diff --git a/research/p112-batch-centre-floor/PROOF.md b/research/p112-batch-centre-floor/PROOF.md new file mode 100644 index 0000000000..1e43f0a920 --- /dev/null +++ b/research/p112-batch-centre-floor/PROOF.md @@ -0,0 +1,191 @@ +# A direct-sum P112 copy-rank theorem + +## 1. Fixed geometry and cost model + +Let X be the 112-by-8 live P112 root matrix in norm-four, frame-adapted +Hamming coordinates. The full set has eight coordinate lines `2 e_j` and +112 signed affine-plane root lines; P112 deletes the coordinate lines. +One representative of each line is chosen. The checker gives an explicit +scaled orthogonal map from standard doubled E8 roots and their deleted +frame, checking `2 h(q) dot h(q') = q dot q'` before line-sign normalization. +The normalized Gram pairing is preserved. + +For the full 120-line set define + + B_ij = 1 + (x_i dot x_j)/2 (mod 2). + +All dot products are even. B is symmetric of rank nine. Its rows define +labels ell_i in a nine-dimensional binary space V, with a well-defined +nondegenerate symmetric pairing `b(ell_i,ell_j)=B_ij`. Indeed symmetry makes +row dependencies column dependencies, and a label pairing to zero with all +generators is the zero row. An arbitrary chosen nine-bit coordinate chart +need not make b the standard dot product. + +For live index sets S,T let `rho(S,T)=rank_F2 B[S,T]`. This depends only on +their label spans and cannot decrease on enlarging either set. Every call +uses this **same** puncture, chart and pairing. + +For p output blocks, q input blocks and `C in C^(p x q)`, require the +all-pairs matrix `G_C=C tensor (X X^T)`. Let `X_p=I_p tensor X` and similarly +X_q. A root-code factorization satisfies + + D T^T = G_C, + col(D) subset col(X_p), col(T) subset col(X_q). + +Coefficients are arbitrary complex numbers and transpose is **ordinary** +transpose. Discard a term if either column is zero. For each active term j, +S_j and T_j are the unions of its nonzero live rows over all output and +input blocks, respectively. Charge the term `rho(S_j,T_j)`, once, regardless +of how many calls it touches. The total charge is their sum. + +This is an explicit algebraic objective that forgets call identity. +Interpreting it as a physical lower bound requires a corresponding +separate-total copy/delivery interface; no assertion about arbitrary +interacting circuits is built into the definition. Inputs in different +blocks are independent variables, even if the requested C is singular. + +## 2. Finite support lemma, with predecessor credit + +The fixed P112 support classification is due to Mihir Srivastava's PR845; +Adam Gall's PR1098 independently certifies it with Gram spectra. Our two +checkers re-establish the following finite premises without importing those +executables or invoking a Weyl-flat classification theorem. + +For each of all 511 nonzero linear functionals h on V, form the matrix of +live roots with `h(ell_i)=1`. Exactly 503 matrices have rational rank eight. +The other eight have rank seven with kernels the eight coordinate lines, +each once. Write E_j for the live support of `X e_j`. Each E_j contains +56 roots and spans eight label dimensions. All 64 ordered pairs satisfy +`rho(E_i,E_j)=8`. + +The rank statements use exact witnesses. Nonzero rational minors remain +nonzero over C, and rational null vectors remain null vectors. Thus all +ranks and one-dimensional kernels hold over C. This enumerates binary +hyperplanes of the **label space**, not all complex coefficient directions. + +**Support lemma.** For any nonzero `a in C^8`, the support of Xa has full +label span V or is exactly E_j for some j. + +X has rank eight, so Xa is nonzero. If its support span is proper, choose +a nonzero h vanishing on it. Every live root outside ker(h) annihilates a. +A rank-eight complement would imply a=0. Hence this is one of the eight +rank-seven cases, so `a=c e_j` with c nonzero and its support is E_j. +This includes complex isotropic a; positivity or nonisotropy is not used. + +**Pair lemma.** Any two nonzero complex root-code supports pair with rank +at least eight. If both are exceptional this is the 64-entry check. If one +spans V, nondegeneracy of b gives rank equal to the dimension of the other +span, which is eight or nine. + +## 3. All-batch lower bound + +A nonzero column of col(X_p) has a nonzero block Xa. Its union support +contains that block's support. Choose a nonzero block Xc in the other +factor. The two block-support pairing matrix is a submatrix of the two +union-support pairing matrix. The pair lemma and rank monotonicity show +that **every active term costs at least eight**, even when the chosen +blocks have different call indices. + +Exact geometry gives `X^T X=56 I_8`, hence `rank(XX^T)=8` over C. For +r=rank(C), the Kronecker-product rank identity yields `rank(G_C)=8r`. +Equivalently, use the full-rank left/right inverses of X_p and X_q on +`G_C=X_p (C tensor I_8) X_q^T`. A sum of m rank-one outer products has +rank at most m. Thus + + m >= 8r, total charge >= 8m >= 64r. (1) + +For independent calls C=I_k, this is m>=8k and charge at least 64k. +This is an all-size proof after a fixed finite premise; finite fixtures do +not substitute for the quantified argument. + +## 4. Attainment and exact coupling-rank law + +Choose a rank factorization `C=sum_(l=1..r) u_l v_l^T`. For every l and +j=1,...,8 take + + d_(l,j) = u_l tensor (X e_j), + t_(l,j) = v_l tensor (X e_j). + +Their outer products sum to G_C. Each u_l and v_l is nonzero, so both union +supports are exactly E_j. Each term costs eight. These 8r terms attain 64r; +for C=0 the empty factorization attains zero. Therefore + + minimum copied-pairing-rank charge = 64 rank(C). (2) + +**Exact charge and equality cases.