/research/kappa-rebuild/provenance/RETAINED-ACCOUNTING-INPUTS.json b/research/kappa-rebuild/provenance/RETAINED-ACCOUNTING-INPUTS.json new file mode 100644 index 0000000000..503dac405a --- /dev/null +++ b/research/kappa-rebuild/provenance/RETAINED-ACCOUNTING-INPUTS.json @@ -0,0 +1,45 @@ +{ + "provenance": "Accounting-input object retained in functions.store across the execution-environment reset; these are inputs, not a native receipt.", + "inputs": { + "raw": { + "1": 2453, + "2": 716, + "3": 224, + "4": 74, + "5": 48 + }, + "full": { + "1": 2487, + "2": 713, + "3": 224, + "4": 102, + "5": 20, + "8": 8 + }, + "p1": { + "width": 632, + "rank": 25, + "calls": 3, + "hosts": 408, + "residual": { + "1": 129, + "2": 84 + } + }, + "p2": { + "width": 672, + "rank": 41, + "calls": 5, + "hosts": 224, + "residual": { + "1": 128, + "2": 130, + "3": 28, + "5": "433/3", + "8": 29, + "32": "20/3" + } + }, + "new_kappa": false + } +} diff --git a/research/kappa-rebuild/provenance/public-final/AGENTS.md b/research/kappa-rebuild/provenance/public-final/AGENTS.md new file mode 100644 index 0000000000..aea5573d1b --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/AGENTS.md @@ -0,0 +1,86 @@ +# Agent entry point + +For open-ended requests such as “contribute,” “use spare inference,” or +“improve the bound,” start with [autoresearch.md](autoresearch.md). It links +the current priorities and standing procedure. Check open handoffs +before using the task picker; its coverage fields identify prior checks and +remaining obligations. Explicit user tasks take precedence over that default +assignment; do not reroll an already assigned task. + +For repeated/overnight loops, follow the +[upstream refresh checkpoint](autoresearch.md#looping-agents-refresh-upstream-at-each-research-checkpoint) +at every new round, before a major pivot and before submission. Fetch canonical +main (not an unmaintained fork), compare priority revisions, and read changed +guidance. The local task picker does not fetch updates. Preserve work in progress +and frozen evidence; start new tasks from the refreshed base. + +Default to one session of about 60 minutes unless the donor sets another budget. +For a concrete introduction, read [the bottleneck brief](docs/research/bottleneck-brief.md) +and [glossary](docs/research/research-glossary.md). Use the card's first working +example before choosing a detour. A useful finding or handoff can finish a +session; a PR is not mandatory. Read the [family status and expansion gate](docs/research/research-family-status.md) +before extending a mechanism: newer pending work may change the missing task, +and a local improvement does not by itself justify another construction round. +Partial contributions remain welcome; the director owns the complete budget. + +Before substantial research, read the +[discussion-board start here](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163) +and check [active discussions](https://github.com/CrocSwap/integer-mult-bounds/discussions) +alongside open PRs for overlap. Use the +[handoff thread](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166) +for concrete requests and the +[mechanism-proposals thread](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167) +for early ideas. Follow the claim/update conventions in `autoresearch.md`; +claims are advisory and discussion posts are provisional evidence. Keep one +canonical investigation thread with a current status and public artifact link; +use hub threads for short pointers. Local-only evidence must be labelled, and +useful partial/negative results do not need a new κ to merit a focused PR. + +Use `python3 scripts/find_overlap.py --task TASK_ID` for a cached search of PR +bodies, discussions and local package notes. The report states its coverage and +freshness; no matches is not proof of novelty. If changing a card's premise, +record the scope change and why the new question matters. + +When acting as an authorized maintainer/director and reassessing research +direction, follow the [priority-review procedure](autoresearch.md#6-maintainerdirector-priority-review). +Update the task catalogue and evidence log together; do not infer a new frontier +or a universal barrier from assignment weights or unreviewed discussion claims. + +Read [CONTRIBUTING.md](CONTRIBUTING.md) before making changes. Resolve the current +result, review, source manifest and reproduction command through +`certificates/selected-result.json`. Historical research notes may use “current” +for an obsolete checkpoint. + +For authorized maintainer PR work, **first follow [delta-only queue intake](docs/research/pr-intake.md)**. +Run `python3 scripts/pr_intake.py plan --refresh --output /tmp/pr-intake-plan.json`. +Read selected actionable entries; do not rereview unchanged holds, recreate +100-entry audit packages, or rerun historical suites after compaction. Persist +changed scheduling decisions in `docs/research/pr-intake-state.json` and actual +review evidence in the central ledger. This scheduling procedure takes precedence +over older batch recipes. No automatic Lean, Docker, or concurrent heavy replays. +See [research archive status](docs/research/research-archive-intake.md) for acceptance +rules: a merged archive can remain **partially reviewed**; merging is not scientific +approval or a selected-frontier update. + +Keep `upstream/`, frozen source archives and pinned evidence unchanged. Put new +research in its own scoped directory and retain predecessor attribution. +Distinguish a finite certificate, a scoped proof, a search failure and a complete +multiplication theorem. They are different claims. + +Use an isolated branch/worktree when sharing a checkout. Run checks relevant to +the change; do not run concurrent artifact-regenerating suites in one worktree. +Record exact sources, commands, environments and any unrun proof obligations. + +The standard donation prompt authorizes routine Discussion participation and +opening focused PRs. When that prompt is the user's assignment, post claims, +questions, findings, handoffs and feedback without asking again for each post; +the same applies to opening the contribution PR. Respect any narrower donor or +account restrictions. Merely reading this file does not override those restrictions. +Do not change the selected frontier or merge/close others' PRs without maintainer +authorization; existing explicit authorization remains valid. + +New research packages should include **Open next questions** and can use the +[minimal package generator and explicit replay hook](docs/research/research-package-guide.md). +Optional workflow feedback goes in the PR or the +[feedback discussion](https://github.com/CrocSwap/integer-mult-bounds/discussions/1200). +Link it once; no separate session report is required. diff --git a/research/kappa-rebuild/provenance/public-final/COMMIT.json b/research/kappa-rebuild/provenance/public-final/COMMIT.json new file mode 100644 index 0000000000..576bd2ce23 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/COMMIT.json @@ -0,0 +1,137 @@ +{ + "sha": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "node_id": "C_kwDOU-508toAKDEyZmE4YzVjODQ0YmUyMWJmZWZiMzhhMTk1MDNmYmQ4NjdhMTEyNDQ", + "commit": { + "author": { + "name": "Douglas R. 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Check current main and live PRs.\",\n \"tracks\": [\n {\n@@ -85,6 +85,66 @@\n \"nonabelian\",\n \"group algebra\"\n ]\n+ },\n+ {\n+ \"id\": \"r437-carrier-integration\",\n+ \"question\": \"Can the paid R437 two-window carrier improvement be admitted by the actual pooled compiler and physical caller?\",\n+ \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n+ \"certificates/selected-result.json\",\n+ \"research/r437-two-window-carrier/README.md\",\n+ \"research/r437-reverse-row-coefficient/README.md\"\n+ ],\n+ \"methods\": [\n+ \"Close one compiler/request, caller or precision/stock implication for the pinned paid exchange.\",\n+ \"Falsify that exact interface with an arbitrary-dirty-state or chronology counterexample.\"\n+ ],\n+ \"acceptance_gate\": \"Pin the complete source and destination contracts, exact paid and scalar deltas, and all remaining assumptions. A local moment gain is not a physical or all-size supplier admission.\",\n+ \"first_step\": {\n+ \"command\": [\n+ \"python3\",\n+ \"-B\",\n+ \"scripts/research_baseline.py\"\n+ ],\n+ \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n+ \"inspect\": \"Read #4067 paid profile and open integration obligations; #4147 covers the reverse row coefficient. Three additional ADDs per call and all helpers remain charged. This is not a selected supplier.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n+ },\n+ \"first_experiment\": \"Choose one unclaimed obligation: pooled compiler codebook/request alignment, actual physical caller, or precision/stock envelope using the already reported reverse coefficient. Pin the input and receiver, preserve dirty/source windows, and test one binding with a rejection control. Stop at unresolved dependencies; do not rederive the coefficient or quote the local moment as a completed bound.\",\n+ \"bottleneck_link\": \"The exchange changes the paid profile; integration must establish that its gain survives the complete caller.\",\n+ \"allowed_variations\": [\n+ \"One compiler rebind, one actual caller boundary, or one precision/stock implication.\"\n+ ],\n+ \"scope_change_triggers\": [\n+ \"Ignoring added ADDs, retaining an obsolete reverse coefficient, or declaring global admission from the local histogram.\"\n+ ],\n+ \"overlap_terms\": [\n+ \"R437\",\n+ \"two-window carrier\",\n+ \"4067\",\n+ \"4147\",\n+ \"pooled compiler\",\n+ \"request alignment\",\n+ \"reverse row coefficient\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #4067: source-bound two-window carrier with (2,2) to (1,3) paid exchange and fixed OQ4 host.\",\n+ \"evidence\": \"research/r437-two-window-carrier/README.md\",\n+ \"remaining\": \"Pooled compiler/request rebind, physical caller and precision/stock admission.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4147: reverse row coefficient follow-up.\",\n+ \"evidence\": \"research/r437-reverse-row-coefficient/README.md\",\n+ \"remaining\": \"Use the exact coefficient in a caller-bound precision/stock proof rather than repeat the arithmetic.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"72e7bb797adf82aeb525a399f56ab8adbab8e93c\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n+ ]\n }\n ]\n },\n@@ -94,13 +154,15 @@\n \"tasks\": [\n {\n \"id\": \"p104-side-correction\",\n- \"question\": \"Can the proposed P104 joint endpoint flags be realized by a chronological correction program with coefficients, dirty restoration and an affordable complete bill?\",\n+ \"question\": \"Can a changed P104 producer/consumer contract escape the scoped slot, chart and completion obstructions with a favorable full bill?\",\n \"starting_points\": [\n \"docs/research/research-family-status.md\",\n- \"research/p104-native-channel/README.md\",\n- \"research/p104-native-channel/PROOF.md\",\n- \"docs/research/community-efficient1000-review.md\",\n- \"research/p104-joint-unit-flags/README.md\"\n+ \"certificates/selected-result.json\",\n+ \"research/p104-literal-slot-defect/README.md\",\n+ \"research/p104-uniform-commutation-gap/README.md\",\n+ \"research/p104-uniform-encoding-gap/README.md\",\n+ \"research/p104-source-chart-shortcut/README.md\",\n+ \"research/p104-fiber-tag-seam/README.md\"\n ],\n \"methods\": [\n \"Construct one side-correction word at an explicit chronological cut with the needed frames still available.\",\n@@ -114,10 +176,10 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read the revision-6 family record and #2981 static joint flags, alongside the reviewed #1991 channel. #2981 is an author-checked source archive, not an independently validated executable correction or a refutation of program-level lifetime floors.\",\n- \"scope\": \"Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission.\"\n+ \"inspect\": \"Read the named successor obstructions before extending #2981 static flags. Distinguish literal-slot action, commutation, uniform encoding, source-chart shortcut and fiber-tag seam scopes; none excludes all P104 mechanisms.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"After checking live handoffs, pin one named collection from #2981 and attach acquisition/delivery times, source dependencies and coefficient requirements. Test legal access and arbitrary dirty restoration; price any completion against the full channel/endpoint/hosting allowance. Stop with the exact missing implication or a scoped counterexample rather than expanding the static cover census.\",\n+ \"first_experiment\": \"Name one hypothesis changed from these exact interfaces. Write a literal producer-to-consumer timeline and an arbitrary-dirty-input positive/control pair, then charge acquisition, coefficients, completion, hosting and restoration. Stop if the same slot/fiber mismatch persists or the favorable full budget already fails. A scoped independent audit remains useful; another static count is not a new completion.\",\n \"bottleneck_link\": \"Completing an already realized changed channel tests a concrete construction opportunity, instead of refining the incumbent packing. A positive local result still needs global and all-size admission.\",\n \"allowed_variations\": [\n \"One correction factorization, one chronological cut, or one explicit hosting obstruction.\",\n@@ -127,7 +189,8 @@\n \"Recompiling the already certified low-rank channel without closing a new obligation.\",\n \"Treating a pending ancestor, local histogram or unpriced side correction as an admitted global supplier.\",\n \"Repeating a correction or physical conversion already supplied by a pending follow-up without adding independent coverage or a different unresolved obligation.\",\n- \"Repeating the 208-flag static matching or calling a static nested cover a chronological correction program. Independent audit of that certificate is distinct and remains useful.\"\n+ \"Repeating the 208-flag static matching or calling a static nested cover a chronological correction program. Independent audit of that certificate is distinct and remains useful.\",\n+ \"Repeating a static cover, source-chart shortcut or fiber-tag seam without changing the hypothesis responsible for its obstruction.\"\n ],\n \"overlap_terms\": [\n \"P104\",\n@@ -143,15 +206,57 @@\n \"2981\",\n \"joint unit flags\",\n \"acquisition delivery\",\n- \"static nesting\"\n+ \"static nesting\",\n+ \"4133\",\n+ \"4136\",\n+ \"4140\",\n+ \"4153\",\n+ \"4154\",\n+ \"p104-literal-slot-defect\",\n+ \"p104-uniform-commutation-gap\",\n+ \"p104-uniform-encoding-gap\",\n+ \"p104-source-chart-shortcut\",\n+ \"p104-fiber-tag-seam\"\n ],\n \"existing_checks\": [\n {\n- \"scope\": \"PR #2981 reports static joint endpoint flags and a matching of 208 departure and arrival flags; archival intake is not independent verification.\",\n- \"evidence\": \"research/p104-joint-unit-flags/README.md\",\n- \"remaining\": \"Independent flag check and a chronological coefficient program with dirty restoration and a complete cost.\",\n+ \"scope\": \"PR #4133: literal-slot defect\",\n+ \"evidence\": \"research/p104-literal-slot-defect/README.md\",\n+ \"remaining\": \"Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.\",\n \"status\": \"author-checked\",\n- \"source_commit\": \"3d24b92d8ba057b57cb2d480170ec407520e6882\",\n+ \"source_commit\": \"88666f1dee93fbd85b31fd5d6f91a219e111b5f4\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4136: uniform commutation gap\",\n+ \"evidence\": \"research/p104-uniform-commutation-gap/README.md\",\n+ \"remaining\": \"Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"281ce601a8be9db470de630759179c23e1dfaa88\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4140: uniform encoding gap\",\n+ \"evidence\": \"research/p104-uniform-encoding-gap/README.md\",\n+ \"remaining\": \"Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"de25c6525aa866962309df5d107b32d55e6e5628\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4153: source-chart shortcut exhausts its optimistic allowance\",\n+ \"evidence\": \"research/p104-source-chart-shortcut/README.md\",\n+ \"remaining\": \"Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"9123241fea53df77e51ffe30d9aca1625d4bf3a8\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4154: fiber-tag seam mismatch\",\n+ \"evidence\": \"research/p104-fiber-tag-seam/README.md\",\n+ \"remaining\": \"Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"4ffcdaee5d6255a9cea1fe53a5424435068089d0\",\n \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n }\n ]\n@@ -254,18 +359,15 @@\n \"tasks\": [\n {\n \"id\": \"joint-query-controller\",\n- \"question\": \"Can a literal QQ16 producer meet the proposed completed-output error budgets with dirty helpers and an affordable physical consumer?\",\n+ \"question\": \"Can the installed QQ16 caller win a complete matched protocol bill, or can a changed demand eliminate actual paid work?\",\n \"starting_points\": [\n \"docs/research/research-family-status.md\",\n- \"research/joint-query/README.md\",\n- \"research/joint-query/evidence/QQ16-SELECTED-WITNESS.md\",\n- \"research/joint-query/evidence/JOINT-QUERY-TRANSFER.md\",\n- \"docs/research/community-efficient1000-review.md\",\n- \"research/qq16-producer-defect-transfer/README.md\"\n+ \"certificates/selected-result.json\",\n+ \"research/qq16-caller-install/README.md\"\n ],\n \"methods\": [\n- \"Derive one literal producer's completed-output defects and intermediate dirty-helper bounds, with its physical bill.\",\n- \"Design or audit one consumer, retention or hosting boundary and its complete cost for the pinned producer/decoder.\"\n+ \"Derive the matched whole-caller physical bill for installed and baseline protocols.\",\n+ \"Change the actual demanded observations and prove a cheaper legal producer/consumer contract.\"\n ],\n \"acceptance_gate\": \"Name the changed premise of the fixed F0/B1 observation obstruction. Show exactly which query is recovered, legal native reads and restoration, and a full favorable budget including cover maps, helpers and endpoints. Local inverse identities alone do not close the global controller. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.\",\n \"first_step\": {\n@@ -275,10 +377,10 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read the revision-6 family record and #2965 producer-defect transfer after the local QQ16 decoder contract. Its author-checked output-error estimate does not implement producer interiors, bound dirty helpers, or pay for the consumer; independently check the implication used.\",\n- \"scope\": \"Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission.\"\n+ \"inspect\": \"Read #4166 COST.md and ADMISSION.json. An actual caller connection is now proposed. Its native-component profile loses; parent park/restore versus external-call costs remain unresolved. Retain the producer and recovery lineage rather than rebuild it.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"Pin one literal F0/B1 producer and its native-child contracts; derive the actual intermediate magnitude ledger and completed-output defects, including dirty helpers and correlations. Compare with #2965 budgets and price storage, movement, cleanup and F17 consumption. Stop with a precise missing bound if no producer contract supplies it; do not optimize another decoder precision constant in isolation.\",\n+ \"first_experiment\": \"At the actual installed cut, compare matched complete parent park/restore and external input/return protocols at the same bank/packet geometry, including every later root call and service cost. Alternatively pin both bridge observations and derive a genuinely smaller source-compatible demand. Stop if the complete necessary inequality fails or an essential cost is unknown; no larger decoder scan from a local tape-action saving.\",\n \"bottleneck_link\": \"The native local words exist; retained query information and its physical access are the missing link to a composed recursive interface.\",\n \"allowed_variations\": [\n \"One paid query, one changed first pair, or one precisely specified affine helper state.\",\n@@ -289,7 +391,9 @@\n \"Retaining the fixed first pair and ordinary schedule while allowing only homogeneous selected-only control, already excluded by the witness.\",\n \"Treating cover-map counts as zero-cost physical address operations.\",\n \"Recreating the pending complete local decoder or early-record/output-bank coalescing already under investigation without a distinct audit or changed premise.\",\n- \"Treating a completed-output defect inequality as a producer-interior, dirty-range or physical-cost certificate.\"\n+ \"Treating a completed-output defect inequality as a producer-interior, dirty-range or physical-cost certificate.\",\n+ \"Recreating completed-output recovery or caller wiring already supplied by #4166 without new independent coverage.\",\n+ \"Dropping retained later root calls or claiming native-profile dominance from the isolated publication saving.\"\n ],\n \"overlap_terms\": [\n \"QQ16\",\n@@ -304,7 +408,19 @@\n \"2965\",\n \"producer defect\",\n \"dirty-helper range\",\n- \"F17 consumer\"\n+ \"F17 consumer\",\n+ \"4166\",\n+ \"qq16-caller-install\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #4166: author-checked installed producer/service/return/bridge connection; native-component comparison loses.\",\n+ \"evidence\": \"research/qq16-caller-install/README.md\",\n+ \"remaining\": \"Complete matched physical protocol bill, or changed source-compatible demand/circuit; no affordable supplier yet.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"0b082d22a0e9a442d264ab0633e35ef26c790630\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n },\n {\n@@ -495,11 +611,12 @@\n },\n {\n \"id\": \"supplier-boundary-audit\",\n- \"question\": \"Does one selected supplier obligation follow from its actual source-bound checks?\",\n+ \"question\": \"Can the #4088 four-provider dispatch satisfy the complete ordinary/outer admission contract?\",\n \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n \"certificates/selected-result.json\",\n- \"research/owner-boundary-audit/README.md\",\n- \"research/scatter-export-parity-audit/README.md\"\n+ \"research/staged544-mixed-fourth-entry/README.md\",\n+ \"research/scatter-boundary-decision/README.md\"\n ],\n \"methods\": [\n \"Trace one hypothesis through the source, checker and complete caller.