** Call a nonzero block-root-code column +coordinate-aligned if it is `u tensor (X e_j)` for some nonzero call vector u +and one root coordinate j. The support lemma and injectivity of X show that +its union support spans an exceptional hyperplane exactly in this case. +Otherwise some block has full span V, or two blocks have distinct +exceptional hyperplanes, whose union spans V. The eight hyperplanes are +distinct because the finite census identifies each with a distinct nonzero +functional. Taking unions does not cancel coefficients across calls. + +The complete pairing table is therefore 8 for exceptional/exceptional or +exceptional/full, and 9 for full/full. For any factorization in the stated +root-code model, let b count active terms where **neither** factor column +is coordinate-aligned. Its exact cost is + + total charge = 8m + b. (2a) + +Among exact factorizations of G_C with r>0, optimum 64r holds if and only if +`m=8r` and b=0. Either side may supply the aligned column term by term; +both sides need not align. Alignment is not a substitute for checking the +full endpoint identity. The fixture with one two-sided nonaligned term +has exact cost 65, while a mixed two-call equality fixture still costs 128. + +This is algebraic attainment only. Arbitrary complex call coefficients do +not come with free preparation, arithmetic, routing or restoration. + +For C=I_2 the outputs request `XX^T x_0` and `XX^T x_1`, separately, and +require 128. For the two-by-two all-ones C, both request `XX^T(x_0+x_1)`, +and the minimum is 64. Substituting the latter for the former introduces +nonzero off-diagonal blocks; the verifier rejects this endpoint mutation. + +## 5. Minimum-term factors are automatically in the code + +Suppose r>0 and an unrestricted exact factorization of G_C has exactly 8r +columns. Both factors have rank 8r. Since `col(G_C) subset col(D)` and +their dimensions coincide, the two spaces are equal; likewise +`col(G_C^T)=col(T)`. These spaces lie in col(X_p), col(X_q), respectively. +Both code hypotheses follow, with no conjugate transpose argument needed. + +Thus (1) covers every minimum-term factorization, including arbitrary +cross-call basis changes. This implication fails for overcomplete factors; +their two code hypotheses remain explicit. + +## 6. Moment consequence and necessary escape invoice + +For independent calls, under a one-child-per-total interface at its pairing +rank, every nonnegative increasing per-child function f obeys + + sum_j f(rho_j) >= m f(8) >= 8k f(8). + +This includes `f(s)=(s/M)^(1-a)` for fixed parent width M and a<1. Changing +the root-code factorization cannot improve the centre-only moment. Changes +to volume, other local children, coverage or the physical contract require +separate pricing; positive added interface costs cannot rescue this route. + +An unrestricted independent-call decoder with charge at most 64k-1 must +have m>=8k+1, at least one column outside its respective block-root code, +and + + sum_j (8-rho_j) >= 8(m-8k)+1. (3) + +The last line just rearranges `sum rho_j <=64k-1`. At least one term has +rank at most seven; ranks above eight count against the savings. For k=2, +m=17 requires at least nine net units. These are necessary conditions, +not sufficient ones, and do not exclude general outside-code factorizations. + +## 7. Limits + +The fixed P112 puncture, common chart, all-pairs endpoint and separate-total +union price are material. Masked identities, changed charts, temporal +source relations, interacting delivery, arbitrary frames and outside-code +overcomplete factors are not covered. Singular C changes the requested +map; it does not merge independent endpoint obligations for free. + +The finite premises have exact executable witnesses and separate checks. +This all-size argument is a written proof, not a Lean theorem or human +peer review. No selected global schedule or frontier is modified. Source +and predecessor credit are in README.md and SOURCES.json. diff --git a/research/p112-batch-centre-floor/README.md b/research/p112-batch-centre-floor/README.md new file mode 100644 index 0000000000..6a779efc83 --- /dev/null +++ b/research/p112-batch-centre-floor/README.md @@ -0,0 +1,216 @@ +# Independent P112 batches cannot amortize the root-code copy bill + +**Scoped mathematical result with exact finite premises. No new supplier or kappa.