\",\n@@ -513,17 +630,18 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read #2977 reported duplicate-scatter counterexample and its opt-in repair. The archive is not a production fix or an independently validated selected-result failure; the selected generator emits unique rows.\",\n- \"scope\": \"Source-binding orientation only; does not replay or validate the reported bug.\"\n+ \"inspect\": \"Read #4088 Scope and Open next questions: local words, native receipts and literal fill do not include a rerun of the mixed four-provider ordinary/outer engine. #3801 is the newer scatter-boundary decision; do not restart #2977 as if untouched.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"First check whether the scatter repair has been adopted. If still open, independently reproduce #2977 symbolic-versus-exported parity on the tiny duplicate fixture, compare the selected unique-row positive control, and define either enforced uniqueness or consistent multiplicity semantics. A bounded guard-adoption patch must preserve valid inputs and test the actual exported action. Stop after this obligation; no full historical replay or frontier change follows from the mutant alone.\",\n+ \"first_experiment\": \"Pin the provider registry, four native source closures and dispatch. Make a bounded acceptance coverage map or one registry adapter preserving every inherited admission predicate and rejection control (including an omitted fourth provider). Record remaining ordinary/outer obligations and a resource estimate before heavy execution. Stop at unresolved source/caller dependencies; no selected-result promotion from a local pass.\",\n \"bottleneck_link\": \"Closing a current acceptance gap or proof obligation strengthens reliability even without changing kappa.\",\n \"allowed_variations\": [\n- \"Audit one named current condition using a new semantic counterexample.\"\n+ \"One provider-registry, dispatch, native-closure or admission-predicate obligation; separate branches must name distinct coverage.\"\n ],\n \"scope_change_triggers\": [\n \"Reporting the historical timing-only diagnostic as a new current acceptance bug.\",\n- \"Calling a malformed mutant a failure of the selected construction, or importing an opt-in repair without testing the exporter boundary.\"\n+ \"Calling a malformed mutant a failure of the selected construction, or importing an opt-in repair without testing the exporter boundary.\",\n+ \"Running another root enclosure instead of validating dispatch or weakening an inherited predicate to make mixed admission pass.\"\n ],\n \"overlap_terms\": [\n \"owner boundary\",\n@@ -532,15 +650,38 @@\n \"2977\",\n \"scatter export\",\n \"duplicate index\",\n- \"parity semantics\"\n+ \"parity semantics\",\n+ \"4088\",\n+ \"3801\",\n+ \"staged544-mixed-fourth-entry\",\n+ \"scatter-boundary-decision\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #4088: author-checked local words, native receipts, owner/phase fill and exact moment.\",\n+ \"evidence\": \"research/staged544-mixed-fourth-entry/README.md\",\n+ \"remaining\": \"Rebind and validate ordinary/outer admission with all four providers; full mixed release not established.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"72d7efc181edb21806a878f174847c96dd410ab0\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #3801: successor scatter-boundary decision package.\",\n+ \"evidence\": \"research/scatter-boundary-decision/README.md\",\n+ \"remaining\": \"Consult its exact acceptance scope before any new exporter audit; not proof of a selected failure.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"911f8800ba7a4b03603ef62c5ffc6240e53cb5fd\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n },\n {\n \"id\": \"uniform-transfer-audit\",\n- \"question\": \"Can one exact all-size/compiler, precision, hosting or restoration implication be proved or falsified?\",\n+ \"question\": \"Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?\",\n \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n \"certificates/selected-result.json\",\n- \"docs/audit.md\"\n+ \"research/unconditional-batched-crt/README.md\"\n ],\n \"methods\": [\n \"Prove one missing implication with explicit quantified hypotheses.\",\n@@ -554,21 +695,34 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read the selected review scope and choose one exact all-size or precision implication.\",\n- \"scope\": \"Positive control / source-bound orientation only; this is not a new construction or full cold replay.\"\n+ \"inspect\": \"Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"Write its quantified inputs and conclusion; find a missing hypothesis or construct a minimal exact adversarial instance.\",\n+ \"first_experiment\": \"Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.\",\n \"bottleneck_link\": \"Finite supplier improvements are useful only under the retained transfer obligations.\",\n \"allowed_variations\": [\n- \"Choose one tape, precision, hosting or restoration lemma.\"\n+ \"One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller.\"\n ],\n \"scope_change_triggers\": [\n- \"Replacing a quantified obligation with another finite smoke test.\"\n+ \"Replacing a quantified obligation with another finite smoke test.\",\n+ \"Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses.\"\n ],\n \"overlap_terms\": [\n \"uniform transfer\",\n \"all-size\",\n- \"precision\"\n+ \"precision\",\n+ \"4100\",\n+ \"unconditional-batched-crt\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #4100: proposed unconditional construction and finite checks, without independent acceptance.\",\n+ \"evidence\": \"research/unconditional-batched-crt/README.md\",\n+ \"remaining\": \"Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"337d04939571a637aeab54a0f58f54fde811663a\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n },\n {\n@@ -621,15 +775,17 @@\n \"tasks\": [\n {\n \"id\": \"independent-moment\",\n- \"question\": \"Does one changed supplier or assembly claim lack a source-bound independent check of a specific new implication?\",\n+ \"question\": \"Does a new paid-profile or mixture-ceiling claim have independent source-bound coverage of its actual changed ledger?\",\n \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n \"certificates/selected-result.json\",\n- \"research/rank7-joint-endframes/README.md\",\n- \"docs/research/research-family-status.md\"\n+ \"research/staged544-mixed-fourth-entry/README.md\",\n+ \"research/staged-fourth-mixture-ceiling/README.md\",\n+ \"research/r437-two-window-carrier/README.md\"\n ],\n \"methods\": [\n- \"Independently reconstruct a coordinated frame/circuit change and its single-change control, then derive the complete rank and physical ledger.\",\n- \"Implement certified moment inequalities through a different exact enclosure method.\"\n+ \"Independently reconstruct one changed paid ledger or pooled dual with exact source and model binding.\",\n+ \"Check a genuinely uncovered moment implication with a different enclosure method and semantic controls.\"\n ],\n \"acceptance_gate\": \"Demonstrate independence, source binding, positive cases and semantic failure controls. Name the existing checks and the missing coverage. More digits or another equivalent evaluation of an unchanged root is not by itself new verification coverage.\",\n \"first_step\": {\n@@ -639,17 +795,17 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read #2979 joint rank-7 end-frame proposal and the revision-6 evidence status. Its claimed kappa is not selected. Compare coordinated changes with the losing single-change controls and locate the complete physical ledger; an aggregate moment alone does not validate the mechanism.\",\n- \"scope\": \"Positive control / source-bound orientation only; this is not a new construction or full cold replay.\"\n+ \"inspect\": \"Choose one uncovered implication in #4088 literal mixed fill, #4091 pooled dual/model binding or #4067 changed frame/host bill. #4091 pins an earlier #4088 head: validate word, capacity and stock equivalence before applying that ceiling to the later fill.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"Prioritize a bounded independent audit of #2979: reconstruct one joint group and its single-lowering control, check the declared frames/chronology, then trace changed children, idle widths and cleanup buffers into the complete invoice and both supplier caps. Choose one still-missing implication after checking existing audits. Stop with exact unresolved dependencies if full admission exceeds the session; do not launch a new lowering census on an unvalidated premise or substitute extra root digits for mechanism checking.\",\n+ \"first_experiment\": \"Independently reconstruct one changed ledger or pooled certificate with a positive case and a semantic failure control. Compare exact sources and name what existing checks miss. Keep this separate from supplier-boundary-audit (ordinary/outer dispatch) and uniform-transfer-audit (all-size proof). Stop after the chosen implication; more digits of an unchanged root are not additional mechanism coverage.\",\n \"bottleneck_link\": \"Independent pricing prevents unsupported frontier claims and can reveal genuine accounting slack.\",\n \"allowed_variations\": [\n \"Reconstruct either a source ledger or a numeric enclosure with a different method.\"\n ],\n \"scope_change_triggers\": [\n \"Importing the primary function and calling its output independent verification.\",\n- \"Promoting #2979 from merged source archive to validated supplier solely from author PASS receipts or aggregate pricing.\"\n+ \"Calling an aggregate histogram or source archive an independently accepted physical supplier.\"\n ],\n \"overlap_terms\": [\n \"independent moment\",\n@@ -658,15 +814,50 @@\n \"2979\",\n \"rank7 joint endframes\",\n \"paired extraction\",\n- \"joint lowering\"\n+ \"joint lowering\",\n+ \"4088\",\n+ \"4091\",\n+ \"4067\",\n+ \"staged544-mixed-fourth-entry\",\n+ \"staged-fourth-mixture-ceiling\",\n+ \"r437-two-window-carrier\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #4088: mixed local/owner/phase ledger and moment reported.\",\n+ \"evidence\": \"research/staged544-mixed-fourth-entry/README.md\",\n+ \"remaining\": \"Independent binding of changed construction to paid histogram, distinct from full admission.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"72d7efc181edb21806a878f174847c96dd410ab0\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4091: sharp pooled ceiling proposed for fixed THREE/FOUR words and inventory.\",\n+ \"evidence\": \"research/staged-fourth-mixture-ceiling/README.md\",\n+ \"remaining\": \"Validate dual, exhaustive pattern scope and exact source equivalence before applying to newer fills.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"a29bdd8f56cd18b805917775cdec09ff935eddc4\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4067: paid (2,2) to (1,3) exchange and fixed OQ4 host reported.\",\n+ \"evidence\": \"research/r437-two-window-carrier/README.md\",\n+ \"remaining\": \"Independent full cost delta; remaining caller/precision/stock obligations are separate.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n },\n {\n \"id\": \"formal-interface-lemma\",\n- \"question\": \"Can one small, precisely scoped interface lemma be formalized with its actual hypotheses?\",\n+ \"question\": \"Can one uncovered interface implication be formalized with exact caller and certificate bindings?\",\n \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n \"certificates/selected-result.json\",\n- \"research/hyperplane-end-lean/PROOF.md\"\n+ \"research/gate-operand-parity-lean/README.md\",\n+ \"research/staged544-certified-endpoint/README.md\",\n+ \"research/unconditional-batched-crt/README.md\"\n ],\n \"methods\": [\n \"Formalize a missing boundary lemma identified by the selected review.\",\n@@ -680,10 +871,10 @@\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Read the exact hypotheses in the hyperplane-end proof and distinguish Lean theorems from finite supplier assumptions.\",\n- \"scope\": \"Positive control / source-bound orientation only; this is not a new construction or full cold replay.\"\n+ \"inspect\": \"Inventory the gate-operand parity lemma (#3807) and certified endpoint arithmetic (#3857). Neither is a full multiplication proof. For #4100, audit the compact-switch statement and its hypotheses before formalizing it.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"State one missing boundary lemma in full before starting a proof-assistant build.\",\n+ \"first_experiment\": \"Write one missing statement with quantified hypotheses, actual caller and existing coverage. A compact-switch lemma is a candidate only after its mathematical contract is reviewed. Establish a bounded proof plan before a separately budgeted proof-assistant build; do not rerun large Lean jobs as orientation.\",\n \"bottleneck_link\": \"A missing interface proof can reduce the conditional gap even if it does not change kappa.\",\n \"allowed_variations\": [\n \"Narrow to one formal lemma or one theorem-to-certificate binding.\"\n@@ -694,7 +885,39 @@\n \"overlap_terms\": [\n \"hyperplane end\",\n \"formal interface\",\n- \"Lean binding\"\n+ \"Lean binding\",\n+ \"3807\",\n+ \"3857\",\n+ \"4100\",\n+ \"gate-operand-parity-lean\",\n+ \"staged544-certified-endpoint\",\n+ \"unconditional-batched-crt\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #3807: scoped gate-operand parity formalization.\",\n+ \"evidence\": \"research/gate-operand-parity-lean/README.md\",\n+ \"remaining\": \"Exact theorem-to-caller binding and genuinely uncovered interface lemmas.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"4c3530f42a6abcc48e7a6bcffd6c901911dc8269\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #3857: certified real-power moment/endpoint arithmetic.\",\n+ \"evidence\": \"research/staged544-certified-endpoint/README.md\",\n+ \"remaining\": \"Uniform algorithmic implications, not another endpoint digit.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"ef4315b8caa605e47a2515004845eb2d2ed71a86\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4100: compact-switch formalization proposed as future validation.\",\n+ \"evidence\": \"research/unconditional-batched-crt/README.md\",\n+ \"remaining\": \"Audit the statement and assumptions before a formal proof campaign.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"337d04939571a637aeab54a0f58f54fde811663a\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n }\n ]\n@@ -863,28 +1086,29 @@\n \"tasks\": [\n {\n \"id\": \"selected-packing\",\n- \"question\": \"Is there a genuinely unsearched, explicitly bounded improvement to the selected packing?\",\n+ \"question\": \"Does a named residual packing task have meaningful uncovered headroom after its exact fixed-inventory ceiling?\",\n \"starting_points\": [\n+ \"docs/research/research-family-status.md\",\n \"certificates/selected-result.json\",\n- \"research/heterogeneous-owner-assembly/README.md\",\n- \"research/complex-local-word-ceiling/README.md\"\n+ \"research/staged-fourth-mixture-ceiling/README.md\",\n+ \"research/staged544-mixed-fourth-entry/README.md\"\n ],\n \"methods\": [\n- \"Search a named unclaimed canonical candidate domain with a reproducible partition.\",\n- \"Prove a stronger scoped relaxation or exact optimum for one named residual subproblem.\"\n+ \"Check a fixed-inventory ceiling and source equivalence before any explicitly bounded residual search.\",\n+ \"Price a genuinely changed word/interface under a newly applicable model.\"\n ],\n- \"acceptance_gate\": \"Check live PR overlap first, price both suppliers and distinguish a search best from an optimum. Report the word's packing cap from ceiling.ceiling() and the fraction of it realized; label packing gains as such.\",\n+ \"acceptance_gate\": \"Check live overlap and exact model/source binding first. Distinguish a chronological construction from a pooled optimum. State the applicable fixed-inventory ceiling and remaining headroom; changed words or interfaces require a new fully charged model. Do not launch a residual search without an explicit scope and time cap.\",\n \"first_step\": {\n \"command\": [\n \"python3\",\n \"-B\",\n \"scripts/research_baseline.py\"\n ],\n \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n- \"inspect\": \"Locate the selected packing source in the review and check live PRs for the same residual problem.\",\n- \"scope\": \"Positive control / source-bound orientation only; this is not a new construction or full cold replay.\"\n+ \"inspect\": \"Read #4091 fixed-word pooled ceiling and #4088 later 1090-call literal fill. Bind the unchanged local invoices/capacities/stock before transferring the ceiling; its earlier 1080-call residual-bin recipe is historical.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"Define an unclaimed canonical domain and a favorable bound before starting a new search.\",\n+ \"first_experiment\": \"Default to a bounded headroom/source-equivalence check. Only start residual search with an explicit unclaimed domain, time cap and reason the tiny remaining gain matters. Otherwise hand off or select a changed-word mechanism. A changed word or interface needs a fresh cost model, not the old ceiling. Stop at the fixed-inventory cap; do not extend by new random seeds.\",\n \"bottleneck_link\": \"An actual paid-profile gain can improve the frontier up to the other supplier cap.\",\n \"allowed_variations\": [\n \"One bounded residual packing instance or exact relaxation.\",\n@@ -896,25 +1120,41 @@\n \"overlap_terms\": [\n \"T16\",\n \"owner exchange\",\n- \"packing\"\n+ \"packing\",\n+ \"4091\",\n+ \"4088\",\n+ \"staged-fourth-mixture-ceiling\",\n+ \"staged544-mixed-fourth-entry\"\n ],\n \"existing_checks\": [\n {\n- \"scope\": \"Packing-independent ceiling for the selected paid profile under the one-unpaid-coordinate coverage and halving guard.\",\n- \"evidence\": \"docs/research/community-efficient1000-review.md\",\n- \"remaining\": \"A changed word or model requires new pricing; a local packing search must name uncovered scope.\",\n- \"status\": \"reviewed\",\n- \"source_commit\": \"836b19bfa309ceafbc8ee32a15e8c4b10dde7e11\",\n+ \"scope\": \"PR #4091: pooled optimum at one-third staging for pinned fixed words; not a chronological fill.\",\n+ \"evidence\": \"research/staged-fourth-mixture-ceiling/README.md\",\n+ \"remaining\": \"Independent scope/source check; any residual optimization is explicitly bounded and optional.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"a29bdd8f56cd18b805917775cdec09ff935eddc4\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4088: later 1090-call literal fill improves the earlier 1080-call witness.\",\n+ \"evidence\": \"research/staged544-mixed-fourth-entry/README.md\",\n+ \"remaining\": \"Complete admission remains separate; old residual-bin instructions must not be copied blindly.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"72d7efc181edb21806a878f174847c96dd410ab0\",\n \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n }\n ]\n },\n {\n \"id\": \"scalar-to-moment\",\n- \"question\": \"Can one known scalar simplification be converted into an actual paid-profile improvement?\",\n+ \"question\": \"Can a changed frame itinerary remove a paid transition beyond the already covered scalar A2 contractions?\",\n \"starting_points\": [\n- \"research/r439-bsection-stars-seed01/README.md\",\n- \"certificates/selected-result.json\"\n+ \"docs/research/research-family-status.md\",\n+ \"certificates/selected-result.json\",\n+ \"research/staged544-a2-graph-rewrite/README.md\",\n+ \"research/staged544-separated-a2/README.md\",\n+ \"research/staged544-a2-composition-audit/README.md\",\n+ \"research/staged544-a1858-contraction-audit/README.md\"\n ],\n \"methods\": [\n \"Derive the changed legal frame itinerary and exact child histogram.\",\n@@ -925,13 +1165,13 @@\n \"command\": [\n \"python3\",\n \"-B\",\n- \"research/r439-bsection-stars-seed01/verify.py\"\n+ \"scripts/research_baseline.py\"\n ],\n- \"expected\": \"PASS_R439_BSECTION_STARS\",\n- \"inspect\": \"Observe that the positive control improves scalar counts while retaining the paid histogram.\",\n- \"scope\": \"Positive control / source-bound orientation only; this is not a new construction or full cold replay.\"\n+ \"expected\": \"PASS_RESEARCH_BASELINE_BINDINGS_ONLY\",\n+ \"inspect\": \"Read the A2 and separated-A2 constructions and their composition/contraction audits. Scalar contractions can preserve the complete recursive histogram; fewer ADDs alone do not justify further scans.\",\n+ \"scope\": \"Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately.\"\n },\n- \"first_experiment\": \"Choose one simplification and derive the complete changed frame itinerary, or prove why it cannot change the histogram.\",\n+ \"first_experiment\": \"Identify an actual paid transition to remove and construct a legal replacement frame itinerary, charging restoration, source windows and changed helpers. Compare complete before/after histograms. Stop if only scalar count changes; report that benefit honestly, without expanding the same scalar-only census.\",\n \"bottleneck_link\": \"A legal change to the histogram, not addition count alone, is the bridge to an exponent gain.\",\n \"allowed_variations\": [\n \"One existing scalar simplification and one proposed frame-path change.\"\n@@ -942,7 +1182,49 @@\n \"overlap_terms\": [\n \"scalar simplification\",\n \"paid profile\",\n- \"R439\"\n+ \"R439\",\n+ \"3998\",\n+ \"4019\",\n+ \"4112\",\n+ \"4152\",\n+ \"staged544-a2-graph-rewrite\",\n+ \"staged544-separated-a2\",\n+ \"staged544-a2-composition-audit\",\n+ \"staged544-a1858-contraction-audit\"\n+ ],\n+ \"existing_checks\": [\n+ {\n+ \"scope\": \"PR #3998: consecutive A2 scalar contraction\",\n+ \"evidence\": \"research/staged544-a2-graph-rewrite/README.md\",\n+ \"remaining\": \"A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"389268a6b5e0373a3125d5c6fcd6995f4ea324f5\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4019: separated A2 sites with unchanged recursive histogram\",\n+ \"evidence\": \"research/staged544-separated-a2/README.md\",\n+ \"remaining\": \"A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"f408e09df300b575235f7f77f81948071397e16d\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4112: composition audit\",\n+ \"evidence\": \"research/staged544-a2-composition-audit/README.md\",\n+ \"remaining\": \"A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"205b7380656ba4b6316a8a493b1a3c679b8f7a0e\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ },\n+ {\n+ \"scope\": \"PR #4152: A1858 contraction audit\",\n+ \"evidence\": \"research/staged544-a1858-contraction-audit/README.md\",\n+ \"remaining\": \"A genuinely changed paid itinerary or a distinct independent audit; no default scalar-only expansion.\",\n+ \"status\": \"author-checked\",\n+ \"source_commit\": \"caaa14cde5962adc372fdca71f63d0f29e73c8b2\",\n+ \"selected_record_sha256\": \"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129\"\n+ }\n ]\n }\n ]" + }, + { + "sha": "1adbc50d224e140d77c122498b601be2d9af2158", + "filename": "docs/research/current-priorities.md", + "status": "modified", + "additions": 68, + "deletions": 79, + "changes": 147, + "blob_url": "https://github.com/CrocSwap/integer-mult-bounds/blob/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fcurrent-priorities.md", + "raw_url": "https://github.com/CrocSwap/integer-mult-bounds/raw/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fcurrent-priorities.md", + "contents_url": "https://api.github.com/repos/CrocSwap/integer-mult-bounds/contents/docs%2Fresearch%2Fcurrent-priorities.md?ref=12fa8c5c844be21bfefb38a19503fbd867a11244", + "patch": "@@ -1,85 +1,74 @@\n # Current research priorities\n \n This is the changing diagnosis and allocation; [autoresearch.md](../../autoresearch.md)\n-contains the standing session procedure. Refresh this file and the task catalogue\n-at each research checkpoint. Priority revision 6 and all track weights are unchanged\n-by the workflow split.\n-\n-\n-**Priority revision 6 \u2014 October 16, 2026.** The selected conditional result remains\n-\u03ba = 0.001672660197076918 (#1094). The complex side binds; both suppliers must\n-advance for a large jump. The [review through #2027](community-efficient1000-review.md)\n-adds checked components and scoped obstructions, not a new selected exponent.\n-Recheck the selected record and live overlap before starting.\n-\n-**Keep the mechanism portfolio; refresh the exact missing task.** Start from\n-[the research-family status and expansion gate](research-family-status.md).\n-It distinguishes reviewed predecessors, newer pending claims and the maintainer's\n-local-only product checkpoint. This revision does not admit any of those newer\n-claims or change the frontier.\n-\n-| Card | Starting evidence | Next decision |\n+links the standing session procedure. Refresh this file and the task catalogue\n+at each research checkpoint.\n+\n+**Priority revision 7 \u2014 October 20, 2026**, assessed against main\n+`9200177938843d92957bbfe6f1fdbe88768d8cc5`.\n+The [selected conditional result](../../certificates/selected-result.json) remains\n+\u03ba = **0.001672660197076918** (#1094). Broad track weights are unchanged.\n+Recent intake retained source archives; it did not scientifically admit their\n+claims. In particular, #4088's conditional construction and #4100's proposed\n+unconditional proof are **not selected results**.\n+\n+**Keep broad mechanism exploration; update the exact missing work.** Roughly\n+60% of the portfolio remains on bit mechanisms, complex mechanisms and recursive\n+interfaces. The latest packages justify fresher assignments, not a measured\n+claim that any track has a higher success rate. Use the\n+[family record](research-family-status.md#revision-7-evidence-and-assignments)\n+for source pins, attribution, remaining obligations and stop rules. Check live\n+[handoffs](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166)\n+before selecting a card.\n+\n+| Card | Current evidence, author-checked unless stated otherwise | Next bounded decision |\n |---|---|---|\n-| `p104-side-correction` | Reviewed #1991 channel; source-archived #2981 static joint endpoint flags; see [Oct. 17 support/chronology handoff](p104-live-support-map.md) | Review the submitted #3259 fixed-cover support cut and its contract. If changing the contract, bind the change to a real source/consumer path, exact coefficients, dirty restoration and a complete physical bill. |\n-| `joint-query-controller` | Reviewed #1969 words; author-checked #2965 producer-defect transfer | Can an actual producer meet the completed-output error budgets, including dirty helpers, and feed an affordable consumer? |\n-| `selected-stock-envelope` | Reviewed stock obligations; author-checked #2972 cutoff chain | Derive the actual selected stock coefficient; cutoff comparisons alone do not supply it. |\n-| `independent-moment` | Author-checked #2979 joint rank-7 frame/circuit proposal | Independently check one coordinated group and its losing single-change control, then trace the changed costs through the complete invoice. |\n-| `supplier-boundary-audit` | Author-checked #2977 scatter/export mismatch | Reproduce the tiny semantic mismatch and evaluate the proposed guard at the actual exporter boundary. |\n-| `census-pruning-audit` | Author-checked #2973 parity-pruning correction | Check the tiny counterexample and identify precisely which predecessor exclusions need revalidation. |\n-\n-**These newer packages are source archives, not scientific admissions.** This\n-guidance pass inspected their claims and remaining obligations without replaying\n-their mathematics. In particular, #2979's claimed \u03ba \u2248 0.001729 is not the selected\n-result. Its reported joint win motivates a bounded independent audit, not a mass\n-search for more lowerings. Exact heads, attribution and stop rules are in the\n-[family record](research-family-status.md#revision-6-handoffs-source-archives-awaiting-independent-validation).\n-\n-Read and pin the current dependency before assigning its missing work. A pending\n-local completion changes the useful question even though it has not been admitted\n-as a supplier. Audit its claim or close a specific remaining obligation; do not\n-silently treat it as verified, and do not reconstruct it just because the old card\n-started earlier. Live claims are advisory, not exclusive reservations.\n-\n-**A local contribution and a family expansion have different acceptance gates.**\n-A checked partial construction, interface lemma or scoped obstruction remains\n-valuable. Contributors need not solve the whole supplier. Before expanding it\n-into another large search or dependent implementation round, the director must\n-update the complete boundary and cost ledger: proved costs, constructive upper\n-bounds, unknowns, and the consumer that avoids repaying the saving. Unknown costs\n-are not zero, and passing an optimistic relaxation does not establish feasibility.\n-Use the linked family record's five-point gate. This is an evidence checkpoint,\n-not an approval queue: a solo agent can record the decision in its existing\n-task/thread. A useful handoff needs no extra session report.\n-\n-The recent local product work motivates this discipline but does not justify\n-routing the community into that branch or declaring a global obstruction. Pause\n-the maintainer's fixed preparation recipe; different source/consumer contracts,\n-coupled passes and coding mechanisms remain open. Its local evidence and narrow\n-scope are explicitly labelled in the family record.\n-\n-**Stop rediscovering covered cases.** #1995/#2013 bound terminal-frame and\n-monotone-frame changes within the fixed selected scalar word and cleanup\n-contract. #2017 proves sharp `C=40` in its Eisenstein binary scalar-rectangle\n-and fixed-support block-linear additive pairing model. #1974 completes its\n-two-register GL3 census and a bounded three-register probe. Name a changed\n-hypothesis before reopening those domains; none is a universal barrier.\n-The catalogue links the precise sources and retains broader exploration.\n-\n-Verification should target an unproved implication, changed source binding or\n-new failure mode. Inventory existing independent checks before another root\n-calculation. A handoff on a central missing lemma is preferable to inventing a\n-peripheral task merely to finish a PR. Incremental and independent contributions\n-remain welcome with their scope stated.\n-\n-Before extending a static cover search, precision calculation or source-count\n-census, name the unresolved obligation it closes or the active research decision\n-it changes. Do not rebuild #2981's static matching, #2965's decoder error constants\n-or #2972's cutoff comparisons merely because they are easy to run. Independent\n-checks of those claims remain useful when they supply missing coverage. For the\n-#2542/#2634/#2691 census chain, first audit the pruning objection in #2973; neither\n-the old exclusions nor their proposed replacement should silently close a route.\n-\n-**Retained revision-3 quantitative diagnosis (scoped arithmetic independently replayed for #1240 and #1264).** On the binding complex side, a \u2248 D/E:\n+| `uniform-transfer-audit` | #4100 proposed all-size construction | Audit **one** quantified implication at its actual caller: CRT total repair, native/free-algebra lift, precision/conditioning, or fixed-tape cost. Independent acceptance is absent. |\n+| `supplier-boundary-audit` | #4088 mixed four-provider local construction | Rebind the ordinary/outer admission engine to all four providers and dispatch, preserving every predicate and rejection control. The full mixed engine has not been rerun. |\n+| `joint-query-controller` | #4166 actual QQ16 caller installation | Compare the complete matched parent park/restore and external-call protocols, retaining all later calls; or change the actual demand to remove paid work. The native-component profile loses. |\n+| `r437-carrier-integration` | #4067 paid two-window carrier; #4147 reverse coefficient | Close one pooled compiler/request, physical caller, or precision/stock obligation. Do not rederive the covered coefficient. |\n+| `p104-side-correction` | #4133/#4136/#4140/#4153/#4154 scoped slot, commutation, encoding, chart and fiber-seam objections | Name a changed producer/consumer premise, then test literal action, dirty restoration and the fully charged boundary. Static flags alone do not supply it. |\n+| `independent-moment` | #4088 changed ledger; #4091 proposed fixed-inventory ceiling; #4067 paid exchange | Independently bind a changed mechanism to its bill or check the scoped pooled dual. More digits of the same root are not additional mechanism coverage. |\n+| `formal-interface-lemma` | #3807 gate-operand and #3857 endpoint formalizations | Target an uncovered interface statement. Audit the proposed compact-switch contract before a formalization campaign. |\n+\n+**Separate admission from proof audit.** #4088 needs finite/compiler integration;\n+#4100 claims to discharge all-size premises. Those are different tasks, and neither\n+is closed by checking the other's numerical root. Partition work by proposition,\n+caller and domain, with one canonical thread and exact source pins. A short session\n+can produce an acceptance coverage map, a counterexample or a single proved lemma.\n+A complete ordinary/outer replay or Lean build needs a separate resource budget;\n+no heavy run is part of default orientation.\n+\n+**Pause repetitive searches within a fixed invoice.** #4091 proposes a sharp\n+pooled ceiling for the fixed THREE/FOUR words and inventory. It is pinned to an\n+earlier #4088 revision: verify local-word, capacity and stock equivalence before\n+applying it to the later 1090-call fill. Its pooled witness is not a chronological\n+implementation. The `selected-packing` card now starts with this scope/headroom\n+check; any tiny residual search is explicitly bounded and optional. Changed words\n+or interfaces require a fresh model, not the old ceiling.\n+\n+Likewise, #3998/#4019 and #4112/#4152 cover scalar A2 contractions and related\n+audits. The `scalar-to-moment` card requires an actual removed paid transition,\n+a legal replacement itinerary and the complete cost delta. Stop if only scalar\n+ADD counts change; that may have a separate benefit, but does not justify a larger\n+\u03ba-oriented scan of the same domain.\n+\n+**Local contributions remain welcome.** A checked partial identity, interface\n+lemma or scoped obstruction can be valuable without a complete supplier. Before\n+expanding a family, update its full producer/consumer ledger: proved costs,\n+constructive upper bounds, unknowns, and why the consumer does not repay the gain.\n+Unknown costs are not zero. Use the [five-point expansion gate](research-family-status.md#contribution-acceptance-and-family-expansion-are-different-gates).\n+This is an evidence checkpoint, not an approval queue or an extra reporting task.\n+\n+The selected stock coefficients remain a targeted audit obligation; cutoff\n+arithmetic does not provide them. For census and exporter work, check successor\n+corrections and their exact scope before repeating a tiny example. Independent\n+verification must name missing coverage. The reviewed #1995/#2013 fixed-word\n+frame bounds, #2017 fixed-model C=40 result and #1974 bounded census remain scoped\n+predecessors, not universal barriers. The maintainer's older product checkpoint\n+also remains local-only evidence as labelled in the family record.\n+\n+**Retained selected-profile diagnosis (revision 3; scoped arithmetic independently replayed for #1240 and #1264).** This paragraph and its lever table concern the frozen selected profile, not the changed #4088 words or every possible supplier. On the binding complex side, a \u2248 D/E:\n \n - **D is small.** Each call has about 99.6 live families. The coverage identity leaves one unpaid\n coordinate per family, and the centre path then takes back 64. Only about 35.6 of 7,232 paid\n@@ -128,7 +117,7 @@ valuable even when the other supplier prevents an immediate headline change.\n \n The helper allocates effort as follows. These are portfolio weights, not success\n probabilities; the catalogue is the machine-readable source for assignments.\n-Revision 6 keeps the track weights: current evidence supports fresher tasks and\n+Revision 7 keeps the track weights: current evidence supports fresher tasks and\n stronger expansion decisions, not a measured reason to reallocate the portfolio.\n Prefer a live, unclaimed central handoff over a generic card when it matches your track or expertise;\n record the scoped override and check overlap." + }, + { + "sha": "abdd2792e6a5596b3fa6be3dc9db882bd8e3ef89", + "filename": "docs/research/research-family-status.md", + "status": "modified", + "additions": 51, + "deletions": 20, + "changes": 71, + "blob_url": "https://github.com/CrocSwap/integer-mult-bounds/blob/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fresearch-family-status.md", + "raw_url": "https://github.com/CrocSwap/integer-mult-bounds/raw/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fresearch-family-status.md", + "contents_url": "https://api.github.com/repos/CrocSwap/integer-mult-bounds/contents/docs%2Fresearch%2Fresearch-family-status.md?ref=12fa8c5c844be21bfefb38a19503fbd867a11244", + "patch": "@@ -1,29 +1,60 @@\n # Research family status and expansion decisions\n \n-Revision 6, assessed 2026-10-16 against main\n-`b3f3ee0392b71b118fd086a6eaf6b4cab54ef455`.\n+Revision 7, assessed 2026-10-20 against main\n+`9200177938843d92957bbfe6f1fdbe88768d8cc5`.\n This is a navigation and decision record, not supplier admission or a complete\n-survey. The selected conditional kappa is still **0.001672660197076918**.\n-Resolve its evidence and assumptions through\n-[selected-result.json](../../certificates/selected-result.json).\n+survey. The selected conditional \u03ba remains **0.001672660197076918**; resolve its\n+sources and assumptions through [selected-result.json](../../certificates/selected-result.json).\n \n-Check live heads and claims before choosing a task. The revision-6 package claims\n-and intake records were read for this update; their newer mathematical\n-claims were **not independently replayed in this guidance pass**. Pin any pending\n-dependency and validate the part your work relies on. A pending claim can guide\n-an audit without being treated as an established theorem or a reservation.\n+The newer package claims and intake receipts were read for this guidance update.\n+Their mathematics was **not independently replayed in this pass**. Merged archives\n+retain contributors' work; they do not certify the claims. Pin and validate any\n+pending dependency your work uses. Check live heads and handoffs before assignment.\n \n ## Current leads\n \n-| Family / decision | Established starting point | Newer evidence status | Next useful decision |\n-|---|---|---|---|\n-| P104 completion / narrow to chronological realization | [Reviewed #1991 channel](../../research/p104-native-channel/README.md), [review](community-efficient1000-review.md) | [Source-archived #2981 static joint flags](../../research/p104-joint-unit-flags/README.md); see the [Oct. 17 fixed-cover support and chronology map](p104-live-support-map.md) for newer source-archived proposals | The fixed-cover route now has a submitted pooled support obstruction at #3259. Review its proof and assumptions before treating that contract as closed; a different route needs an explicit changed source/frame contract, exact correction, restoration and complete bill. The archives are not supplier admissions. |\n-| QQ16 controller / narrow to actual producer and consumer | [Reviewed local words and scoped witness](../../research/joint-query/README.md), [review](community-efficient1000-review.md) | [#2965 defect transfer](../../research/qq16-producer-defect-transfer/README.md), author-checked source archive | Derive actual producer-output defects and dirty-helper ranges, then price physical consumption. Another local inverse or isolated precision constant is not the missing implementation. |\n-| Selected stock / retain targeted audit | [Stock audit](../../research/selected-bootstrap-stock-audit/README.md), [review](community-efficient1000-review.md) | [#2972 cutoff chain](../../research/stock-margin-chain/README.md), author-checked source archive; actual coefficients remain unresolved | Derive one actual selected stage reserve/depth coefficient from its producer. Cutoff arithmetic does not supply it. |\n-| Joint frame/circuit changes / audit before expansion | Selected supplier and complete paid-profile obligations | [#2979 coordinated lowerings and extractions](../../research/rank7-joint-endframes/README.md), author-checked source archive | Independently check a joint group against a single-change control and trace its entire cost delta; stop at unresolved admission dependencies before searching more groups. |\n-| Semantic acceptance and census reliability / bounded audits | Existing selected-source and export checks | [#2977 scatter/export mismatch](../../research/scatter-export-parity-audit/README.md) and [#2973 pruning correction](../../research/f7-weight-parity-census-correction/README.md), author-checked source archives | Validate tiny counterexamples, then map precisely affected acceptance conditions or exclusions. Neither report automatically invalidates the selected construction. |\n-| New coding, geometry and recursive boundaries / retain exploration | Existing scope-specific barriers and [budget tool](../../research/complex-local-word-ceiling/README.md) | No new general mechanism admitted by this update | Specify which obligation disappears and why a complete consumer does not repay it. Choose a genuinely different contract or mechanism, not another seed. |\n-| Product preparation / pause the maintainer's fixed recipe | Local checkpoint described below; not integrated scientific evidence | Maintainer-local checked packages, not independently reviewed or available from this main checkout | Audit the exact source when published, or change production **and** its consumer, the two-pass interface, or the central/side construction together. Do not interpret the local stop as a universal product obstruction. |\n+| Family | Decision and next useful work | Stop / reopening condition |\n+|---|---|---|\n+| Proposed all-size proof #4100 | **Audit**, partitioned into CRT repair, native/free-algebra lift, precision/conditioning and tape costs. Name one proposition and actual caller. | Stop at a precise missing implication or proved scoped result. Finite checks do not discharge universal premises. Broaden only after that dependency is resolved. |\n+| Mixed four-provider construction #4088 | **Complete admission**, beginning with registry/dispatch and coverage of every ordinary/outer predicate and mutation control. | No frontier promotion from a local/root pass. Budget full execution separately after source/caller bindings are concrete. |\n+| QQ16 #4166 | **Narrow** to matched whole-caller protocol cost or changed source-compatible demand. Caller installation exists as an author-checked proposal; native-component comparison loses. | Stop if the full necessary inequality fails or remains unpriced. Reopen with a consumer/demand that removes paid work, retaining all later root calls. |\n+| R437 #4067/#4147 | **Integrate or audit** the paid carrier exchange at compiler/request, caller and precision/stock boundaries. | Reverse coefficient is already covered. No claim of global admission from local moment or fixed-host checks alone. |\n+| P104 successor interfaces | **Change premise or audit scope**: literal slots, commutation, encoding, source charts and fiber seams are now the relevant obstacles. | Stop repeated static counts or the same mismatched seam. Reopen with a changed producer/consumer and fully charged completion, not a new seed. |\n+| Fixed-inventory packing / scalar contractions | **Narrow** to source-bound ceiling checks or actual paid-transition changes. | No default fixed-invoice repacking or scalar-only scan. Residual packing is an explicitly bounded opt-in after its headroom is established. |\n+| New geometry and recursive boundaries | **Continue broad exploration**, roughly 60% of allocation. | Name the obligation that disappears, price the completed consumer and test a discriminating example before expansion. No universal exclusion follows from these families. |\n+| Selected stock and formal interfaces | **Retain targeted audits** of actual coefficients and uncovered caller-bound lemmas. | Cutoff comparisons and formal endpoint arithmetic do not replace uniform implementation proofs. |\n+\n+## Revision-7 evidence and assignments\n+\n+Each row is **author-checked source evidence**, not independent maintainer scientific\n+acceptance. Exact heads identify the retained contributions; package notices retain\n+full predecessor credit. The catalogue carries the same pins and selected-record hash.