** + +For **every number k of independent standard P112 all-pairs demands**, scalar +coding across calls cannot reduce the copied-pairing-rank bill below **64k** +when both factors stay in the direct-sum root code. This allows any finite +number of terms, arbitrary complex coefficients and cross-call mixing. It +even prices a shared term only once on its **union of labels across calls**, +without a separate charge for each call it touches. + +The exact charge is **8m+b**: m is the number of active terms, and b counts +terms whose reader and source columns both mix root-coordinate directions. +An optimal decoder has the minimum term count and at least one +coordinate-aligned side in every term. Cross-call mixing within one root +coordinate is allowed and can attain equality; it does not lower the bill. + +For a factorization with the minimum **8k terms**, the root-code restriction +is automatic. Changing the complete basis of a minimum-size batched decoder +therefore cannot evade the bound. Every additional active root-code term +costs at least another eight units. + +More generally, for a rectangular call-coupling matrix C, the minimum in this +model is **64 rank(C)** for the requested matrix `C tensor (X X^T)`. X is +the fixed norm-four 112-by-8 P112 root matrix. For two independent demands +C=I2, the minimum is 128. For C=all-ones(2,2), it is 64, but **both output +blocks then request the image of the sum of the two inputs**. These are +different endpoints. Attainment is a decoder identity, not an implemented +shared centre or a complete physical cost claim. + +## Why this is useful + +The selected complex supplier's 64-per-call centre charge is a central +bottleneck. The [existing lever screen](../complex-local-word-ceiling/README.md) +prices ideal two-call sharing at about 1.90 times the current primitive-saving +ceiling, replacing 128 centre-rank units by 64. The present theorem rules +out that replacement by root-code scalar mixing of independent P112 demands, +even before positive preparation, restoration, routing, hosting or precision +charges. + +Under the further interface that each scalar total pays one child at its +pairing rank, at unchanged parent width and volume, the centre moment cannot +improve either: at least 8k terms each have rank at least eight. Every +nonnegative increasing per-child price remains at least `8k * price(8)`. +This comparison does not install a replacement coverage law or claim that +every physical circuit has this separate-total accounting. + +The new result is the **all-batch coupling-rank law**, including the +unrestricted minimum-term corollary and a quantitative necessary budget for +outside-code escapes. The one-call support census belongs to **Mihir +Srivastava's PR845**, independently corroborated by **Adam Gall's PR1098**. +The one-call arbitrary-term factor-linear floor belongs to **woklerx's +PR1180**. We recheck their finite premises by two separate implementations; +we do not claim their census or support lemma as new. This fixed-P112 proof +does not require the more general Weyl-flat classification used elsewhere. + +## Exact model and evidence + +[PROOF.md](PROOF.md) proves the result for every complex C. Transpose is +ordinary transpose, not conjugate transpose. All calls retain the **same** +P112 puncture, binary label embedding and pairing. A term's reader and +source supports are unions over its nonzero output/input blocks. Its cost +is the F2 rank of their label-pairing matrix. For overcomplete factorizations, +both factors must be in their respective block-root codes. Exactness is +required on **all** pairs, including the cross-call blocks. + +| Rechecked finite premise | Result | +|---|---| +| All 511 nonzero binary label functionals: rational rank of roots outside their hyperplane | 503 have rank 8; eight have rank 7 | +| Eight deficient complements | Their nullspaces are the eight removed frame directions | +| Live support of each exceptional direction | 56 roots; binary label span 8 | +| Ordered exceptional support pairs | All 64 have F2 rank 8 | + +The rank-seven nullspaces remain one-dimensional over C. Thus any nonzero +complex root-code column has full label span nine or one of those eight +exceptional supports. Every support pair costs at least eight. Unions cannot +lower a submatrix rank. Matrix rank forces at least `8 rank(C)` terms, and +a rank factorization of C with the coordinate decoder attains the bound. + +## Reproduce + +Python 3.12, its standard library, and Git with the pinned base commit +available suffice. From the repository root: + +```sh +python3.12 -I -O -B research/p112-batch-centre-floor/verify.py +python3.12 -I -O -B research/p112-batch-centre-floor/audit.py +python3.12 -I -O -B research/p112-batch-centre-floor/test_package.py +python3.12 scripts/research_package.py --package research/p112-batch-centre-floor +``` + +The primary checker reconstructs the Hamming geometry and exact integer +minor witnesses. The separate auditor reconstructs standard doubled E8 +roots, uses set-based binary elimination, and proves rational ranks