\n+\n+| PR / source | Retained head | Covered claim and remaining task |\n+|---|---|---|\n+| #4100, Cinna Davis \u2014 [batched CRT](../../research/unconditional-batched-crt/README.md) | `337d04939571a637aeab54a0f58f54fde811663a` | Proposed unconditional construction. Start independent audit at CRT.md total repair or COMPLEX.md free-algebra lifting if unclaimed; separate precision and tape-cost assignments. |\n+| #4088, Cinna Davis \u2014 [mixed fourth entry](../../research/staged544-mixed-fourth-entry/README.md) | `72d7efc181edb21806a878f174847c96dd410ab0` | 1090 fourth-entry calls with local/native/owner/phase evidence and conditional pricing. Full ordinary/outer four-provider admission remains open. |\n+| #4091, Ross Richardson \u2014 [pooled mixture ceiling](../../research/staged-fourth-mixture-ceiling/README.md) | `a29bdd8f56cd18b805917775cdec09ff935eddc4` | Proposed sharp fixed-inventory root ceiling near 0.001730980256. Pins an earlier #4088 head; check model equivalence before using it on the later fill. No chronological fill follows from its pooled witness. |\n+| #4166, Chris R / Clear1nsight \u2014 [QQ16 caller](../../research/qq16-caller-install/README.md) | `0b082d22a0e9a442d264ab0633e35ef26c790630` | Actual caller connection, with losing native-component bill and unresolved matched physical comparison. Preserve parent parking, external protocols, all retained calls and service costs. |\n+| #4067, Chris R / Clear1nsight \u2014 [R437 carrier](../../research/r437-two-window-carrier/README.md) | `a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e` | Paid (2,2)\u2192(1,3) exchange, three added ADDs per call and fixed OQ4 host. Pooled compiler/request, physical caller, precision and stock remain. |\n+| #4147 \u2014 [reverse coefficient](../../research/r437-reverse-row-coefficient/README.md) | `72e7bb797adf82aeb525a399f56ab8adbab8e93c` | Use the covered reverse coefficient in the missing source-bound admission proof; do not repeat it as a new task. |\n+| #4133 \u2014 [literal slot](../../research/p104-literal-slot-defect/README.md) | `88666f1dee93fbd85b31fd5d6f91a219e111b5f4` | Scoped slot defect; changed literal action or independent audit required. |\n+| #4136 \u2014 [commutation](../../research/p104-uniform-commutation-gap/README.md) | `281ce601a8be9db470de630759179c23e1dfaa88` | Scoped uniform commutation gap; identify the changed interface. |\n+| #4140 \u2014 [encoding](../../research/p104-uniform-encoding-gap/README.md) | `de25c6525aa866962309df5d107b32d55e6e5628` | Scoped encoding gap; static representation is not uniform completion. |\n+| #4153 \u2014 [source chart](../../research/p104-source-chart-shortcut/README.md) | `9123241fea53df77e51ffe30d9aca1625d4bf3a8` | Shortcut exhausts its favorable allowance before other costs. Change the source/consumer premise to reopen. |\n+| #4154 \u2014 [fiber seam](../../research/p104-fiber-tag-seam/README.md) | `4ffcdaee5d6255a9cea1fe53a5424435068089d0` | Scoped tag/fiber mismatch, not a universal P104 obstruction. |\n+| #3998 \u2014 [A2 rewrite](../../research/staged544-a2-graph-rewrite/README.md); #4019 \u2014 [separated A2](../../research/staged544-separated-a2/README.md) | `389268a6b5e0373a3125d5c6fcd6995f4ea324f5`; `f408e09df300b575235f7f77f81948071397e16d` | Scalar contraction is not a paid-profile gain. Next work must remove an actual paid transition. |\n+| #4112 \u2014 [composition audit](../../research/staged544-a2-composition-audit/README.md); #4152 \u2014 [contraction audit](../../research/staged544-a1858-contraction-audit/README.md) | `205b7380656ba4b6316a8a493b1a3c679b8f7a0e`; `caaa14cde5962adc372fdca71f63d0f29e73c8b2` | Check existing audit coverage before rediscovering the same local identity. |\n+| #3807 \u2014 [gate operand formalization](../../research/gate-operand-parity-lean/README.md); #3857 \u2014 [endpoint formalization](../../research/staged544-certified-endpoint/README.md) | `4c3530f42a6abcc48e7a6bcffd6c901911dc8269`; `ef4315b8caa605e47a2515004845eb2d2ed71a86` | Scoped formal results; choose an uncovered interface lemma, not another endpoint digit. |\n+| #3801 \u2014 [scatter decision](../../research/scatter-boundary-decision/README.md) | `911f8800ba7a4b03603ef62c5ffc6240e53cb5fd` | Successor acceptance-boundary work; consult it before repeating #2977. |\n+\n+**Assignment separation.** A #4100 proof auditor selects a proposition and caller;\n+a #4088 admission contributor selects a registry/predicate obligation; an independent\n+ledger reviewer selects a changed cost or dual certificate. These are distinct even\n+when they share a package. Check live claims, and do not launch full native, Lean,\n+Docker or all-history replays as orientation. Source-binding commands are not\n+scientific validation. Reassess after an independently resolved implication,\n+a complete-cost decision, changed supplier, or substantial new intake.\n \n Historical revision-5 inspected heads (preserved as predecessor context, not a\n fresh statement of their GitHub status):\n@@ -51,7 +82,7 @@ the same question. Neither thread is evidence of a stronger selected result.\n \n ## Revision-6 handoffs: source archives awaiting independent validation\n \n-The following exact heads are retained on the assessed main. Their intake\n+Historical revision-6 assignments below are retained for provenance; the revision-7 table supersedes their default next tasks. The following exact heads are retained on the assessed main. Their intake\n records certify source retention, not their mathematics. Credit stays with the\n named contributors and all predecessor attribution in their packages.\n " + }, + { + "sha": "fcbba3a441cbaa16364ef066412ce6309055c2a6", + "filename": "docs/research/research-priority-log.md", + "status": "modified", + "additions": 84, + "deletions": 0, + "changes": 84, + "blob_url": "https://github.com/CrocSwap/integer-mult-bounds/blob/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fresearch-priority-log.md", + "raw_url": "https://github.com/CrocSwap/integer-mult-bounds/raw/12fa8c5c844be21bfefb38a19503fbd867a11244/docs%2Fresearch%2Fresearch-priority-log.md", + "contents_url": "https://api.github.com/repos/CrocSwap/integer-mult-bounds/contents/docs%2Fresearch%2Fresearch-priority-log.md?ref=12fa8c5c844be21bfefb38a19503fbd867a11244", + "patch": "@@ -556,3 +556,87 @@ one actionable construction cluster. Add coverage records as actual review\n evidence arrives; expand profile intake only if it reduces repeated review.\n Do not create a survey PR per session or replace missing scientific evidence\n with structured labels.\n+\n+## Priority revision 7 \u2014 2026-10-20: current admission gaps and changed contracts\n+\n+**Assessment base:** `9200177938843d92957bbfe6f1fdbe88768d8cc5`.\n+The selected #1094 record remains \u03ba = `0.001672660197076918`, SHA256\n+`bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129`.\n+Its source manifest remains\n+`research/community-frontier1094-audit/SELECTED-SOURCE.json`; no scientific\n+input, certificate or selected-record binding is changed by this revision.\n+\n+**Evidence and limits.** Read retained package claims, open questions and exact-head\n+intake receipts from the October 20 batches, especially\n+[batch d](pr-intake-evidence/queue-oct20d-20261020.json),\n+[batch e](pr-intake-evidence/queue-oct20e-20261020.json) and\n+[batch f](pr-intake-evidence/queue-oct20f-20261020.json).\n+The [revision-7 family table](research-family-status.md#revision-7-evidence-and-assignments)\n+and catalogue pin the relevant heads and preserve package attribution.\n+This was a guidance assessment, not a scientific replay: newer claims remain\n+**author-checked**, including claims whose packages contain their own independent\n+checker or worker review. Source retention is not theorem acceptance.\n+\n+A bounded review of the last four comments in\n+[feedback #1200](https://github.com/CrocSwap/integer-mult-bounds/discussions/1200)\n+and last six in [handoffs #1166](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166)\n+found existing tooling feedback and concrete scoped successors (A1858, P104,\n+T488 and rank-8/u2). This is not an exhaustive board census or a measure of donor\n+yield. Existing timeout/explain-plan/snapshot/sparse-inventory feedback has\n+retained tooling follow-ups; it does not justify another duplicate tool assignment.\n+The A1858 and P104 handoffs informed the narrower cards. No measured basis for\n+reweighting the portfolio emerged.\n+\n+**Allocation, old \u2192 new:** bit mechanisms 20\u219220; complex mechanisms 20\u219220;\n+recursive interfaces 20\u219220; adversarial audit 15\u219215; independent verification\n+10\u219210; verifier engineering 8\u21928; consolidation 5\u21925; incumbent optimization 2\u21922.\n+These are effort weights, not success probabilities. Existing task IDs retain\n+their questions' lineage; the distinct R437 caller question gets the new\n+`r437-carrier-integration` card within the bit track. This changes its within-track\n+mix, not the broad allocation.\n+\n+**Decisions and next experiments.**\n+\n+- **Audit #4100's proposed unconditional proof**, selecting one proposition and\n+ caller in CRT total repair, native/free-algebra lifting, precision/conditioning\n+ or tape costs. Stop with the exact implication established or missing. Finite\n+ checks and sharper root digits are not an all-size proof.\n+- **Complete #4088 admission coverage**, starting at four-provider registry and\n+ dispatch bindings, preserving all ordinary/outer predicates and mutation\n+ controls. A full mixed-engine replay is a separately budgeted follow-up;\n+ orientation must not launch it. Admission and all-size proof audit are distinct.\n+- **Narrow QQ16 to the actual #4166 caller**: compare complete matched parent\n+ park/restore and external-call protocols, including all later calls and new\n+ service work. Native-component cost loses. Reopen construction with a genuinely\n+ cheaper source-compatible demand or consumer, not another isolated inverse.\n+- **Integrate R437's #4067 paid exchange** at a compiler/request, physical caller\n+ or precision/stock boundary. #4147 already covers the reverse coefficient.\n+ Preserve extra ADDs and all helper costs; a local paid gain is not admission.\n+- **Refresh P104 to #4133/#4136/#4140/#4153/#4154**. Audit a specific objection\n+ or change the source/consumer premise and fully price completion. Stop when the\n+ same slot or seam mismatch recurs, or the favorable full allowance fails.\n+ These are scoped obstacles, not a universal exclusion.\n+- **Narrow fixed-inventory packing and scalar-only scans.** #4091 pins an earlier\n+ #4088 revision: first verify words, capacities and stock equivalence before\n+ applying its proposed ceiling to the later fill. Its pooled witness does not\n+ establish chronology. Residual search is explicitly bounded and optional.\n+ #3998/#4019/#4112/#4152 motivate requiring an actual removed paid transition\n+ before expanding scalar contraction scans. Reopen with a changed invoice or\n+ interface and an applicable full cost model.\n+- **Retain independent and formal coverage where missing.** Distinguish ledger\n+ audit from dispatch admission and universal proof. #3807/#3857 already supply\n+ scoped formal work; target uncovered caller-bound statements rather than extra\n+ endpoint digits. Selected stock coefficients remain a separate obligation.\n+\n+**Next review trigger:** an independently resolved all-size implication, a\n+completed mixed-provider admission, a complete-cost decision for QQ16/P104/R437,\n+a changed selected supplier, or another substantial intake. Looping agents\n+consume revision 7 at their normal upstream checkpoint and preserve frozen work.\n+No new reporting artifact or heavy replay is required merely to refresh guidance.\n+\n+**Validation:** `python3 -B tests/test_research_assignment.py` passed 16 tests;\n+`python3 -B tests/test_research_coordination.py` passed 8 tests. Changed cards\n+were checked through the task-picker CLI; new coverage pins match exact intake\n+receipts. Documentation/source paths were checked against the working tree or\n+Git tree for sparse archives. Track weights and selected-record bytes are\n+unchanged. No Lean, Docker or scientific replay was run for this guidance patch." + } + ] +} diff --git a/research/kappa-rebuild/provenance/public-final/CONTRIBUTING.md b/research/kappa-rebuild/provenance/public-final/CONTRIBUTING.md new file mode 100644 index 0000000000..fe54f23f19 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/CONTRIBUTING.md @@ -0,0 +1,101 @@ +# Contributing and independent review + +**For donated inference or an open-ended “contribute” task, start with +[autoresearch.md](autoresearch.md).** It provides weighted task suggestions, +semantic overlap checks and bounded stopping criteria. Follow explicit user +assignments instead of drawing a new task. The root [AGENTS.md](AGENTS.md) also +points agents there. Existing external donation prompts should use this +repository-owned guidance for current priorities. + +Before substantial research, check the +[discussion board](https://github.com/CrocSwap/integer-mult-bounds/discussions) +alongside open PRs. The [start-here post](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163) +explains claims and updates; use [handoffs](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166) +for concrete requests and [mechanism proposals](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167) +for early ideas. Existing discussion claims are advisory, and findings remain +provisional until their evidence is checked. + +New research packages should include **Open next questions**, with relevance, +a feasible first step and blockers. Use the optional +[package generator and replay manifest](docs/research/research-package-guide.md) +to reduce packaging work. Keep frozen predecessor packages unchanged. A focused +finding or handoff is also a valid session outcome; a PR is not mandatory. + +The standard donation prompt in autoresearch.md authorizes routine Discussion +posts and opening focused PRs without repeated permission checks. Respect narrower +donor instructions. Optional workflow feedback belongs in the PR or the +[feedback discussion](https://github.com/CrocSwap/integer-mult-bounds/discussions/1200). + + +For maintainer queue work, start with the [delta-only intake procedure](docs/research/pr-intake.md) +and persistent scheduling cache. Do not repeat unchanged blocked reviews. + +Before reviewing an existing PR, consult the [central review ledger](docs/research/pr-ledger.md). +Record the reviewed commit and its evidence separately from publication and GitHub closure. +Recheck the current head before closing incorporated work, link its retained source and attribution, +and leave partial or changed submissions open with the remaining question recorded. Research notes +and verification tools can merit integration even when they do not improve the selected bound. + +Maintainers distinguish **archival intake** from **validated incorporation**. +New isolated research packages may be merged after source/integrity intake while +remaining `partially_reviewed` in the ledger, with the unverified obligations +preserved. Such a merge retains the contribution; it does not verify a theorem, +certify a new κ, or change the selected result. Shared-code changes and known +correctness objections remain in substantive review. See the +[archive policy and exact-head index](docs/research/research-archive-intake.md). + +This is a research draft. Corrections to the mathematics, implementation, +source attribution, and stated scope are welcome through issues or pull requests. + +For a mathematical issue, identify the exact proposition, source label, or +inequality; give a counterexample or the missing implication when possible; +and distinguish numerical validation from an algorithmic proof obligation. +For the current result, start with the +[current status](docs/research/current-status.md), +[selected record](certificates/selected-result.json), and the maintainer review +at that record's `review` path. +The selected record and review identify the actual supplier sources, +reproduction command, evidence and remaining assumptions. Earlier construction +notes are historical checkpoints, even when their text calls a result current. +The [original audit](docs/audit.md) describes the retained upstream assumptions. + +For a parameter improvement, supply exact rational choices, the full dependency +argument, and an updated patch against the pinned source. Explain whether the +change stays within the parameter ceiling or changes one of its hypotheses. + +Keep `upstream/`, frozen source archives and pinned evidence unchanged. When +changing generated artifacts, edit their generators and include the regenerated +certificates and patches in the same change. Run the checks affected by your +changes; do not launch every historical replay for an unrelated contribution. +Changes to the selected construction also need its applicable complete cold +replay: use the `reproduce` command in the selected record rather than copying a +target from a historical note. Source-bound reuse must preserve explicit coverage +and invalidation. Run artifact-regenerating suites in isolated worktrees. +The historical `make gen4-bank-verify` target checks the earlier generation, +not the selected frontier. The proof and bank scheduling supplements are supplied as Markdown; +historical PDF targets belong to their respective checkpoints. Review changes to +claims in the README and note together. Finite tests should address a mathematical +identity or a failure mode, rather than simply restating implementation details. + +**Regenerated artifacts and hashes.** When verification regenerates a compressed artifact, compare its decompressed payload (or a documented canonical representation of every proof-relevant field). Compression headers and streams can differ across Python, zlib and tar versions. Keep exact hashes for committed source archives and other immutable inputs; a regenerated container hash may be recorded with environment details but should not determine mathematical equality across environments. See the [community reproducibility audit](research/gzip-container-reproducibility/README.md) by Maxime Fleury, following Antoine Olivier’s independent replay findings. + +New contributions are under the repository's Apache-2.0 license. Retain source +attribution and disclose substantial AI assistance. Do not describe certificate +success as a formal verification of the full multiplication theorem. + +Before building on pending work, consult the [contribution-review index](docs/research/contribution-review.md), pin its sources, and preserve contributor attribution. + +Parallel approaches, small improvements, independent reproductions, review and +well-scoped negative results are welcome. Please identify predecessor work and +explain what your submission adds, even if a concurrent result has a stronger +headline. We preserve credit for useful contributions that are superseded or not +imported; see [CONTRIBUTORS.md](CONTRIBUTORS.md). Attribution corrections are +welcome too. State which proof obligations remain open rather than presenting +finite certificate checks as a complete theorem audit. + +The historical `integration/community` workflow used `make verify` for its +imported candidate chain; it is not the current selected cold replay. See +[the integration ledger](docs/research/community-integration.md) for that history +and [the batching guide](docs/research/batched-review.md) for the changed recursive +interface. Preserve the historical checkpoint tests. +Imported CC0 proof sources retain their source-specific license and notice. diff --git a/research/kappa-rebuild/provenance/public-final/REFRESH.json b/research/kappa-rebuild/provenance/public-final/REFRESH.json new file mode 100644 index 0000000000..dbe17c0720 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/REFRESH.json @@ -0,0 +1,43 @@ +{ + "canonical_repository": "CrocSwap/integer-mult-bounds", + "checked_utc": "2026-10-21T05:19:47.352891+00:00", + "files": { + "AGENTS.md": { + "bytes": 5580, + "same_as_comparison": true, + "sha256": "eb25d8ec495b80190a4b1a634710819bb74f0fae88d5203285c2b75b6f147347" + }, + "CONTRIBUTING.md": { + "bytes": 7143, + "same_as_comparison": true, + "sha256": "8a49fe97e2ad088a2b2e0c5b222aaab08752783b1f9c1260790d7dd13011d215" + }, + "autoresearch.md": { + "bytes": 5298, + "same_as_comparison": true, + "sha256": "8c98d179208c34a8c6768931e13f3bbeaa1cb63d59c660799ffde05c40935d9b" + }, + "certificates/selected-result.json": { + "bytes": 10725, + "same_as_comparison": true, + "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + "docs/research/agent-procedure.md": { + "bytes": 27753, + "same_as_comparison": true, + "sha256": "b2dd2f64e5746fce6289422351f2e3d1eb224f7db09aeb72f4d6626263190aa1" + }, + "docs/research/current-priorities.md": { + "bytes": 11125, + "same_as_comparison": true, + "sha256": "8f22daeac030282dea99ede86f577bac7451a0063b05df2561d15e36d322b88d" + }, + "docs/research/research-family-status.md": { + "bytes": 18359, + "same_as_comparison": true, + "sha256": "0e61c47dfbc6f16e5f64594fcc7d907d4187568e38dddf10d473f0992419c44f" + } + }, + "main_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "scope": "Commit and seven exact guidance files; no broad PR admission or mathematical replay." +} diff --git a/research/kappa-rebuild/provenance/public-final/autoresearch.md b/research/kappa-rebuild/provenance/public-final/autoresearch.md new file mode 100644 index 0000000000..e5445702e6 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/autoresearch.md @@ -0,0 +1,92 @@ +# Contributing donated research effort + +**Start here if you were told to “contribute.”** Aim for one useful, independently +reviewable result in about **60 minutes**, unless your donor specifies a budget. +A precise partial result, counterexample, missing-lemma audit or handoff can be +valuable; producing a PR or increasing κ is not required. + +1. Read [current priorities](docs/research/current-priorities.md) and the + [selected record](certificates/selected-result.json). Use the + [bottleneck brief](docs/research/bottleneck-brief.md) and + [glossary](docs/research/research-glossary.md) as needed. +2. **Check [handoffs #1166](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166) + before drawing a card.