using +modular lower bounds plus exact integer null-vector upper bounds. Neither +imports a predecessor checker as an oracle. Both check the same canonical +finite receipt [RESULT.json](RESULT.json). Default replay writes nothing; +semantic checks remain active under `-O`. + +Eight independent-call fixtures check 451,584 entries, including 250,880 +off-diagonal call entries. Shared-column examples attain 128/192 for two/three +calls; a 17-term root-code example costs 136. A strict mixing example costs 65. +A rectangular three-output/two-input rank-two demand checks another 75,264 +entries and costs 128. Controls include an exact outside-code cost-129 example +with a cost-one term, changed correlated endpoints, ordinary-versus-Hermitian +transpose, and cross-call crosstalk invisible to diagonal-block checks. + +[VERIFY.json](VERIFY.json) registers bounded replay. [SOURCES.json](SOURCES.json) +pins exact source blobs and hashes, and [TASK.json](TASK.json) records the +assignment and next questions. Selected record and catalogue pins are +historical provenance. Finite batch fixtures are regression evidence; the +all-size conclusion is the written proof, not an enumeration of all k. + +Validation on 2026-10-14 used Windows, CPython 3.12.2 and only the standard +library. All three registered commands passed under isolated, optimized +Python (`-I -O -B`), including the renamed-package positive replay and five +rejections from four on-disk corruption groups. The primary checker has ten +semantic controls; the separate finite-geometry auditor has seven. The +auditor does not independently check the batch fixtures or certify the +written all-size argument. The complete package replay also passed through +the repository's registered-package runner. + +## Scope and overlap + +The `centre-path-sharing` card is refined to its copied-information +lower-bound method. One selector draw was used, nonce +`afb2871f5b82a11465a83416a203340b`, on main +`0b1cccf86e2f5132ed286a26ef0a681a495740ea`. The ceiling control and lever +screen passed. Selected source-binding checks passed using Windows extended +paths; ordinary paths hid seven files beyond MAX_PATH. All 1,579 source +files and 51 proof supplements matched. This was not the selected cold replay. + +Entry overlap covered 382 open PRs including drafts and 333 Discussion +records, complete pagination, fetched at 07:04:07 and 07:05:18 UTC on +2026-10-14; local notes and the ledger were also searched. Coverage excludes +arbitrary code diffs. Claim: +[Discussion1691](https://github.com/CrocSwap/integer-mult-bounds/discussions/1691). + +The pre-submission refresh covered 396 open PRs and 352 Discussion records, +with complete pagination at 07:29:35 and 07:30:48 UTC. Main was still the +pinned base. All six direct predecessor heads and the contextual PR1633/1689 +heads were rechecked. No equivalent batch coupling-rank result was identified +in that coverage; this is not a literature-priority or novelty certificate. + +- PR845/1098: the finite support lemma; PR968/1077: eight-term punctured E8 + floors; PR1180: arbitrary finite root-code term counts for one call. +- PR765: tensor products of a binary full-E8 code. Here the field is C, + the puncture is P112, and the construction is a direct sum of calls. +- ctoth's [Discussion1516](https://github.com/CrocSwap/integer-mult-bounds/discussions/1516) + and PR1643/1664/1684: fixed-word descent trees and dirty-host transport. + Here decoder columns may change across calls, under a different explicit + algebraic price model. +- [Discussion1636](https://github.com/CrocSwap/integer-mult-bounds/discussions/1636) + changes the spectral demand and reports a routing obstruction; that is + outside the independent all-pairs endpoint here. +- PR1671 fixes the selected scalar word and optimizes its gate frames; + this package does neither that census nor a frame-optimality claim. +- Hanson Sun / bbMisk's PR1633 proves sharp outside-code excess-rank budgets + for one-call overcomplete decoders, extending joaqcoo / PR1224. It is + complementary algebraic guidance for the escape question; its code and + witnesses are not imported or represented as this package's results. +- Denis / just-mn's PR1689 studies a two-coordinate masked-Walsh family + outside the root code, with one-call 72/77 bounds. That family and its + finite census are not covered or reused here. +- The [shared-source two-panel claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1516#discussioncomment-18918111) + explicitly assumes identical centre source signals in both calls. It + changes the independent-input premise, so the present theorem does not + exclude its proposed fanout mechanism. + +## Open next questions + +1. **Can an outside-code overcomplete batch beat the floor?** For k=2 the + first possible case has at least 17 active terms. Cost at most 127 requires + `sum_j (8-rho_j) >= 8*(m-16)+1`: at 17 terms, at least nine net units of + rank saving relative to rank eight. First test one specified 17-term + family by exact full-matrix replay and support pricing. The blocker is + an actual exact outside-code family; failed search is not exclusion. +2. **Does an actual caller request a rank-deficient separable C?** Such a + caller has algebraic headroom. First derive its literal block matrix, + retaining independent input variables and cross-call obligations. The + blocker is a realized caller with compatible labels and a complete paid + restoration/hosting/precision budget. +3. **Can changing cross-call label charts help?** First specify two legal + maps, calculate cross-support pair ranks and price all map conversions. + The blocker is a valid new interface; no free relabeling is admitted. + +Masked identities, changed charts, temporal source relations, interacting +delivery and outside-code overcomplete factors remain outside the theorem. +No selected cold replay, native/compiler implementation, owner admission, +precision/tape analysis, all-size physical realization, analytic transfer, +Lean proof or independent human review is claimed. The selected frontier, +frozen archives and predecessor packages remain unchanged. + +## Credit + +**Mihir Srivastava / mihir-s-05 (PR845)** owns the support reduction and +original rank census; **Adam Gall / adamgall (PR1098)** independently +rechecked it with Gram spectra. **themilkman216 (PR968)** supplies frame-pair +geometry and eight-term floors; **woklerx (PR1077/1180)** supplies complex +coefficients and the arbitrary-term root-code extension. **Douglas Colkitt +(PR1240)** supplies the packing ceiling and centre-sharing motivation. + +Retain **DreamingOfClouds / PR717**, **Joel Pulikkan / GamingPuzzled / PR620**, +PR595 format geometry, **icekylinx**, and the E8/compiler lineage named in +the predecessor notices. **Mister-Meeseeks / PR765**, with GamingPuzzled and +**Chi Zhao / PR631**, supplies the related tensor-code precedent. ctoth's +disclosed model reviewers, **Dennis / antunishdPursuit**, **Andrey Mas / an664**, +and **Cinna Davis / 1337isnot1337** have complementary fixed-word audits. +Selected construction credit stays with **Rohan Arun, Jeremy Kothe, +Hao Yan** and the complete lineage in the selected review. + +Prepared for **alephnan** with substantial **OpenAI Codex assistance** in +proof, programming, separate algorithmic checks and adversarial review. +No generic rank-factorization or geometry literature-priority claim is made. +New work is Apache-2.0 under the repository license. Separate AI checks are +not human peer review or proof-assistant certification. diff --git a/research/p112-batch-centre-floor/RESULT.json b/research/p112-batch-centre-floor/RESULT.json new file mode 100644 index 0000000000..62651cbb94 --- /dev/null +++ b/research/p112-batch-centre-floor/RESULT.json @@ -0,0 +1,21841 @@ +{ + "claim": "Common-chart block-root-code copy optimum 64 rank(C); cost 8m+b; independent-call floor 64k", + "controls": { + "Gaussian_unit_phase_entries_checked": 12544, + "correlated_demand": { + "C": [ + [ + 1, + 1 + ], + [ + 1, + 1 + ] + ], + "cost": 64, + "entries_checked": 50176, + "input_blocks": 2, + "output_blocks": 2, + "rank_C": 1, + "terms": 8 + }, + "count": 10, + "crosstalk_passes_diagonal_only": true, + "outside_code": { + "cost": 129, + "entries_checked": 50176, + "factors_sha256": "2dee30e806fe27467b1a945656b1f86b3841c69b97966849e05c521025b91345", + "new_term_cost": 1, + "not_a_cheaper_decoder_claim": true, + "term_costs": [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 1 + ], + "terms": 17 + }, + "rejected": [ + "root norm mutation", + "binary pairing mutation", + "omitted hyperplane", + "duplicate minor row", + "omitted rank-one summand", + "cross-call crosstalk", + "outside-root-code exact factorization", + "correlated demand substituted", + "conjugate transpose substituted", + "boolean replaces integer receipt" + ] + }, + "finite_premises": { + "basis_root_indices": [ + 0, + 1, + 2, + 4, + 8, + 9, + 16, + 24, + 32 + ], + "checked_integer_minors": 511, + "complement_rank_histogram": { + "7": 8, + "8": 503 + }, + "exceptional_functionals": [ + 1, + 2, + 4, + 7, + 8, + 11, + 13, + 14 + ], + "exceptional_pairing_ranks": [ + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 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"Complex-coefficient extension" + }, + { + "pr": 1180, + "head": "75b408646d558147a4eb250fc407d1fbfcb8df4e", + "author": "woklerx", + "scope": "One-call arbitrary-term root-code floor" + }, + { + "pr": 1240, + "head": "1e136b8c4a552050820334ad113628775182f08c", + "author": "Douglas Colkitt / Mister-Meeseeks", + "scope": "Selected-word ceiling and sharing lever" + } + ], + "contextual_pending_prs": [ + { + "pr": 1633, + "head": "652264fa1f70820139b2730363a0056b7fadc689", + "author": "Hanson Sun / bbMisk", + "scope": "Outside-code residual-rank budget and ten-term normal form; not imported" + }, + { + "pr": 1689, + "head": "f87724057cc5fc5192fe2904675a563273612851", + "author": "Denis / just-mn", + "scope": "Two-coordinate outside-code masked-Walsh one-call lower bounds; not imported" + } + ] +} diff --git a/research/p112-batch-centre-floor/TASK.json