** Prefer a suitable open request on your track; check + its artifacts, claims and live PR overlap. Do not reroll an assigned task. +3. If no suitable request fits, run `python3 scripts/choose_research_task.py`. + Inspect the card's `existing_checks` and remaining obligations. Unknown + coverage is not evidence of novelty; old source contexts need reassessment. +4. Follow the [standing procedure](docs/research/agent-procedure.md), + [CONTRIBUTING.md](CONTRIBUTING.md), and the relevant + [family expansion gate](docs/research/research-family-status.md). + Name one concrete question, success/stop test, and bounded verification. +5. Preserve source attribution and report exactly what was established. + [Post updates from a file](docs/research/coordination.md); structured metadata + is optional and advisory. Under the standard donation prompt below, routine + Discussion participation and focused PRs are authorized. Merging, closing + others' PRs and changing the selected frontier require maintainer authorization. + +For short sessions finishing within an hour, an advance CLAIM is optional after +overlap checking. Claim a handoff you take, likely overlapping work and work +continuing across sessions. FINDING/DONE does not imply scientific approval. + +## Looping agents: refresh upstream at each research checkpoint + +At each new round, before a major pivot and before submission, fetch canonical +main and pin its commit. During a longer round, check at a safe point roughly +once per hour. Read changed priorities, catalogue, selected record and relevant +evidence. The picker does not fetch or reserve work. Preserve in-progress work +and frozen inputs; fetching does not update your checkout. See the +[exact refresh procedure](docs/research/agent-procedure.md#looping-agents-refresh-upstream-at-each-research-checkpoint). + +## 1. Orient briefly, then choose a distinct task + +[Handoff-first selection and overlap checks](docs/research/agent-procedure.md#1-orient-briefly-then-choose-a-distinct-task). +An existing overlap cache can surface structured claims and handoffs with +`python3 scripts/choose_research_task.py --coordination-cache /tmp/overlap.json`. +Check the reported freshness and prose-only discussions too. + +## 2. Where effort is most useful + +[Current diagnosis and allocation](docs/research/current-priorities.md), +[machine-readable cards](docs/research/agent-task-catalog.json), and +[priority decisions](docs/research/research-priority-log.md). + +## 3. Make parallel effort meaningfully different + +[Task cards, claims, handoffs and stopping rules](docs/research/agent-procedure.md). + +## 4. Bound the experiment and kill unpromising routes early + +[Session budgets and stopping rules](docs/research/agent-procedure.md#4-bound-the-experiment-and-kill-unpromising-routes-early). + +## 5. Verify and submit one useful result + +[Verification, scope and attribution](docs/research/agent-procedure.md#5-verify-and-submit-one-useful-result). + +## 6. Maintainer/director priority review + +Use the [director review procedure](docs/research/agent-procedure.md#6-maintainerdirector-priority-review). +Update current priorities, affected cards and their coverage, and the evidence +log together. For PR work, use [delta-only intake](docs/research/pr-intake.md) +and its construction-grouped view. Do not rerun unchanged historical reviews. + +## Short prompt for donating a session + +> Contribute one useful, independently reviewable result to +> https://github.com/CrocSwap/integer-mult-bounds. Read AGENTS.md, autoresearch.md +> and CONTRIBUTING.md from the current main branch first. Use the repository's +> task-selection and overlap process unless I assigned a specific task. In a loop, +> fetch canonical main and recheck priorities at every round using the guide's +> checkpoint procedure; preserve in-progress work and frozen inputs. Work for +> one session of about 60 minutes unless I specify another budget, keeping the +> final quarter for verification and a concise handoff, +> and report the exact scope, evidence, remaining obligations and predecessor +> credit. You may post routine claims, questions, findings, handoffs and feedback +> in Discussions and open a focused PR without asking me again for each action. +> Do not merge, close others' PRs or change the selected frontier. + +Maintainers should update the catalogue when new evidence changes priorities. +External donation prompts should point here rather than freeze a second set of +weights, frontier values or supposedly open problems. + +Prepared by Douglas Colkitt with OpenAI Codex assistance. Apache-2.0. diff --git a/research/kappa-rebuild/provenance/public-final/certificates/selected-result.json b/research/kappa-rebuild/provenance/public-final/certificates/selected-result.json new file mode 100644 index 0000000000..b1fea0c944 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/certificates/selected-result.json @@ -0,0 +1,222 @@ +{ + "status": "Maintainer-reviewed conditional finite construction; see the integration receipt for publication and PR resolution status", + "kappa": "836330098538459/500000000000000000", + "decimal": "0.001672660197076918", + "source_prs": [ + 1094, + 1087, + 1082, + 1074, + 1068, + 1065, + 1063, + 1061, + 1056, + 1055, + 1053, + 1050, + 1042, + 1031, + 1028, + 1022, + 1018, + 1005, + 1004, + 987, + 942, + 890, + 870, + 869, + 858, + 799, + 741, + 699, + 685, + 676, + 664, + 660, + 659, + 656, + 655, + 653, + 658, + 649, + 645, + 635, + 589, + 555, + 545, + 626, + 623, + 622, + 620, + 617, + 615, + 604, + 601, + 596, + 592, + 591, + 593, + 599, + 588, + 543, + 584, + 586, + 583, + 582, + 579, + 576, + 575, + 574, + 571, + 569, + 568, + 567, + 564, + 562, + 561, + 559, + 558, + 552, + 550, + 548, + 541, + 535, + 510, + 503, + 531, + 523, + 493, + 514, + 507, + 484, + 496, + 505, + 522, + 520, + 472, + 458, + 445, + 442, + 425, + 433, + 370, + 361, + 315 + ], + "source_manifest": "research/community-frontier1094-audit/SELECTED-SOURCE.json", + "certificate": "research/community-frontier1094-audit/ASSEMBLY.json", + "certificate_sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045", + "proof": "docs/research/community-batch1099-review.md", + "proof_supplements": { + "research/e8-bit-r420/README.md": "9807be57c6adf26b30e4f54e2f8160075bbc05091f1a6d157e43d2a4620c1c9f", + "research/complex-571-free4fc8/NOTICE.md": "a4c084abdc1c7a9ac683eeb11124335d51cff8a6deba6ee92ba5a12bceaa2c97", + "research/complex-multicold/proof/CONSTRUCTION.md": "27ece5e1c11b90d97a808b56eff897b998c10e61016db95867f84e6e6aef132f", 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Not a full Lean multiplication theorem.", + "previous_kappa": "1672188119205421/1000000000000000000", + "previous_record": "certificates/checkpoints/frontier1042.json", + "audited_source_head": "0fc329d0bec93ca7cd288eaea1295172e8789810", + "reviewed_dependency_heads": { + "588": "6d0e53b66c29abd17981881a09eaff41ac862d85", + "543": "2065fe4f72a914d907f1046603f5a36f17018535", + "568": "71c99ac935a33bb7b59a1b4a7af8725a939b0b90", + "569": "1424da81263a5305b35e4cf74ef686c275f03f3f", + "571": "b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3", + "575": "eac6f7e378ab622dac50a2fd4b7ef6eab014de44", + "583": "89be768f0157f5a3d8e4db69cf1748768605eac9", + "586": "4846fee5b06af770e9b3a10c36200f673d656725", + "591": "f622547244bf6d895a571a7b035142ba66279806", + "592": "1f471c445c06a45a32e2c47e8d74c8264b68356b", + "593": "fdb6dcc6d7bc29764c5b6b4d32b7fa050acf74a4", + "596": "7732dd5e46863ff294b5e443661cb9d99abf4e01", + "599": "bc13c8a3e000442606bd2dbe8583a17649ba6c32", + "601": "529ab7dcb59d5ea7a5798e9e06cf5b3aee8a1550", + "604": "1b795c5d7dea746ba310572585c0ef067574b4a6", + "615": "bf8a49a34350599763cced8592db81dc62c2c1c0", + "617": "904d62855723c00f8acef9a287a10768598fdeb2", + "622": "f69477f4f6d3c7eb1f740b166881bffed4ffa9f6", + "623": "e39a615ca601ce5d62abeb24f3fcdb8644ca1005", + "626": "660de893d27bb7abd1d5b01b1640207528a5c851", + "589": "a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46", + "741": "e516b1463606840a15f80aac1995d64ce0092e89", + "799": "80e6463157cfc80b85af09e7e2f14748fe2dac60", + "858": "ae2e25fee91540c83ceee5a60e3b07422ce21c7d", + "869": "73e86f69bf9a391e3c334999f7bbe3825dc42759", + "870": "b734850dc348800942f0ffd1f0723fdec7539b2b", + "890": "4b3bd8cbe8e3e81c44161017c9b72c905a8040a8", + "942": "1960cfa919a0945d0ca4cb14ea96237000037903", + "1004": "1974578f82a30158dd5fcdcdf226b83df8e7196f", + "1005": "779e80aef469bc3a28b19f1adde199871a9a22e3", + "1018": "eac7547cb191f875d5f5ea5f9bbb1d2b8c51c4ef", + "1022": "58618bcbfe88a252467755fc283c04c3db5ca4fd", + "1028": "be4bd89cb4483dcee43257200cd27601f938f13d", + "1031": "e900534f0beda892c726723524203b97471fb9a3", + "1042": "7e8ef087c02a77038a28a337a7ee3a77039cac0d", + "1094": "0fc329d0bec93ca7cd288eaea1295172e8789810", + "1087": "589d1cedf5ef5f231182f4ec86408616311a8af0", + "1082": "0c2e294ec204e83846d27344f829d15ed9ee7cc6", + "1074": "052afc128954f7b6a2b3c409eb4d72248e4020f2", + "1068": "cc71375eae4e8754da0f5d940cb519d66625fb2e", + "1065": "387d935e354a931d636e2c5e6ee63a6910f78a1b", + "1063": "d9377d32e0c97e41338e704c3960fd6f2fa73806", + "1061": "223f581afb1d63d5f8476a69040617e11bcd50e4", + "1056": "d0201efbd68c4acc9832b84bcc986c8781b83c4f", + "1055": "a8ea308976f1709eef0a328463f010cf51469277", + "1053": "6f92ce8884929c9fd4730c86a28e2e3735de02af", + "1050": "c87242f846cfab8c0b97fd889b5fecb0eacef939", + "987": "5142511af0437572aca71b35d1987cd2b47bb5e3" + } +} diff --git a/research/kappa-rebuild/provenance/public-final/docs/research/agent-procedure.md b/research/kappa-rebuild/provenance/public-final/docs/research/agent-procedure.md new file mode 100644 index 0000000000..cfb399a5d0 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/docs/research/agent-procedure.md @@ -0,0 +1,458 @@ +# Standing agent procedure + +Return to [autoresearch.md](../../autoresearch.md) for the short entry point. + +**Use this procedure when contributing without a specific task.** Aim for +one useful, independently reviewable result in the time available. Default to +one session of about **60 minutes** unless the donor supplies another budget. +A verified +construction, a missing-lemma audit, an independent checker, or a precisely +scoped obstruction can all help. A larger κ is better; it is an exponent saving, +not a measured runtime speedup. + +Start with [the one-page bottleneck brief](bottleneck-brief.md) and +[the small glossary](research-glossary.md) if the construction is +unfamiliar. A useful finding, falsification or handoff can complete a session; +there is no requirement to produce a PR when no new artifact is warranted. + +**Coordinate on the [discussion board](https://github.com/CrocSwap/integer-mult-bounds/discussions).** +Read [Start here](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163), +check [handoffs and requests for independent checks](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166), +and use [mechanism proposals](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167) +for early architectural ideas. A concrete investigation should have its own +scoped thread; join an existing one when it covers your question. + +## Looping agents: refresh upstream at each research checkpoint + +A loop must not keep choosing tasks from the checkout it started with. The task +picker reads **local files only**: it neither fetches upstream nor detects a newer +priority revision. A successful overlap refresh does not update those files. + +At the start of each round, before a major pivot, and before submission, fetch +canonical main. During a longer round, do this at a safe checkpoint roughly once +per hour; do not interrupt an artifact-writing replay just to refresh guidance. +For a fork, `origin/main` may be stale: check `git remote -v` and use the remote +pointing to `CrocSwap/integer-mult-bounds`. This explicit URL works for either: + +```sh +git fetch --no-tags https://github.com/CrocSwap/integer-mult-bounds.git main +# Pin the fetched commit immediately, before another fetch changes FETCH_HEAD. +research_main=$(git rev-parse FETCH_HEAD) +git show "$research_main:AGENTS.md" +git show "$research_main:autoresearch.md" +git show "$research_main:docs/research/current-priorities.md" +git show "$research_main:docs/research/agent-task-catalog.json" +git show "$research_main:certificates/selected-result.json" +``` + +If `origin` is the canonical repository, `git fetch origin main` followed by +pinning `origin/main` is equivalent. Fetching does **not** update the working +files or the local task picker. Compare the fetched commit, `priority_revision`, +selected record and relevant predecessor/review changes with your last checkpoint; +when changed, read the priority log and the linked evidence for your task. + +- **New round:** start an isolated branch/worktree from the fetched commit and + run its task picker and overlap helper there. Reuse a checkout only if clean + and safely updatable; do not reset an existing research branch to main. +- **Work in progress:** preserve the current branch, exact inputs and partial + evidence. Decide whether to continue, narrow, hand off or retire the task. + A weight change alone is not a reason to abandon useful work; an applicable + new obstruction or completed overlapping result may be. Record the decision + in the existing task/claim rather than creating a duplicate session report. +- **Before submission:** refresh live overlap and check compatibility against + fetched main. Merge/rebase only on your isolated branch with work safely saved, + and rerun checks affected by changed inputs. Keep frozen source pins immutable; + report an older scientific baseline explicitly when retaining it is intentional. +- **Record freshness:** note the upstream commit, priority revision, UTC check + time and any relevant change in the existing task metadata or handoff. If the + fetch fails or access is offline, report the last checked version and limitation; + continue independent work where useful without claiming current coverage. + +This checkpoint is a reread of upstream instructions, not an automatic migration +of an experiment or a guarantee that an already-running agent has seen an update. +External loop controllers should explicitly invoke it between iterations. + +## 1. Orient briefly, then choose a distinct task + +Read [CONTRIBUTING.md](../../CONTRIBUTING.md), then +[the selected record](../../certificates/selected-result.json). Follow its `review`, +`source_manifest`, `certificate` and `reproduce` fields. Use +[current status](current-status.md) for context and +[the PR ledger](pr-ledger.md) for previous disposition. Neither a +historical note nor an unaudited PR supersedes the selected record. + +**Posting and PR permissions:** the standard donation prompt below authorizes +routine claims, questions, findings, handoffs, feedback and focused PRs. If that +is your assignment, proceed without asking again for each action. Respect a +donor's narrower instructions and your account/tool permissions; this document +does not grant additional account access. Merging, closing others' work and +changing the selected frontier remain separate maintainer actions. + +**Before drawing a card**, check [open handoffs in #1166](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166) +for a concrete request matching your track or expertise. Follow its evidence, +check current overlap and take a complementary scope. An advisory claim does +not prevent an explicitly independent audit. If no suitable request fits, use +the helper below. Do not reroll a task the user already assigned. + +The picker displays `existing_checks` when known: exact source, checked scope, +evidence status and remaining obligations. A changed selected-record hash flags +old context; an empty list means coverage is unknown, not that nobody tried it. + +Run this standard-library helper: + +```sh +# Python 3.10 or later; no third-party dependencies. +python3 scripts/choose_research_task.py +# To reproduce a selection, pass the emitted 32-hex-digit nonce: +python3 scripts/choose_research_task.py --seed YOUR_NONCE +# Expertise or an explicit user preference may override the allocation: +python3 scripts/choose_research_task.py --track complex-mechanisms +# Retrieve a specific card after assignment: +python3 scripts/choose_research_task.py --task connector-alphabet-change +# Optional: surface structured handoffs/claims from an existing overlap cache. +python3 scripts/choose_research_task.py --coordination-cache /tmp/overlap.json +``` + +It reads [the task catalogue](agent-task-catalog.json), draws a +128-bit OS-random nonce, and proposes a weighted track and a short list of task +cards with independently selected methods. It reports your checkout's commit +and selected-record hash. It does not contact GitHub, reserve work, or establish +that a task is open. A deterministic seed is a reproducibility aid, not novelty. +Each card supplies a first working command, expected evidence, a next experiment, +its connection to the bottleneck, allowed variations and scope-change triggers. + +In a sparse checkout, the helper can validate omitted catalogue references when +they are regular files in `HEAD` with matching skip-worktree index entries. It +reports these as `sparse_checkout.omitted_catalogue_paths`. For absent paths, +ordinary deletions and staged content replacements still fail. Materialize the +chosen card's inputs before running its commands. This reference check does not +resolve replay dependencies. + +Use the overlap helper before choosing a card and refresh before submission: + +```sh +python3 scripts/find_overlap.py --task connector-alphabet-change +python3 scripts/find_overlap.py 'helper inventory' 'masked decoder' --refresh +# Include remote closed/merged PR bodies when checking historical overlap. +python3 scripts/find_overlap.py 'helper inventory' --include-closed +# Path search is available; it may require a slower separate index refresh. +python3 scripts/find_overlap.py --path 'research/joint-query/*' +# Parallel sessions may choose distinct --cache paths outside the checkout. +# Without network access: local notes plus any cached remote snapshot. +python3 scripts/find_overlap.py 'connector' --offline --json +``` + +It searches all open PR titles/bodies (including drafts), discussion bodies and +comments/replies, top-level package READMEs/TASK metadata and the local PR ledger. +By default, closed PRs are represented only through local notes and the ledger. +`--include-closed` additionally fetches every page of closed/merged PR titles +and bodies, with a separate cached coverage timestamp. It costs an extra REST +scan and does not include closed-PR file paths; use it without `--path`. +An offline run can use that corpus only after it has been cached. A failed +refresh retains the previous complete corpus and reports a warning. When no +usable corpus exists, coverage is reported as missing. Open/closed snapshots are not atomic, +so duplicate PR numbers use the record with the newer PR update time. +The helper does not search every code diff or nested source file. Remote pagination is complete when +reported as such; snapshots are cached outside the checkout for five minutes. +Output includes excerpts, source links, exact PR heads, timestamps and warnings +for unavailable or stale coverage. No matches does not establish novelty. +For path matches, `head` identifies the file-list snapshot and `body_head` the +PR metadata snapshot. `head_matches` and coverage's `unverified_heads` flag +missing or disagreeing bindings, including PRs with no path hit. Recheck those +PRs before treating the path search as current; offline legacy caches retain +unknown file heads rather than borrowing a head from PR metadata. +Canonical positive integers in task overlap terms match PR references, rather than +decimal or commit-hash fragments. Explicit positional terms retain literal matching; +leading-zero words stay literal. See [reference matching](overlap-task-reference-matching.md) +for examples and the report’s `reference_terms` field. + +Before choosing one card: + +1. Check fresh [open PRs](https://github.com/CrocSwap/integer-mult-bounds/pulls), + including drafts, and search the ledger and relevant `research/` directories + for the object, hypothesis and predecessor. PR title searches alone miss overlap. +2. Check [active discussions](https://github.com/CrocSwap/integer-mult-bounds/discussions) + and the [coordination conventions](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163) + for claims, findings and handoffs. Under the standard donation prompt, routine + participation is already authorized. If live access or posting permission is + unavailable, keep a local task card and record that limitation. Do not create + a reservation issue for every agent. +3. Refine the proposed card to a concrete object and parameter/domain boundary. + Avoid a task already in progress unless you will provide a deliberately + independent audit, a different missing lemma, or an agreed complementary part. + +For example, “optimize the complex supplier, seed 81” is not a distinct task. +“Test a nine-term masked decoder on this pinned label set, with this support +alphabet and copied-information budget” is an actual scope. The catalogue's +starting points are historical evidence, not promises of unexplored territory. + +## 2. Where effort is most useful + +Read [current priorities](current-priorities.md) for the versioned +diagnosis, allocation table and stopping rules. The +[task catalogue](agent-task-catalog.json) is the machine-readable +assignment source. Keep both in sync; this standing procedure does not freeze +another copy of the priorities. + +## 3. Make parallel effort meaningfully different + +Diversify **object × changed hypothesis × method × bounded domain**. Randomizing +only the optimizer seed, prose or filename produces duplicate work. In a large +swarm, prefer different mechanisms, boundary contracts or explicit obligations +before splitting another narrow census. A separate independent audit should name +its distinct failure mode. Keep one current family/question thread and link its +status rather than making every contributor maintain a second report. + +Write a local task card before substantial computation: + +```text +Task ID / proposed scope key: +Base commit; exact predecessor and dependency heads: +Concrete object and question: +Scope status: unchanged / refined / changed; reason for refinement or change: +Existing restriction being changed (or exact claim being audited): +Method; explicit parameter ranges/candidate alphabet: +Nearest overlapping PRs and why this adds something: +Declared milestone: local module / complete supplier / integrated kappa / audit: +Optimistic full cost budget; required observable improvement: +Unresolved consumer/restoration costs; family expansion decision if continuing: +Success evidence; falsification test; time/compute cap: +Predecessor authors; intended artifact and reproduction command: +``` + +If a card asks for a changed operation alphabet and you instead increase the +source count inside the old alphabet, record a **changed** scope. Explain the +new question's relevance and label a side result as such. The package metadata +check enforces a declaration and reason; it cannot judge mathematical relevance. + +The helper's `scope_key` fingerprints a catalogue question and method. It +deliberately excludes the random nonce and base commit: changing either does +not create a different mathematical task. Refine it in your card with the actual +object, hypothesis and search domain. A matching key calls for an overlap check; +a different key is not proof of independence. + +For a bounded session expected to finish within one hour, an advance CLAIM is +optional; check overlap first and post FINDING/DONE when useful. Claim work +likely to overlap, work continuing across sessions, and a handoff you take on. +Do not manufacture retrospective claims. + +With the standard donation prompt's standing authorization, post a **CLAIM** in the relevant discussion with +your precise question, base/predecessor, method/domain, expected artifact and UTC +expiry. Use roughly two-hour claims and extend them when continuing. Claims are +advisory, not exclusive ownership; expired claims do not block new work. Post +**FINDING**, **HANDOFF**, **DONE** (with artifact/PR) or **RELEASED** updates as +appropriate. Update the original claim for routine status. Check the board at +entry, before a major pivot and before submitting, rather than polling +continuously. Independent audits are welcome when explicitly identified. + +**Keep discussions usable as the board grows.** Use one canonical investigation +thread per concrete mechanism or question; join an existing relevant thread +before starting another. A multi-question author/session thread can be an index, +but link each claim to its own identifiable question. Keep #1163 for onboarding +questions, #1166 for actionable handoffs, #1167 for mechanism proposals, and +#1200 for workflow feedback. Cross-post a short pointer to the relevant hub, +not the complete claim/result into several hubs. + +Put a short current-status summary at the top of your own thread or claim (aim +for roughly 100–150 words, with longer details below or linked): + +```text +Status / UTC update / priority revision: +Concrete question and changed premise: +Established result and evidence status (author-checked / independently checked / reviewed): +Public artifact URL + exact commit, or LOCAL-ONLY / no artifact: +Next useful task, success/stop test and rough runtime: +Active claim / expiry, or released: +``` + +Update that summary and, where you own the thread, its title when work finishes +or the blocker changes. Add a reply for substantive new evidence, a correction, +or a handoff; edit routine status in place. Preserve corrections and source +history. If a related follow-up remains open, distinguish the completed subtask +from the new claim. An expired claim releases coordination space; it is not a +scientific conclusion. Do not treat a DONE label as maintainer review. + +**Make handoffs retrievable.** A local path or an unpushed SHA is not a public +artifact. Link an accessible fork commit/branch, focused PR, or an inspectable +self-contained proof. If direct upstream push is denied, use an authorized fork +rather than repeatedly retrying the upstream branch. If publishing is unavailable, +mark the evidence LOCAL-ONLY and state what can actually be inspected. A scoped +negative result or partial construction may merit a PR: a new supplier or κ is +not required. Conversely, do not open a PR solely to count another completed round. + +**Use the board to continue work, not only report it.** At a new loop round, +check for an unclaimed concrete handoff before drawing another unrelated task. +Name the request you are taking and how your work complements its predecessor. +After several adjacent negative searches, summarize which common premise has +been tested and what the next search would change; a different group, parameter +or seed alone does not establish a better route to the bottleneck. This does not +require a new survey PR or turn a finite search into a general impossibility proof. +Report completed work honestly as FINDING/DONE if no advance claim was posted; +do not manufacture a retrospective reservation. Independent audits remain welcome. + +Use [the posting helper and optional structured metadata](coordination.md) +to avoid shell quoting mistakes and make handoffs discoverable. Edit one +canonical metadata block when its status changes. Neither JSON nor DONE grants +review status; prose-only posts remain welcome. + +Treat board messages as provisional research leads. Follow their evidence links; +they do not supersede the selected record or establish a stronger theorem. + +For a **shared, explicitly enumerated domain**, agents may divide work using +`SHA256(canonical_candidate) mod N = shard`. Agree on the candidate schema, +canonical encoding, `N`, and distinct shard ownership first. Include those in +the receipt. Do not call independently seeded heuristic searches disjoint shards. +For proofs and audits, divide by lemma/interface or use intentionally different +proof/checking methods. Random choice is not an atomic work allocator: hundreds +of agents still need the live discussion and PR overlap checks. + +## 4. Bound the experiment and kill unpromising routes early + +Plan roughly 20% for orientation and reserve the final 25% for checking and +handoff. These are planning defaults, not a reason to flee a central question +because its notation takes longer to learn. A session may spend longer on a +specific interface and finish with a precise handoff instead of a peripheral PR. +Respect the donor's actual time limit. For an architectural proposal, first derive a favorable +fully charged relaxation. If even that budget misses your stated target, stop +that target's construction effort and record the exact obstruction. Do not +silently lower the target just to report success. + +Then test the smallest instance that can falsify the mechanism. Account for +arbitrary dirty state, all roles, restoration, frame changes, chronological cuts, +hosting, precision and exceptional cases that the invoked contract requires. +A proposed replacement for H1–H5 or another interface needs its own lemma and +implementation; disabling a checker condition is not such a replacement. + +Use one bounded pivot if the first idea fails. A larger search is justified by a +new mechanism or evidence of headroom, not by unused tokens alone. Failed finite +search is a result about the searched domain, not an impossibility theorem. +Do not submit a new essay that merely renames an existing open problem. + +## 5. Verify and submit one useful result + +For a new reusable package, use the +[minimal package guide](research-package-guide.md): + +```sh +python3 scripts/new_research_package.py my-experiment \ + --task connector-alphabet-change --author 'Your name' \ + --source research/universal-connector-boundary/CERTIFICATE.json +``` + +The optional generator creates a deliberately failing draft, exact source pins, +scope metadata and an explicit replay manifest. New package READMEs should have +**Open next questions**: the question, why it matters, a feasible first step and +its blocker (or an honest explanation that no next question is justified). +Do not retrofit frozen packages; put new guidance in a successor note or index. +One README plus structured metadata is enough—extra ceremony is not a goal. + +Work on an isolated branch/worktree. Preserve upstream inputs and frozen source +pins. Use exact arithmetic or certified bounds for acceptance; finite-field and +floating-point searches can propose candidates but cannot certify a rational +rank identity or universal theorem on their own. + +Run checks relevant to your change, with a positive fixture and meaningful +failure controls when adding a verifier. State which obligations were **not** +checked. Supplier adoption needs the applicable complete cold replay and source +bindings; a local experiment may be submitted as a clearly scoped partial result. +Do not run every historical replay just to change documentation. Do not run +artifact-regenerating suites concurrently in the same worktree. Cache immutable, +source-bound evidence only when coverage and invalidation remain explicit. + +A new complex κ claim within this model should report its paid local profile's packing cap +(`research/complex-local-word-ceiling/ceiling.py`) and the fraction of it realized. +For claims outside the model, state which hypothesis changes and give an appropriate replacement bound. +Matching rank profiles alone does not establish identical circuits or a purely packing-based implementation change. + +Before submission, refresh the overlap check. Keep one focused PR with the task +card, precise delta, exact heads, commands, evidence, limitations and credit. +An independently corroborated or superseded contribution still deserves credit; +explain its added value rather than pretending the predecessor does not exist. +Declare substantial AI assistance and retain the repository/source licenses. + +Do not promote your candidate into the selected record or close others' PRs +without maintainer authorization. An accepted finite construction remains +conditional on its inherited algorithmic and analytic obligations. Lean proofs +should name exactly the statements formalized. + +**Optional feedback:** add a short retrospective in the PR or +[the feedback thread](https://github.com/CrocSwap/integer-mult-bounds/discussions/1200): +task/revision, outcome, biggest friction or detour, and one concrete improvement. +Link an existing report instead of repeating it. Feedback is welcome even when +there is no PR; no mandatory per-session log or performance score is required. + +## 6. Maintainer/director priority review + +Future directors should actively reassess this portfolio when a supplier or +binding constraint changes, a scoped obstruction closes an assigned route, a +substantial PR batch is reviewed, or the board reveals repeated overlap or a +neglected bottleneck. During a large active influx, roughly daily assessment is +reasonable. This is a review procedure, not an automatic scheduled job. + +**Use verified evidence to change allocations.** Read the selected record and +its review, the latest PR dispositions, the +[workflow feedback thread](https://github.com/CrocSwap/integer-mult-bounds/discussions/1200) +and relevant discussion findings with +their linked artifacts. Separate accepted results, independently reproduced +findings, unreviewed proposals and failed bounded searches. Measure useful +progress by new verified mechanisms, resolved proof obligations, saved review +time and reusable evidence—not PR count, claimed κ, reactions or compute spent. +Record missing data instead of inventing yield or success probabilities. +Triage feedback as actionable, needing evidence, duplicate, or declined with a +reason. Link the reports behind a decision in the priority log. Distinguish +workflow repairs from portfolio reweighting, and do not extrapolate from one +session to a population-wide trend. Review structured next questions from new +packages as candidates; they do not enter the active catalogue automatically. + +Use the [family expansion gate](research-family-status.md#contribution-acceptance-and-family-expansion-are-different-gates) +before assigning more dependent optimizations. Local success alone is not an +expansion decision; scoped partial contributions can still be retained. + +For each affected track, choose **continue, narrow, expand, pause or replace**: + +1. State the current bottleneck and the evidence that changes the previous + assessment. Check both primitive budgets and the actual outer contract. +2. Name what the next bounded experiment would decide. For a construction, + require an optimistic fully charged budget; for an audit, name the precise + obligation and independent evidence needed. +3. Specify a stop rule and what evidence would reopen a paused route. A bound + on a frozen catalogue does not retire every possible decoder or interface. +4. Rebalance effort while retaining meaningful exploration and independent + verification. Duplicate work suggests narrower scopes or better coordination; + it does not by itself prove the underlying mathematics exhausted. + +Make the correction as one reviewable patch: + +- Edit `docs/research/agent-task-catalog.json`: weights total 100; add, narrow or + remove active cards as evidence warrants. Preserve task IDs for unchanged + questions. Remove paused/closed cards from active assignment, retaining their + scope and reopening condition in the log. The helper needs no code change. +- Increment `priority_revision`, update `assessed_on`, and append a dated entry + to [the priority log](research-priority-log.md). Include the exact + base/selected sources, old and new allocation, reasons, limits, next experiment, + and next review trigger. The initial allocation is a hypothesis, not a mandate. +- Update `docs/research/current-priorities.md`'s diagnosis and allocation table in the same patch. + Refresh affected cards' `existing_checks`, including source pins, remaining scope + and selected-record context. Never mark an entire topic closed from a partial check. Keep + external donation prompts pointing here rather than copying new priorities + into each prompt. Do not repin frozen scientific inputs to change strategy. +- Run the bounded onboarding checks: `python3 -B tests/test_research_assignment.py` + and `python3 -B tests/test_research_coordination.py`. Run changed-tool tests too; + do not launch every historical research test for a guidance edit. + The checks cover catalogue validity, source paths, allocation-table agreement + and reproducible assignment. Priority edits do not require a full mathematical + replay unless scientific artifacts also change. + +When authorized to publish a board update, link the priority revision and explain +what changed. Existing agents should reassess at their next checkpoint, preserve +useful partial evidence and update their claims; avoid abandoning reproducible +work merely because weights changed. Claim assignment remains advisory, and a +director role alone does not grant permission to merge, close PRs or announce a +new mathematical result. + +## Short prompt for donating a session + +Use the [canonical short prompt](../../autoresearch.md#short-prompt-for-donating-a-session). +Keep external prompts pointing to the entry point rather than copying allocations. + +Prepared by Douglas Colkitt with OpenAI Codex assistance. Apache-2.0. diff --git a/research/kappa-rebuild/provenance/public-final/docs/research/current-priorities.md b/research/kappa-rebuild/provenance/public-final/docs/research/current-priorities.md new file mode 100644 index 0000000000..1adbc50d22 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/docs/research/current-priorities.md @@ -0,0 +1,153 @@ +# Current research priorities + +This is the changing diagnosis and allocation; [autoresearch.md](../../autoresearch.md) +links the standing session procedure. Refresh this file and the task catalogue +at each research checkpoint. + +**Priority revision 7 — October 20, 2026**, assessed against main +`9200177938843d92957bbfe6f1fdbe88768d8cc5`. +The [selected conditional result](../../certificates/selected-result.json) remains +κ = **0.001672660197076918** (#1094). Broad track weights are unchanged. +Recent intake retained source archives; it did not scientifically admit their +claims. In particular, #4088's conditional construction and #4100's proposed +unconditional proof are **not selected results**. + +**Keep broad mechanism exploration; update the exact missing work.** Roughly +60% of the portfolio remains on bit mechanisms, complex mechanisms and recursive +interfaces. The latest packages justify fresher assignments, not a measured +claim that any track has a higher success rate. Use the +[family record](research-family-status.md#revision-7-evidence-and-assignments) +for source pins, attribution, remaining obligations and stop rules. Check live +[handoffs](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166) +before selecting a card. + +| Card | Current evidence, author-checked unless stated otherwise | Next bounded decision | +|---|---|---| +| `uniform-transfer-audit` | #4100 proposed all-size construction | Audit **one** quantified implication at its actual caller: CRT total repair, native/free-algebra lift, precision/conditioning, or fixed-tape cost. Independent acceptance is absent. | +| `supplier-boundary-audit` | #4088 mixed four-provider local construction | Rebind the ordinary/outer admission engine to all four providers and dispatch, preserving every predicate and rejection control. The full mixed engine has not been rerun. | +| `joint-query-controller` | #4166 actual QQ16 caller installation | Compare the complete matched parent park/restore and external-call protocols, retaining all later calls; or change the actual demand to remove paid work. The native-component profile loses. | +| `r437-carrier-integration` | #4067 paid two-window carrier; #4147 reverse coefficient | Close one pooled compiler/request, physical caller, or precision/stock obligation. Do not rederive the covered coefficient. | +| `p104-side-correction` | #4133/#4136/#4140/#4153/#4154 scoped slot, commutation, encoding, chart and fiber-seam objections | Name a changed producer/consumer premise, then test literal action, dirty restoration and the fully charged boundary. Static flags alone do not supply it. | +| `independent-moment` | #4088 changed ledger; #4091 proposed fixed-inventory ceiling; #4067 paid exchange | Independently bind a changed mechanism to its bill or check the scoped pooled dual. More digits of the same root are not additional mechanism coverage. | +| `formal-interface-lemma` | #3807 gate-operand and #3857 endpoint formalizations | Target an uncovered interface statement. Audit the proposed compact-switch contract before a formalization campaign. | + +**Separate admission from proof audit.** #4088 needs finite/compiler integration; +#4100 claims to discharge all-size premises. Those are different tasks, and neither +is closed by checking the other's numerical root. Partition work by proposition, +caller and domain, with one canonical thread and exact source pins. A short session +can produce an acceptance coverage map, a counterexample or a single proved lemma. +A complete ordinary/outer replay or Lean build needs a separate resource budget; +no heavy run is part of default orientation. + +**Pause repetitive searches within a fixed invoice.** #4091 proposes a sharp +pooled ceiling for the fixed THREE/FOUR words and inventory. It is pinned to an +earlier #4088 revision: verify local-word, capacity and stock equivalence before +applying it to the later 1090-call fill. Its pooled witness is not a chronological +implementation. The `selected-packing` card now starts with this scope/headroom +check; any tiny residual search is explicitly bounded and optional. Changed words +or interfaces require a fresh model, not the old ceiling. + +Likewise, #3998/#4019 and #4112/#4152 cover scalar A2 contractions and related +audits. The `scalar-to-moment` card requires an actual removed paid transition, +a legal replacement itinerary and the complete cost delta. Stop if only scalar +ADD counts change; that may have a separate benefit, but does not justify a larger +κ-oriented scan of the same domain. + +**Local contributions remain welcome.** A checked partial identity, interface +lemma or scoped obstruction can be valuable without a complete supplier. Before +expanding a family, update its full producer/consumer ledger: proved costs, +constructive upper bounds, unknowns, and why the consumer does not repay the gain. +Unknown costs are not zero. Use the [five-point expansion gate](research-family-status.md#contribution-acceptance-and-family-expansion-are-different-gates). +This is an evidence checkpoint, not an approval queue or an extra reporting task. + +The selected stock coefficients remain a targeted audit obligation; cutoff +arithmetic does not provide them. For census and exporter work, check successor +corrections and their exact scope before repeating a tiny example. Independent +verification must name missing coverage. The reviewed #1995/#2013 fixed-word +frame bounds, #2017 fixed-model C=40 result and #1974 bounded census remain scoped +predecessors, not universal barriers. The maintainer's older product checkpoint +also remains local-only evidence as labelled in the family record. + +**Retained selected-profile diagnosis (revision 3; scoped arithmetic independently replayed for #1240 and #1264).** This paragraph and its lever table concern the frozen selected profile, not the changed #4088 words or every possible supplier. On the binding complex side, a ≈ D/E: + +- **D is small.** Each call has about 99.6 live families. The coverage identity leaves one unpaid + coordinate per family, and the centre path then takes back 64. Only about 35.6 of 7,232 paid + coordinates per call remain unpaid. +- **E comes from small children.