b/research/p112-batch-centre-floor/TASK.json new file mode 100644 index 0000000000..38a7086d84 --- /dev/null +++ b/research/p112-batch-centre-floor/TASK.json @@ -0,0 +1,103 @@ +{ + "schema_version": 1, + "task_id": "centre-path-sharing", + "priority_revision": 3, + "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea", + "original_card": { + "id": "centre-path-sharing", + "question": "Can the 64-per-call centre path (eight rank-8 COPY/ERASE totals through the private temporary) be shared across calls or shrunk?", + "starting_points": [ + "research/complex-local-word-ceiling/README.md", + "research/split-centres-global/proofs/PR386-BIT-GEOMETRY.txt", + "certificates/selected-result.json" + ], + "methods": [ + "Derive a fully charged favorable relaxation for sharing one centre path across k calls, including restoration, hosting and precision.", + "Prove a lower bound on copied information per call for the selected decoder contract." + ], + "acceptance_gate": "Show the changed centre charge and copy profile per call with every added restoration or hosting cost charged, and its ceiling via ceiling.ceiling(centre=...).", + "first_step": { + "command": [ + "python3", + "-B", + "research/complex-local-word-ceiling/verify.py" + ], + "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING", + "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.", + "scope": "Positive control and planning budget only; this is not a new construction or full cold replay." + }, + "first_experiment": "Write the per-call centre charge and copy children for sharing across k = 2 calls, charge the sharing overhead, and screen it; stop this route if even the ideal sharing gains under 10%. Ideal budgets: free 2.80x, shared by two 1.90x (levers.py).", + "bottleneck_link": "The centre path consumes 64 of about 99.6 unpaid coordinates per call, i.e. about 64% of the raw deficit.", + "allowed_variations": [ + "One sharing pattern across a fixed number of calls.", + "One reduced copied-total contract for the existing decoder.", + "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope." + ], + "scope_change_triggers": [ + "Removing copies without a replacement contract for the decoder's copied totals." + ], + "overlap_terms": [ + "centre path", + "copied totals", + "private temporary", + "shared centre", + "batching" + ] + }, + "scope_status": "refined", + "actual_scope": "Exact minimum 64 rank(C) for all-pairs P112 batch demands C tensor XX^T in the common-chart block-root-code union-support model; 64k for independent calls, with unrestricted minimum-term corollary and outside-code escape invoice. Independent exact replay of credited PR845/1098 finite premises. Exact cost 8m+b and characterization of optimum among exact root-code factorizations.", + "scope_change_reason": "Refine the centre-sharing question to its copied-information lower-bound method. Replace a proposed sharing construction by a favorable independent-batch model before building a global schedule; extend to separable call coupling only to state the exact sharing condition. Existing one-call census is credited, not claimed new.", + "bottleneck_link": "The ideal k=2 shared-centre budget improves the packing cap by 1.90x, but assumes 64 total centre rank instead of 128. The proposed floor rules out that saving in the specified direct-sum root-code model, even before positive restoration, routing, hosting or precision costs.", + "budget_minutes": 60, + "stop_rule": "One 60-minute session starting 2026-10-14T07:01:15Z; reserve final 15 minutes for verification and handoff. Stop the sharing construction if the exact union-support floor is 64k. Publish only after independent proof scrutiny and exact geometry/fixture/mutation checks; otherwise hand off the precise gap.", + "outcome": "completed", + "evidence": "PROOF.md gives the all-size 64 rank(C) optimum, exact 8m+b charge, equality characterization and escape invoice. RESULT.json records 511 exact hyperplanes/minors, 64 pairings, 451584 independent-call entries plus 75264 rectangular entries and controls. verify.py and code-disjoint audit.py use exact arithmetic; physical implementation and selected cold replay remain unrun.", + "author": "alephnan with substantial OpenAI Codex assistance", + "ai_assistance": "Substantial OpenAI Codex assistance, including separate geometry implementation and adversarial mathematical review by collaborating agents. No independent human or proof-assistant verification is implied.", + "selection_nonce": "afb2871f5b82a11465a83416a203340b", + "catalogue_scope_key": "184695337cedab03152e", + "concrete_scope_key": "p112-independent-direct-sum-root-code-union-support-floor", + "open_questions": [ + { + "question": "Can a specified outside-code overcomplete two-call decoder cost below 128?", + "why_it_matters": "Every cheaper decoder must escape the proven block-root-code floor. A 17-term candidate needs at least nine net units below rank 8.", + "first_step": "Specify one exact 17-term family; replay all 224-by-224 entries and compute every support-pairing