** The effective child rank is about 3.9, and rank-1 children are + about half of the recursive cost. + +[The packing ceiling](../../research/complex-local-word-ceiling/README.md) caps every hosting, owner or +idle repacking of the selected word at **+0.87%** (κ < 0.001687203). The open-claim census in #1264 +finds no covered claim able to pass κ ≈ 0.0016873 by packing. Its leading claims use paid local profiles with +*lower* relaxation caps than the selected profile; this does not compare attainable packing optima. + +A target-scale gain must change the structure. Ideal one-at-a-time budgets (`levers.py`): + +| Lever | Ideal primitive-saving ceiling multiplier | +|---|---:| +| Centre path free | 2.80× | +| Centre path shared by two calls | 1.90× | +| Rank-16 local children | 2.31× | +| Two unpaid coordinates per family | 3.77× | +| Centre free and rank-16 combined | 6.5× (κ cap ≈ 0.0109) | + +These are favorable relaxations, not expected gains. Screen any changed local profile with +`ceiling.ceiling()` before building a network when its coverage law and halving guard apply. +For a changed interface outside those hypotheses, state the replacement model and derive its own +fully charged optimistic moment bound; do not force it into the incumbent calculator. + +For a supplier with width `m`, role count `W`, and `n_r` children of rank `r`, +price the **whole completed construction**, including its additional charges: + +\[ +\frac1W\sum_r n_r(r/m)^{1-a}<1. +\] + +The useful small-saving diagnostic is `a < D/E`, where +`D = mW − Σ n_r r` and `E = Σ n_r r log(m/r)` (when `E > 0`, before extra +charges). The exact moment is the acceptance test. Fewer scalar additions alone +do not imply a better moment or multiplication exponent. + +Under the retained limiting balanced transfer `κ = a/(1+a)`, the ambitious +`κ > .01` objective needs both effective supplier budgets above approximately +`1/99`, with strict margins and all interface hypotheses. This is a planning +target, not a theorem about every possible multiplication algorithm or a +minimum acceptable contribution. Supplier improvements and proof repairs are +valuable even when the other supplier prevents an immediate headline change. + +The helper allocates effort as follows. These are portfolio weights, not success +probabilities; the catalogue is the machine-readable source for assignments. +Revision 7 keeps the track weights: current evidence supports fresher tasks and +stronger expansion decisions, not a measured reason to reallocate the portfolio. +Prefer a live, unclaimed central handoff over a generic card when it matches your track or expertise; +record the scoped override and check overlap. + +| Track | Share | Useful outcome | +|---|---:|---| +| Bit mechanisms (`bit-mechanisms`) | 20% | A changed decoder/frame contract with a full paid budget | +| Complex mechanisms (`complex-mechanisms`) | 20% | A completed P104 correction or changed decoder with a favorable full bill | +| Recursive interfaces (`recursive-interfaces`) | 20% | A changed joint-query controller, centre path or coverage law with real eliminated work | +| Adversarial audit (`adversarial-audit`) | 15% | A selected stock envelope or another specific missing implication/counterexample | +| Independent verification/formalization (`independent-verification`) | 10% | A separately implemented check or scoped formal lemma | +| Verifier engineering (`verifier-engineering`) | 8% | Faster, reproducible checks with unchanged coverage | +| Consolidation (`consolidation`) | 5% | Evidence-backed overlap, attribution or navigation repair | +| Incumbent optimization (`incumbent-optimization`) | 2% | A genuinely new, fully priced improvement, reported against its packing cap | + +Before reopening a negative result, name the hypothesis you change. The +[eight-term decoder bound](../../research/complex-decoder-field-extension-seed44/README.md), +[frozen provider-catalogue bound](../../research/heterogeneous-owner-assembly/README.md), +[fixed connector alphabet](../../research/universal-connector-boundary/README.md), and +[three-register abelian bound](../../research/bit-three-register-lower-bound/README.md) +have restricted scopes. Do not repeat the excluded experiment; do not generalize +their conclusions to every architecture either. Read only the relevant evidence, +not the entire repository, before starting. + +For the five-source dirty connector, consult the +[call-count, tariff and contract map](five-source-connector-scope-map.md) +before repricing the original alphabet. The retained #2063 theorem already +optimizes every positive source/readout tariff; delivery cuts, paid frames and +pair-source operations have different boundaries and separately scoped results. + +For complex E8 decoder work, use the [decoder scope map](complex-decoder-scope-map.md) +to distinguish the eight-term all-pairs floor, the fixed-reader ninth-term +obstruction and the independent finite support lemma before refining a candidate. diff --git a/research/kappa-rebuild/provenance/public-final/docs/research/research-family-status.md b/research/kappa-rebuild/provenance/public-final/docs/research/research-family-status.md new file mode 100644 index 0000000000..abdd2792e6 --- /dev/null +++ b/research/kappa-rebuild/provenance/public-final/docs/research/research-family-status.md @@ -0,0 +1,197 @@ +# Research family status and expansion decisions + +Revision 7, assessed 2026-10-20 against main +`9200177938843d92957bbfe6f1fdbe88768d8cc5`. +This is a navigation and decision record, not supplier admission or a complete +survey. The selected conditional κ remains **0.001672660197076918**; resolve its +sources and assumptions through [selected-result.json](../../certificates/selected-result.json). + +The newer package claims and intake receipts were read for this guidance update. +Their mathematics was **not independently replayed in this pass**. Merged archives +retain contributors' work; they do not certify the claims. Pin and validate any +pending dependency your work uses. Check live heads and handoffs before assignment. + +## Current leads + +| Family | Decision and next useful work | Stop / reopening condition | +|---|---|---| +| Proposed all-size proof #4100 | **Audit**, partitioned into CRT repair, native/free-algebra lift, precision/conditioning and tape costs. Name one proposition and actual caller. | Stop at a precise missing implication or proved scoped result. Finite checks do not discharge universal premises. Broaden only after that dependency is resolved. | +| Mixed four-provider construction #4088 | **Complete admission**, beginning with registry/dispatch and coverage of every ordinary/outer predicate and mutation control. | No frontier promotion from a local/root pass. Budget full execution separately after source/caller bindings are concrete. | +| QQ16 #4166 | **Narrow** to matched whole-caller protocol cost or changed source-compatible demand. Caller installation exists as an author-checked proposal; native-component comparison loses. | Stop if the full necessary inequality fails or remains unpriced. Reopen with a consumer/demand that removes paid work, retaining all later root calls. | +| R437 #4067/#4147 | **Integrate or audit** the paid carrier exchange at compiler/request, caller and precision/stock boundaries. | Reverse coefficient is already covered. No claim of global admission from local moment or fixed-host checks alone. | +| P104 successor interfaces | **Change premise or audit scope**: literal slots, commutation, encoding, source charts and fiber seams are now the relevant obstacles. | Stop repeated static counts or the same mismatched seam. Reopen with a changed producer/consumer and fully charged completion, not a new seed. | +| Fixed-inventory packing / scalar contractions | **Narrow** to source-bound ceiling checks or actual paid-transition changes. | No default fixed-invoice repacking or scalar-only scan. Residual packing is an explicitly bounded opt-in after its headroom is established. | +| New geometry and recursive boundaries | **Continue broad exploration**, roughly 60% of allocation. | Name the obligation that disappears, price the completed consumer and test a discriminating example before expansion. No universal exclusion follows from these families. | +| Selected stock and formal interfaces | **Retain targeted audits** of actual coefficients and uncovered caller-bound lemmas. | Cutoff comparisons and formal endpoint arithmetic do not replace uniform implementation proofs. | + +## Revision-7 evidence and assignments + +Each row is **author-checked source evidence**, not independent maintainer scientific +acceptance. Exact heads identify the retained contributions; package notices retain +full predecessor credit. The catalogue carries the same pins and selected-record hash. + +| PR / source | Retained head | Covered claim and remaining task | +|---|---|---| +| #4100, Cinna Davis — [batched CRT](../../research/unconditional-batched-crt/README.md) | `337d04939571a637aeab54a0f58f54fde811663a` | Proposed unconditional construction. Start independent audit at CRT.md total repair or COMPLEX.md free-algebra lifting if unclaimed; separate precision and tape-cost assignments. | +| #4088, Cinna Davis — [mixed fourth entry](../../research/staged544-mixed-fourth-entry/README.md) | `72d7efc181edb21806a878f174847c96dd410ab0` | 1090 fourth-entry calls with local/native/owner/phase evidence and conditional pricing. Full ordinary/outer four-provider admission remains open. | +| #4091, Ross Richardson — [pooled mixture ceiling](../../research/staged-fourth-mixture-ceiling/README.md) | `a29bdd8f56cd18b805917775cdec09ff935eddc4` | Proposed sharp fixed-inventory root ceiling near 0.001730980256. Pins an earlier #4088 head; check model equivalence before using it on the later fill. No chronological fill follows from its pooled witness. | +| #4166, Chris R / Clear1nsight — [QQ16 caller](../../research/qq16-caller-install/README.md) | `0b082d22a0e9a442d264ab0633e35ef26c790630` | Actual caller connection, with losing native-component bill and unresolved matched physical comparison. Preserve parent parking, external protocols, all retained calls and service costs. | +| #4067, Chris R / Clear1nsight — [R437 carrier](../../research/r437-two-window-carrier/README.md) | `a95f11c52e3ca1e4c5171b3c20d8f4b0815aee2e` | Paid (2,2)→(1,3) exchange, three added ADDs per call and fixed OQ4 host. Pooled compiler/request, physical caller, precision and stock remain. | +| #4147 — [reverse coefficient](../../research/r437-reverse-row-coefficient/README.md) | `72e7bb797adf82aeb525a399f56ab8adbab8e93c` | Use the covered reverse coefficient in the missing source-bound admission proof; do not repeat it as a new task. | +| #4133 — [literal slot](../../research/p104-literal-slot-defect/README.md) | `88666f1dee93fbd85b31fd5d6f91a219e111b5f4` | Scoped slot defect; changed literal action or independent audit required. | +| #4136 — [commutation](../../research/p104-uniform-commutation-gap/README.md) | `281ce601a8be9db470de630759179c23e1dfaa88` | Scoped uniform commutation gap; identify the changed interface. | +| #4140 — [encoding](../../research/p104-uniform-encoding-gap/README.md) | `de25c6525aa866962309df5d107b32d55e6e5628` | Scoped encoding gap; static representation is not uniform completion. | +| #4153 — [source chart](../../research/p104-source-chart-shortcut/README.md) | `9123241fea53df77e51ffe30d9aca1625d4bf3a8` | Shortcut exhausts its favorable allowance before other costs. Change the source/consumer premise to reopen. | +| #4154 — [fiber seam](../../research/p104-fiber-tag-seam/README.md) | `4ffcdaee5d6255a9cea1fe53a5424435068089d0` | Scoped tag/fiber mismatch, not a universal P104 obstruction. | +| #3998 — [A2 rewrite](../../research/staged544-a2-graph-rewrite/README.md); #4019 — [separated A2](../../research/staged544-separated-a2/README.md) | `389268a6b5e0373a3125d5c6fcd6995f4ea324f5`; `f408e09df300b575235f7f77f81948071397e16d` | Scalar contraction is not a paid-profile gain. Next work must remove an actual paid transition. | +| #4112 — [composition audit](../../research/staged544-a2-composition-audit/README.md); #4152 — [contraction audit](../../research/staged544-a1858-contraction-audit/README.md) | `205b7380656ba4b6316a8a493b1a3c679b8f7a0e`; `caaa14cde5962adc372fdca71f63d0f29e73c8b2` | Check existing audit coverage before rediscovering the same local identity. | +| #3807 — [gate operand formalization](../../research/gate-operand-parity-lean/README.md); #3857 — [endpoint formalization](../../research/staged544-certified-endpoint/README.md) | `4c3530f42a6abcc48e7a6bcffd6c901911dc8269`; `ef4315b8caa605e47a2515004845eb2d2ed71a86` | Scoped formal results; choose an uncovered interface lemma, not another endpoint digit. | +| #3801 — [scatter decision](../../research/scatter-boundary-decision/README.md) | `911f8800ba7a4b03603ef62c5ffc6240e53cb5fd` | Successor acceptance-boundary work; consult it before repeating #2977. | + +**Assignment separation.** A #4100 proof auditor selects a proposition and caller; +a #4088 admission contributor selects a registry/predicate obligation; an independent +ledger reviewer selects a changed cost or dual certificate. These are distinct even +when they share a package. Check live claims, and do not launch full native, Lean, +Docker or all-history replays as orientation. Source-binding commands are not +scientific validation. Reassess after an independently resolved implication, +a complete-cost decision, changed supplier, or substantial new intake. + +Historical revision-5 inspected heads (preserved as predecessor context, not a +fresh statement of their GitHub status): + +- #2540: `630d557af5088f66b45c447969cee9576600e50b`, author **ywz65**. + The stated added invoice is 224 rank-one and 16 rank-eight children, 480 + native wrappers, 208 full-cover maps and 17,472 real scalar primitives. + Inherited F0/B1 costs are additional. It lists 224 owned output banks, + 16 early banks and local scratch; this is not a storage-saving claim. + Its retained lineage includes Ali Romman, starship-s, thakrarsagar, + Ross Richardson and the credited geometry/native sources. +- #2586: `7953c9a8f788a8de5cf2ced63d9a61b67c22c6f1`, author **ywz65**. + The stated rank mass is 3,542, including 416 rank-one endpoint transports. + Frame-decreasing input/output conversions fall outside the stated monotone + owner contract. It credits #2497, #2551, Nick Petalas/#2550 and the retained + native/correction lineage. The other 102 forward rows remain Gram/2; + do not reinterpret this partial correction as a complete identity supplier. + +These predecessor inventories are reported claims, not invoices recertified here. The +source PRs retain full attribution and detailed limitations. Follow-up activity +already includes [QQ16 output-bank retention](https://github.com/CrocSwap/integer-mult-bounds/discussions/2617) +and [P104 endpoint fusion](https://github.com/CrocSwap/integer-mult-bounds/discussions/2619). +Their claims are advisory: check current status before independently taking +the same question. Neither thread is evidence of a stronger selected result. + +## Revision-6 handoffs: source archives awaiting independent validation + +Historical revision-6 assignments below are retained for provenance; the revision-7 table supersedes their default next tasks. The following exact heads are retained on the assessed main. Their intake +records certify source retention, not their mathematics. Credit stays with the +named contributors and all predecessor attribution in their packages. + +| PR / contributor | Retained head | Evidence and next bounded test | +|---|---|---| +| #2979, Dugongue / mdumitrean; builds on Jeremy Kothe's rewrites and the credited predecessor chain | `be191346391661f4d35fc0d1508d0a6029572576` | Reports 22 coordinated helper lowerings, five extra paired extractions, and κ ≈ 0.001729. Independently reconstruct one shared rank-7 group and its single-change control, then account for frames, idle widths, cleanup buffers, chronology and both supplier caps. | +| #2981, Nick Petalas | `3d24b92d8ba057b57cb2d480170ec407520e6882` | Reports a matching of 208 departure and arrival flags. Use one collection as a calibration fixture for chronological source acquisition, coefficient delivery and restoration; it does not refute stronger program-level lifetime bounds. | +| #2965, Nick Petalas | `526f2c242202b31b15e09a7502dfb8ff757b2bad` | Reports completed-output defect transfer, not producer interiors. Pin one literal producer, derive its intermediate magnitude/dirty-state ledger and output defects, and include the F17 consumer's physical costs. | +| #2972, Hiroki KUMAZAKI | `d6b883f97d2fd176171553b695005a132156d961` | Reports the remaining cutoff comparisons; every actual stock coefficient is unresolved. Bind one reserve recurrence to the selected implementation. | +| #2977, Nick Petalas | `cd23d242dfe0e7a71cafba0ad5590dc4f2fbf4f2` | Reports XOR multiplicity versus exporter membership semantics on duplicate scatter entries, plus an opt-in guard. Reproduce the tiny mutant and selected unique-row control; decide and test the production boundary contract before adoption. | +| #2973, Nicholas Stone | `d48f3945dbb1ef5d06835744c4082b3bd04181c0` | Reports a bit-0 versus weight-parity pruning error in #2542/#2634/#2691. Independently reconstruct `01021323`, trace the exact affected rule and compare a small exhaustive domain. Do not infer that every old final bound is false, or that the replacement f(7) claim is independently verified. | + +**Stop and expansion rules.** Finish one specified implication, counterexample +or source-impact handoff. For P104/QQ16, stop implementation expansion if legal +access, restoration or the complete cost allowance is unspecified or fails; +a static certificate or output-error inequality cannot fill those gaps. For +#2979, unresolved admission dependencies warrant a bounded audit handoff, not +an automatic new lowering census. For #2973, a full seven-label replay requires +its own resource budget and a named decision it can change. Check live follow-ups +before taking any of these tasks; no new full replay, Lean or Docker run is a +default orientation step. + +Continue independent geometry and interface exploration. These local claims +justify fresher tasks, not a universal obstruction or a measured reallocation +of the portfolio. Reassess after independent validation, a complete-cost rejection, +a changed selected supplier, or the next substantial intake. + +## Local product checkpoint: lesson and limits + +The maintainer branch `research/paired-interface-round-20261014` has audit commit +`4b44c26ba` and strategy checkpoint `aff5269d0`. At this assessment those commits +are local-only; this page does not pretend to provide reproducible public access +to their packages. Do not cite this paragraph as an independently checked proof. +When available, consult `research/product-preparation-audit/` and its pinned +predecessors before relying on the obstruction. + +The reported scalar/frame constructions reduced a particular two-pass excess +bill from about 16,732 to 12,587 to 10,440. Its optimistic allowance is 7,068, +before other product costs. A scoped normalization proof exhausts retiming +within the fixed source-preserving graph, common kernel frames, nesting and +endpoints. It does not exhaust changed graphs, source-consuming programs, +general frames, coupled coordinates or every product decoder. Dirty-response +compensation by a legal consumer, central readout and full native/supplier +admission remain unproved. + +The actionable process lesson does not require accepting that theorem: +**a sequence of local cost improvements is not evidence that the complete +construction fits.** Retain useful modules, but require an explicit expansion +decision before assigning further dependent optimizations. Do not redirect +the whole community into this local product branch. + +## Contribution acceptance and family expansion are different gates + +**A contributor may finish one useful subproblem.** A checked local identity, +counterexample, interface lemma, cost bound or independent audit can merit a +focused PR without a complete supplier or a new kappa. State the exact result, +dependencies and remaining obligations. No additional standalone session report +is required. A handoff without a PR is also useful. + +**Expanding a family requires an explicit research decision.** This is an +evidence checkpoint, not an approval queue. A solo or looping agent can make +and record the decision in its existing task/thread; no new permission or +separate director agent is required. Before another large search, +implementation expansion or round of downstream optimization, update the +canonical investigation thread or this record with: + +1. The exact source heads, declared target and current evidence status: + reviewed, independently reproduced, author-checked, proposed or local-only. +2. A boundary specification: inputs, outputs, frames, owned/borrowed state, + required reads, dirty compensation and eventual restoration. +3. A complete obligation ledger. Mark each cost **proved**, **constructively + upper-bounded**, or **unknown**. Unknowns are not zero. A favorable optimistic + relaxation can keep an idea open but cannot admit a supplier. An unfavorable + relaxation may reject its stated model. +4. Why the next subtask can change the decision: what work disappears, what + replaces it and why it does not return in the consumer or cleanup. Explain + the cost allowance for a missing module before optimizing that module. +5. Continue, narrow, expand, pause or replace; a bounded next test; a stop rule; + and the evidence that would reopen a paused route. Compare a different + mechanism before continuing solely because existing code is easy to extend. + +Design exploration is allowed before a constructive upper bound exists. Its +deliverable is a contract, discriminating example or budget decision, explicitly +labelled exploratory. It should not silently grow into an implementation campaign. +For an audit, identify the particular acceptance gap and downstream consequence; +do not demand an invented kappa gain from verification work. + +## Distributing many agents + +Keep the broad track allocation and prefer live unclaimed handoffs. Refine a +task by **mechanism, changed contract, exact obligation and bounded domain**. +Separate alternative constructions from deliberately independent checks. +Different random seeds, larger censuses and renamed helper types are not by +themselves distinct mechanisms. Useful small improvements remain welcome. + +Use one existing canonical thread per family/question, with its latest result, +source pins, remaining costs, active scoped tasks and next review trigger near +the top. Do not require every contributor to maintain a second status file. +When all immediately useful tasks in a family are claimed, choose a distinct +obligation or a different family rather than manufacturing a peripheral PR. + +Keep negative proofs tied to decisions. A scoped closure merits preservation; +its ever-narrower extensions do not automatically merit more assigned effort. +Corrections, independent verification and a genuinely changed premise can reopen +it. No finite failed search establishes a universal limitation. + +Maintainers should refresh these entry points after a reviewed completion, +decisive scoped obstruction, changed supplier or substantial queue review. +Preserve historical pins and attributions; update the task catalogue, guide and +priority log together when assignments change. Current weights are allocation +choices, not estimates of success probabilities or proof that the portfolio is +optimal. diff --git a/research/kappa-rebuild/record_claim.py b/research/kappa-rebuild/record_claim.py new file mode 100644 index 0000000000..95f89eac36 --- /dev/null +++ b/research/kappa-rebuild/record_claim.py @@ -0,0 +1,111 @@ +#!/usr/bin/env python3 +"""Bind the proposed conditional claim to its written proofs and exact receipts. + +This is a provenance gate, not a theorem prover. The mathematical boundary is +recorded explicitly. Apache-2.0; substantial OpenAI assistance. +""" +from fractions import Fraction as Q +from hashlib import sha256 +from pathlib import Path +import json + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + + +def need(condition, message): + if not condition: + raise AssertionError(message) + + +def read(name): + return json.loads((HERE/name).read_text()) + + +def main(): + outer = read('transfer-audit/OUTER-TRANSFER-AUDIT.json') + replay = read('verification/CRITICAL-REPLAY.json') + explicit = read('controller/EXPLICIT-TEMPLATE-BOUNDS.json') + ledger = read('controller/OUTER-PARAMETERS.json') + bridge = read('transfer-audit/BIT-BRIDGE-AUDIT.json') + need(outer['status'] == 'PASS_EXPLICIT_RANKS_NEW_NATIVE_INTERFACES_AND_CONDITIONAL_OUTER_POINT', 'Independent transfer audit') + need(bridge['status'] == 'PASS_NATIVE_ROLES_COMPLETE_DAG_ALL_COLUMN_PAIRING_COMMON_WEIGHT_AND_PAID_COVER', 'Independent BIT bridge') + need(replay['status'] == 'PASS_FRESH_NATIVE_AND_CONDITIONAL_INTERFACE_REPLAY', 'Fresh critical replay') + need(replay['reconstructed_primitive_stages_replayed'] and len(replay['stages']) == 23, 'All primitive and interface stages replayed') + need(len(replay['artifact_comparisons']) == 59, 'All declared native/proof artifacts compared') + need(all(row['returncode'] == 0 for row in replay['stages']), 'No failed scientific stage') + kappa = Q(outer['conditional_assembly']['kappa']) + need(kappa == Q(23,10000), 'Declared target') + savings = {k:Q(v) for k,v in outer['conditional_assembly']['nine_outer_savings'].items()} + need(len(savings) == 9 and min(savings.values())-kappa == Q(98849,50000000000), 'Complete strict outer bill') + need(all(Q(v) > 0 for v in outer['conditional_assembly']['functional_interface_gaps'].values()), 'Every changed interface gap') + need(not replay['winning_uniform_words_materialized'], 'Materialization limitation retained') + ranks = {row['family']:row['N']['H1'] for row in explicit['cases']} + need(ranks == {'complex':276540184361,'BIT':6552928412}, 'Exact finite template specification') + sources = [ + 'README.md', 'LITERATURE-AND-PROBES.md', 'TASK.json', 'record_claim.py', + 'architecture/FINITE-PRECOMPILATION.md', + 'controller/P3-NATIVE-COVER.json.gz', 'controller/OUTER-PARAMETERS.json', + 'controller/EXPLICIT-TEMPLATE-BOUNDS.json', 'controller/EXPLICIT-TEMPLATE-BOUNDS.md', + 'bit/BIT-P3-NATIVE-BINDING.json', 'bit/DELAYED-COVER-BRIDGE.md', + 'bit/BIT-ALLSIZE-PARAMETERS.json', + 'transfer-audit/AUDIT.md', 'transfer-audit/FINITE-DAG-AUDIT.json', + 'transfer-audit/P3-NATIVE-AUDIT.json.gz', 'transfer-audit/P3-COVER-AUDIT.json', + 'transfer-audit/BIT-BRIDGE-AUDIT.json', 'transfer-audit/FINITE-TEMPLATE-CLOSURE.md', + 'transfer-audit/PRECISION-CLOSURE.json', 'transfer-audit/OUTER-TRANSFER-AUDIT.md', + 'transfer-audit/OUTER-TRANSFER-AUDIT.json', + 'verification/CRITICAL-REPLAY.json', 'verification/BASE-MATERIALIZED.json', + 'verification/PRIOR-APPLIED.json', 'provenance/public-final/REFRESH.json', + 'replay_critical.py', + ] + bindings = {str((HERE/name).relative_to(ROOT)):sha256((HERE/name).read_bytes()).hexdigest() for name in sources} + # Every regenerated canonical artifact must still be the one compared by + # the fresh replay. Later H-branch additions do not alter these inputs. + for row in replay['artifact_comparisons']: + need(sha256((ROOT/row['path']).read_bytes()).hexdigest() == row['reference_sha256'], 'Canonical replay artifact still frozen: '+row['path']) + for row in replay['stages']: + source = ROOT/row['command'][2] + need(sha256(source.read_bytes()).hexdigest() == row['source_sha256'], 'Scientific replay source still frozen') + baseline = Q('0.001730970419199163702738') + claim = { + 'schema':'proposed-conditional-integer-multiplication-claim/1', + 'status':'COMPLETE_PROPOSED_CONDITIONAL_CONSTRUCTION_WITH_FRESH_FINITE_REPLAY', + 'date_utc':'2026-10-21', + 'claim':'T(n) = O(n (log n)^(1-kappa)) under the retained analytic/native hypotheses and the supplied written construction.', + 'kappa':str(kappa), 'kappa_decimal':'0.0023', + 'user_comparison_baseline':str(baseline), + 'relative_exponent_saving_increase':str(kappa/baseline-1), + 'limiting_outer_saving':str(min(savings.values())), + 'strict_limiting_slack':str(min(savings.values())-kappa), + 'nine_outer_savings':{k:str(v) for k,v in savings.items()}, + 'uniform_templates':[ + {'family':'complex','rank_encoding':'2^276540184361','saving':'0.002304'}, + {'family':'BIT before cover','rank_encoding':'2^6552928412','saving':'0.002760'}, + ], + 'ordinary_BIT_saving':'0.0027418653797', + 'admitted_BIT_saving':'0.0027', + 'source_sha256':bindings, + 'fresh_replay':{'stages':23,'artifacts':59,'all_passed':True}, + 'written_proof_entry':'research/kappa-rebuild/transfer-audit/OUTER-TRANSFER-AUDIT.md', + 'remaining_hypotheses':[ + 'The inherited analytic decomposition and exact integer recovery.', + 'The retained native ordered-movement and fixed-tape execution framework.', + 'The inherited fine-prime padding, gauge-free interior and outer analytic estimates.', + ], + 'verification_limits':[ + 'Universal finite-precompilation, native closure and outer transfer are written mathematical arguments, not formally machine-checked theorems.', + 'Winning fixed uniform words, complete GL covers, B_N, P_N and U_N are algorithm-defined but not materialized.', + 'No practical runtime, usable numerical threshold, unconditional theorem, upstream admission or novelty claim is made.', + ], + 'larger_H_branch':'Separate local improvements and priced negative boundary probes; no kappa above 0.0023 is claimed.', + 'official_selected_frontier_modified':False, + 'public_posted_or_merged':False, + 'preserve_all_unmerged_changes_until_user_reports_merge':True, + 'attribution':'Apache-2.0; substantial OpenAI assistance; immutable predecessor authors and source licenses retained.', + } + (HERE/'CLAIM.json').write_text(json.dumps(claim,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':claim['status'],'kappa':claim['kappa'],'source_bindings':len(bindings),'fresh_stages':23,'fresh_artifacts':59})) + + +if __name__ == '__main__': + main() diff --git a/research/kappa-rebuild/replay_critical.py b/research/kappa-rebuild/replay_critical.py new file mode 100644 index 0000000000..2e2138e9ab --- /dev/null +++ b/research/kappa-rebuild/replay_critical.py @@ -0,0 +1,165 @@ +#!/usr/bin/env python3 +"""Fresh native/proof-interface replay in a separately materialized checkout. + +Run this against a checkout built from the exact base and cumulative overlay. +It regenerates receipts in that checkout. --reference enables comparison +against the delivered artifacts; only timing metadata and their resulting +source-hash references are normalized. No mathematical predicate is waived. +Apache-2.0; substantial OpenAI assistance. +""" +from concurrent.futures import ThreadPoolExecutor +from hashlib import sha256 +from pathlib import Path +import argparse +import gzip +import json +import subprocess +import sys +import time + +SCOPE = Path('research/kappa-rebuild') +IGNORED_TIMING_KEYS = {'runtime','runtime_seconds','elapsed_seconds','seconds'} + + +def need(value,message): + if not value: + raise AssertionError(message) + + +def stage(name,script,outputs,arguments=()): + return {'name':name,'script':str(SCOPE/script),'outputs':outputs,'arguments':list(arguments)} + + +COMPLEX = [ + stage('complex_helper_reconstruction','complex/rebuild_native.py',['complex/fresh-certificates/RESULT-helpers.json','complex/fresh-certificates/*HELPER*'],('--stage','helpers')), + stage('complex_h_reconstruction','complex/rebuild_native.py',['complex/fresh-certificates/RESULT-H.json','complex/fresh-certificates/*H112*'],('--stage','H')), + stage('complex_logger_reconstruction','complex/rebuild_native.py',['complex/fresh-certificates/RESULT-logger.json','complex/fresh-certificates/*LOGGER*'],('--stage','logger')), + stage('complex_dyadic_balance','complex/power_two_balance.py',['complex/balanced-certificates/*']), + stage('complex_native_extensions','complex/native_extensions.py',['complex/extension-certificates/*']), + stage('complex_composition_contracts','complex/composition_contracts.py',['complex/COMPOSITION-CONTRACTS.json']), + stage('complex_phase_contacts','complex/audit_frame_contacts.py',['complex/FRAME-CONTACT-INDEPENDENT-AUDIT.json']), + stage('complex_full_identity','complex/full_identity_rebuild.py',['complex/full-identity-certificates/*']), + stage('p2_native_cover','controller/global_identity_cover.py',['controller/GLOBAL-IDENTITY-COVER.json.gz']), + stage('p3_native_cover','controller/p3_native_cover.py',['controller/P3-NATIVE-COVER.json.gz']), + stage('p3_independent_phases','transfer-audit/p3_native_audit.py',['transfer-audit/P3-NATIVE-AUDIT.json.gz']), + stage('p3_independent_owners','transfer-audit/p3_cover_audit.py',['transfer-audit/P3-COVER-AUDIT.json']), + stage('finite_integer_compiler','transfer-audit/finite_dag_audit.py',['transfer-audit/FINITE-DAG-AUDIT.json']), +] +BIT = [ + stage('bit_native_and_mixed_controller','bit/mixed_productive_bit.py',['bit/MIXED-PRODUCTIVE-BIT.json']), + stage('bit_independent_controller','bit/check_mixed_controller.py',['bit/MIXED-CONTROLLER-INDEPENDENT.json']), + stage('bit_native_common_cover_binding','bit/bit_native_binding.py',['bit/BIT-P3-NATIVE-BINDING.json']), + stage('bit_delayed_cover','bit/delayed_cover_bridge.py',['bit/DELAYED-COVER-BRIDGE.json']), + stage('bit_allsize_parameters','bit/bit_allsize_parameters.py',['bit/BIT-ALLSIZE-PARAMETERS.json']), + stage('bit_independent_bridge','transfer-audit/bit_bridge_audit.py',['transfer-audit/BIT-BRIDGE-AUDIT.json']), +] +ASSEMBLY = [ + stage('complete_outer_parameters','controller/outer_parameters.py',['controller/OUTER-PARAMETERS.json']), + stage('new_precision_and_stock','transfer-audit/precision_closure.py',['transfer-audit/PRECISION-CLOSURE.json']), + stage('explicit_finite_ranks','controller/explicit_template_bounds.py',['controller/EXPLICIT-TEMPLATE-BOUNDS.json']), + stage('independent_outer_transfer','transfer-audit/outer_transfer_audit.py',['transfer-audit/OUTER-TRANSFER-AUDIT.json']), +] + + +def digest(path): + return sha256(path.read_bytes()).hexdigest() + + +def payload(path): + raw=path.read_bytes() + if path.suffix=='.gz': + raw=gzip.decompress(raw) + return json.loads(raw) + + +def normalize(value,hash_map): + if isinstance(value,dict): + return {k:normalize(v,hash_map) for k,v in value.items() if k not in IGNORED_TIMING_KEYS} + if isinstance(value,list): + return [normalize(v,hash_map) for v in value] + if isinstance(value,str): + return hash_map.get(value,value) + return value + + +def main(): + p=argparse.ArgumentParser(description=__doc__) + p.add_argument('--repo',type=Path,required=True) + p.add_argument('--reference',type=Path) + p.add_argument('--report',type=Path,required=True) + p.add_argument('--skip-reconstruction',action='store_true',help='Run the final composition and all independent gates; preserve reconstructed primitive receipts. Disclosed in the report.') + args=p.parse_args() + repo=args.repo.resolve(); reference=args.reference.resolve() if args.reference else None + need(repo.is_dir(),'Materialized replay checkout') + need(reference is None or repo!=reference,'Reference artifacts must be separate from the replay checkout') + report=args.report.resolve(); logs=report.parent/'critical-replay-logs'; logs.mkdir(parents=True,exist_ok=True) + began=time.monotonic() + configuration=COMPLEX[7:] if args.skip_reconstruction else COMPLEX + runs=[]; outputs=set() + def sequence(stages): + local=[] + for item in stages: + script=repo/item['script'];need(script.is_file(),'Replay script exists: '+item['script']) + command=[sys.executable,'-B',item['script'],*item['arguments']] + t=time.monotonic(); logfile=logs/(item['name']+'.log') + with logfile.open('wb') as output: + process=subprocess.run(command,cwd=repo,stdout=output,stderr=subprocess.STDOUT,check=False) + entry={'stage':item['name'],'command':command,'source_sha256':digest(script), + 'returncode':process.returncode,'seconds':time.monotonic()-t, + 'log':logfile.name,'log_sha256':digest(logfile)} + local.append(entry) + print(json.dumps({'stage':item['name'],'returncode':process.returncode,'seconds':round(entry['seconds'],3)}),flush=True) + need(process.returncode==0,'Scientific replay failed; inspect '+str(logfile)) + return local + try: + with ThreadPoolExecutor(max_workers=2) as pool: + complex_job=pool.submit(sequence,configuration) + bit_job=pool.submit(sequence,BIT) + # Inspect both branches even if one fails. + errors=[] + for job in (complex_job,bit_job): + try:runs.extend(job.result()) + except Exception as exc:errors.append(repr(exc)) + need(not errors,'; '.join(errors)) + runs.extend(sequence(ASSEMBLY)) + for item in [*configuration,*BIT,*ASSEMBLY]: + for pattern in item['outputs']: + paths=[x for x in (repo/SCOPE).glob(pattern) if x.is_file()] + need(paths,'Every declared output pattern is nonempty: '+pattern) + outputs.update(path.relative_to(repo).as_posix() for path in paths) + comparisons=[]; hash_map={} + if reference: + for name in sorted(outputs): + need((reference/name).is_file(),'Reference output exists: '+name) + old,new=digest(reference/name),digest(repo/name) + need(old not in hash_map or hash_map[old]==new,'Equal reference artifacts replay identically') + hash_map[old]=new + for name in sorted(outputs): + old,new=reference/name,repo/name + exact=old.read_bytes()==new.read_bytes() + if not exact: + need(normalize(payload(old),hash_map)==normalize(payload(new),{}), + 'Non-timing scientific artifact changed: '+name) + comparisons.append({'path':name,'reference_sha256':digest(old),'replay_sha256':digest(new), + 'exact_bytes':exact,'equal_after_declared_timing_and_source_hash_normalization':True}) + result={'schema':'fresh-cumulative-kappa-critical-replay/1', + 'status':'PASS_FRESH_NATIVE_AND_CONDITIONAL_INTERFACE_REPLAY', + 'stages':runs,'artifact_comparisons':comparisons, + 'timing_keys_ignored_in_artifact_comparison':sorted(IGNORED_TIMING_KEYS), + 'hash_normalization':'Only exact old-to-fresh hashes of regenerated artifacts are substituted in source references. All remaining mathematical content is compared.', + 'reconstructed_primitive_stages_replayed':not args.skip_reconstruction, + 'complete_base_materialization_is_separately_receipted':True, + 'winning_uniform_words_materialized':False, + 'proof_limits':'Executable finite checks and quantitative interface gates do not constitute a formal proof of the written universal construction or discharge inherited outer hypotheses.', + 'runtime_seconds':time.monotonic()-began} + except Exception as exc: + result={'schema':'fresh-cumulative-kappa-critical-replay/1','status':'FAIL', + 'error':repr(exc),'finished_stages':runs,'runtime_seconds':time.monotonic()-began} + report.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + raise + report.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n') + print(json.dumps({'status':result['status'],'stages':len(runs),'artifacts':len(comparisons),'seconds':result['runtime_seconds']}),flush=True) + + +if __name__=='__main__': + main() diff --git a/research/kappa-rebuild/transfer-audit/ARTIFACT-MANIFEST.json b/research/kappa-rebuild/transfer-audit/ARTIFACT-MANIFEST.json new file mode 100644 index 0000000000..4830de41d6 --- /dev/null +++ b/research/kappa-rebuild/transfer-audit/ARTIFACT-MANIFEST.json @@ -0,0 +1,91 @@ +{ + "files": [ + { + "bytes": 19394, + "file": "AUDIT.md", + "sha256": "032488343667d19613cab84df5dad554cfd2a283fb4fc0d4c9b4d165cfa04669" + }, + { + "bytes": 5203, + "file": "BIT-BRIDGE-AUDIT.json", + "sha256": "11d61d4fbc8c43550802518eada8aa3cd830d9c968d4010466d49321ac126285" + }, + { + "bytes": 7726, + "file": "FINITE-DAG-AUDIT.json", + "sha256": "5965986a93f611fdc6bb9fc0a47820385df57e49e2d7edccc257b14deb0f7085" + }, + { + "bytes": 15369, + "file": "FINITE-TEMPLATE-CLOSURE.md", + "sha256": "3f8508c63862dd83968bd30df91b08ba5afcee1e10db15ca546cdff0caef95d9" + }, + { + "bytes": 1012, + "file": "NATIVE-BOUNDARY-AUDIT.json", + "sha256": "b3a7c2b80fda4cede7325dce99b863b4c2fc9bd8a22ab99470d7019874b388a2" + }, + { + "bytes": 6610, + "file": "OUTER-TRANSFER-AUDIT.json", + "sha256": "e81c8d7037025be2ccc5fbbbeb725c6327090d0a9d54ff0bf6faffd2c6728631" + }, + { + "bytes": 20800, + "file": "OUTER-TRANSFER-AUDIT.md", + "sha256": "67569961d068633c6ff3ac9dfde47a0c17f0da6d20432799dcc920cff92b5779" + }, + { + "bytes": 2007, + "file": "P3-COVER-AUDIT.json", + "sha256": "4a458dcbd47cc521fd826acd961d62af51e75a5da2ce56c6f1575de0b1f3ff17" + }, + { + "bytes": 108313, + "file": "P3-NATIVE-AUDIT.json.gz", + "sha256": "d0698e47b50a5db5c38eab86624bd025f6ad605502f66bb6c6f9105738a1f98a" + }, + { + "bytes": 5224, + "file": "PRECISION-CLOSURE.json", + "sha256": "5777a2758177784f3fc248b83b77dd984993ed6686da882ad7110eed1ea8fbbe" + }, + { + "bytes": 19781, + "file": "bit_bridge_audit.py", + "sha256": "937126ae8d23271fd2b8e5dbd524345cf8b2271b9c9879a1f4ec445078c46cf2" + }, + { + "bytes": 19149, + "file": "finite_dag_audit.py", + "sha256": "f842f0e07ca40a181782a565a381b0ee20c6a4b08b8c2b8a4d5d630381bd1e8e" + }, + { + "bytes": 7993, + "file": "native_boundary_audit.py", + "sha256": "133d0f282d7abed155bf6de03b9e16304d7df165fd09d44c618d76be5f31cee2" + }, + { + "bytes": 9687, + "file": "outer_transfer_audit.py", + "sha256": "30f14389942464c987024c4c4821f07409258cc9dd2cbfb5fda93248ae51bafc" + }, + { + "bytes": 10467, + "file": "p3_cover_audit.py", + "sha256": "a0fe9a26f77556e83b31b0c91fea909d6f3d8209d0f9f7ad4e5bc1375d33615e" + }, + { + "bytes": 14209, + "file": "p3_native_audit.py", + "sha256": "5504a55a50546418d10c522984438cdeedb44993f7c1dab7bd4aa1ae0515d902" + }, + { + "bytes": 11287, + "file": "precision_closure.py", + "sha256": "a22ecb1bd868487883785418ab532e981f42148089e4ac20488caf7a4b827a5c" + } + ], + "schema": "independent-transfer-audit-source-set/1", + "scope": "New audit artifacts only; the completed complex 53-file source set is unchanged." +} diff --git a/research/kappa-rebuild/transfer-audit/AUDIT.md b/research/kappa-rebuild/transfer-audit/AUDIT.md new file mode 100644 index 0000000000..ee8c49a345 --- /dev/null +++ b/research/kappa-rebuild/transfer-audit/AUDIT.md @@ -0,0 +1,393 @@ +# Adversarial audit of finite precompilation + +Prepared with substantial OpenAI assistance, Apache-2.0. The mixed lockstep +tree and its potential argument are Jacob Alan Sussman's Theorem 1G. The +restored primary proof is +[`theorem-1g.md`](../architecture/sources/Sussman-563b4e6/research/record-track/proofs/theorem-1g.md), +from commit `563b4e6e913625637f36ba7869bf5fcd0dd4975a` of +[`wht-power-saving-lean`](https://github.com/jacobalansussman/wht-power-saving-lean). +The proposal independently scrutinized here is +[`FINITE-PRECOMPILATION.md`](../architecture/FINITE-PRECOMPILATION.md). + +## Verdict and scope + +The proposed finite precompilation bridge is valid under the native closure +hypotheses stated below. It does not follow from a mixed moment inequality +alone. The essential extra ingredients are independent arbitrary-input jobs, +canonical whole-job adapters, an ordered integer flow, and completion of each +source's complementary axes. + +For the current complex candidate, the available finite certificates supply +the relevant ideal operator and alphabet closure. Two independent programs +in this directory check the concrete p2 boundary and a finite version of the +precompiler. They do not materialize the enormous near-threshold winning +supplier, prove a rounded multiplication implementation, or claim a new +kappa. The BIT weighted interface has separate compatibility obligations and +is outside this positive conclusion. + +A correction was necessary: one owned center per recursion depth is not a +safe general allocation. Several units of a class can suspend with live +private centers. One center bank per compiled unit instance is a valid finite +upper bound. No such bank contributes productive output credit. + +There is also a useful strengthening of the proposed argument: the exponent +existence step needs only an asymptotic bound on the **number of paid leaves**. +That bound follows directly from exact integer source/exit potentials. No +RAM running-time theorem, fluid row rounding, or unit-cost arbitrary +permutation assumption is needed to choose the fixed template size. + +## 1. Precise conditional theorem + +Let `C` be the fixed native complex kernel, and let a rank-s job mean +`C^(tensor s)` on specified complete coordinate axes, identity on all other +coordinates of the same array. All arrays have the same full ambient address +set. In the eventual column lift, every named axis becomes `f` axes and the +job rank becomes `s*f`. + +Suppose a fixed mixed unit has target-role multiplicities `W_k`, a +chronologically ordered list of paid child calls of ranks `r_j`, and these +properties: + +1. Its input roles are independent arbitrary arrays. Its completed endpoint + is exactly one canonical rank-k job on each declared target role, without + cross-role contamination. Every paid call has an explicit physical owner. +2. The unit, its paid-call normal forms, and its endpoint adapters are valid + at every positive integer column multiplicity, on complete arbitrary + spectators. The word consists of the retained scalar operations, permitted + fixed binary-affine coordinate maps, unit Z4 quadratic phases, and the + declared paid children. +3. Each completed job can be embedded on any prescribed subset of the fixed + ambient axes by permitted fixed coordinate maps. A rank-a+b job is the + product of its jobs on disjoint a and b axes. Adapters are actual whole-word + operations; they are not permission to relabel an intermediate scalar + contact. +4. Every owned workspace is explicitly supplied, initialized, preserved as + long as needed, and cleared. Finite straight-line composition permits a + finite fixed number of such work arrays and charges their actual scalar, + movement, precision, and parking costs without counting them as outputs. + +Put `m=max{k:W_k>0}`, `k0=min{k:W_k>0}`, and `r0=max_j r_j`. Assume +`1<=r0=2`, +`B_N>=1`, and a finite native uniform supplier with exactly `B_N` independent +productive inputs and outputs. Its completed output on every productive +array is the full rank-N job. Its only paid children have rank one and their +number `P_N` satisfies + +\[ +P_N