rank.", + "blocked_by": "No exact outside-code family attaining the necessary saving is supplied." + }, + { + "question": "Does a realized caller request a rank-deficient separable call coupling C?", + "why_it_matters": "The exact algebraic bill is 64 rank(C), so a lower-rank demand has possible sharing headroom.", + "first_step": "Derive the literal complete block endpoint with independent input variables and every off-diagonal obligation.", + "blocked_by": "A valid caller with common labels, chronology, and full preparation/restoration/hosting/precision costs." + }, + { + "question": "Can legally changed cross-call label charts lower union support ranks?", + "why_it_matters": "The common-chart premise is essential and changing it is outside the theorem.", + "first_step": "Specify two admissible charts, recompute cross-support ranks, and charge all conversions.", + "blocked_by": "A replacement common-label/physical interface that permits and pays for the maps." + } + ], + "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1691", + "entry_overlap": { + "open_prs": 382, + "discussions_comments_replies": 333, + "open_prs_fetched_at": "2026-10-14T07:04:07.799861+00:00", + "discussions_fetched_at": "2026-10-14T07:05:18.769683+00:00", + "complete_pagination": true, + "limits": "Open PR titles/bodies, Discussion content, local top-level package notes and ledger; not all historical diffs." + }, + "submission_overlap": { + "open_prs": 396, + "discussions_comments_replies": 352, + "open_prs_fetched_at": "2026-10-14T07:29:35.058112+00:00", + "discussions_fetched_at": "2026-10-14T07:30:48.560791+00:00", + "complete_pagination": true, + "main_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea", + "direct_predecessor_heads_rechecked_at": "2026-10-14T07:31:33Z", + "limits": "Open PR titles/bodies, Discussion content, local top-level package notes and ledger; not all historical diffs.", + "new_context": [ + "PR1689: one-call outside-code two-coordinate masked-Walsh family; no code or witnesses imported.", + "Discussion1516 comment18918111: shared-source two-panel fanout explicitly changes independent inputs." + ] + } +} diff --git a/research/p112-batch-centre-floor/VERIFY.json b/research/p112-batch-centre-floor/VERIFY.json new file mode 100644 index 0000000000..e0596aed68 --- /dev/null +++ b/research/p112-batch-centre-floor/VERIFY.json @@ -0,0 +1,55 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/complex-decoder-field-extension-seed44/NOTICE.md", + "research/complex-decoder-field-extension-seed44/README.md", + "research/complex-frame-pair-decoders/README.md", + "research/complex-frame-pair-decoders/verify.py", + "research/complex-local-word-ceiling/README.md", + "research/e8-factor-linear-overcomplete47/NOTICE.md", + "research/e8-factor-linear-overcomplete47/README.md", + "research/p112-gram-support-seed80/README.md", + "research/p112-single-lift/NOTICE.md", + "research/p112-single-lift/PROOF.md", + "research/p112-single-lift/reference-geometry.json", + "research/tensor-e8-copy-floor/README.md" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 120, + "checks": "Exact source/geometry binding; 511 hyperplanes and integer minors; all 64 support pairs; independent-call and correlated-demand full-matrix fixtures; semantic mutation controls; full canonical receipt equality." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/audit.py" + ], + "timeout_seconds": 120, + "checks": "Separate standard-root and set-elimination implementation; modular rank lower bounds plus exact null upper bounds; all 511 receipt supports/ranks/minors checked by subset-expansion determinants; seven mutations." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-B", + "{package}/test_package.py" + ], + "timeout_seconds": 120, + "checks": "Positive renamed-package replay and default read-only check; on-disk corruptions of an exact minor, immutable-input blob ID, rectangular coupling cost and duplicate JSON key must fail with intended diagnostics." + } + ] +} diff --git a/research/p112-batch-centre-floor/audit.py b/research/p112-batch-centre-floor/audit.py new file mode 100644 index 0000000000..f5fe0436f9 --- /dev/null +++ b/research/p112-batch-centre-floor/audit.py @@ -0,0 +1,334 @@ +#!/usr/bin/env python3 +"""Independent finite geometry audit; no imports from primary/predecessor code. + +Uses row sets for binary elimination and modular lower bounds plus exact +integer normal vectors for rational ranks. This is a finite audit, not a proof +assistant formalization of the all-k statement in README.md. + +alephnan with substantial OpenAI Codex assistance. Apache-2.0. +""" + +from collections import Counter +from itertools import combinations, product +from pathlib import Path +import argparse +import copy +import hashlib +import json + + +def need(value, reason): + if not value: + raise ValueError(reason) + + +def direct_geometry(): + """One positive-first vector from each antipodal doubled E8 root pair.""" + lines = set() + for first, second in combinations(range(8), 2): + for sign in (-1, 1): + line = [0] * 8 + line[first], line[second] = 2, 2 * sign + lines.add(tuple(line)) + for tail in product((-1, 1), repeat=7): + line = (1,) + tail + if sum(x < 0 for x in line) % 2 == 0: + lines.add(line) + frame = set() + for first in (0, 2, 4, 6): + for sign in (-1, 1): + line = [0] * 8 + line[first], line[first + 1] = 2, sign * 2 + frame.add(tuple(line)) + need(len(lines) == 120 and len(frame) == 8, "root/frame enumeration") + need(frame <= lines, "frame consists of roots") + need(all(dot(x, x) == 8 for x in lines), "doubled root norm") + need(all(dot(x, y) == 0 for x, y in combinations(frame, 2)), + "frame orthogonality") + return sorted(lines), sorted(frame), sorted(lines - frame) + + +def dot(left, right): + return sum(x * y for x, y in zip(left, right)) + + +def binary_basis(rows): + """Gaussian elimination as symmetric differences of coordinate sets.""" + pivots = {} + for source in rows: + row = set(source) + while row: + lead = min(row) + if lead in pivots: + row.symmetric_difference_update(pivots[lead]) + else: + pivots[lead] = row + break + return list(pivots.values()) + + +def binary_rank(rows): + return len(binary_basis(rows)) + + +def prime_rank(rows, prime=101): + """Rank over F_101, hence a rigorous lower bound for integer matrix rank/Q.""" + matrix = [[x % prime for x in row] for row in rows] + if not matrix: + return 0 + rank = 0 + for column in range(len(matrix[0])): + pivot = next((i for i in range(rank, len(matrix)) + if matrix[i][column]), None) + if pivot is None: + continue + matrix[rank], matrix[pivot] = matrix[pivot], matrix[rank] + factor = pow(matrix[rank][column], -1, prime) + matrix[rank] = [x * factor % prime for x in matrix[rank]] + for i in range(rank + 1, len(matrix)): + factor = matrix[i][column] + matrix[i] = [(x - factor * y) % prime + for x, y in zip(matrix[i], matrix[rank])] + rank += 1 + if rank == len(matrix[0]): + break + return rank + + +def pairing_rows(ports): + """B(r,s) = ( + 1) mod 2 in undoubled coordinates.""" + rows = [] + for left in ports: + row = set() + for j, right in enumerate(ports): + numerator = dot(left, right) + need(numerator % 4 == 0, "integral root inner product") + if (numerator // 4 + 1) % 2: + row.add(j) + rows.append(row) + return rows + + +def all_nonzero_patterns(rows): + basis = binary_basis(rows) + need(len(basis) == 9, "full binary pairing rank must be nine") + patterns = {frozenset()} + for row in basis: + patterns |= {pattern ^ frozenset(row) for pattern in patterns} + need(len(patterns) == 512, "complete binary functional space") + patterns.remove(frozenset()) + return sorted(patterns, key=lambda values: tuple(sorted(values))) + + +def audit_geometry(): + all_lines, frame, ports = direct_geometry() + need(len(ports) == 112, "live port count") + rows = pairing_rows(ports) + patterns = all_nonzero_patterns(rows) + rational_ranks = Counter() + size_ranks = Counter() + deficient = [] + canonical_patterns = [] + for pattern in patterns: + vectors = [ports[i] for i in sorted(pattern)] + lower = prime_rank(vectors) + normals = [normal for normal in frame + if all(dot(vector, normal) == 0 for vector in vectors)] + if normals: + need(len(normals) == 1 and lower == 7, + "a unique exact normal and modular rank seven") + rank = 7 # exact nonzero normal gives <=7; mod101 minor gives >=7. + deficient.append({"support": sorted(pattern), + "normal_doubled": list(normals[0])}) + else: + need(lower == 8, "full rational rank requires modular rank eight") + rank = 8 # there are exactly eight columns. + rational_ranks[rank] += 1 + size_ranks[(len(pattern), rank)] += 1 + canonical_patterns.append({"support": sorted(pattern), "rank": rank}) + need(rational_ranks == {7: 8, 8: 503}, "hyperplane rank census") + need({tuple(row["normal_doubled"]) for row in deficient} == set(frame), + "all deficient normals are exactly the omitted frame") + supports = [{i for i, root in enumerate(ports) if dot(root, normal) != 0} + for normal in frame] + need(all(len(support) == 56 for support in supports), "frame support size") + support_pair_ranks = [[binary_rank([rows[i] & right for i in left]) + for right in supports] for left in supports] + need(all(rank == 8 for row in support_pair_ranks for rank in row), + "all 64 frame support-pair ranks must be eight") + encoded = json.dumps(canonical_patterns, separators=(",", ":"), + sort_keys=True).encode("ascii") + return { + "root_lines": len(all_lines), "ports": len(ports), + "pairing_rank": binary_rank(rows), "functionals": len(patterns), + "rational_rank_histogram": {str(k): v for k, v in sorted(rational_ranks.items())}, + "size_rank_histogram": [[size, rank, count] + for (size, rank), count in sorted(size_ranks.items())], + "frame_doubled": [list(row) for row in frame], + "deficient": deficient, "support_pair_ranks": support_pair_ranks, + "canonical_patterns_sha256": hashlib.sha256(encoded).hexdigest(), + } + + +def mutation_controls(): + """Reject model changes which invalidate the finite premises.""" + _, frame, ports = direct_geometry() + rows = pairing_rows(ports) + changed = [set(row) for row in rows] + changed[0].symmetric_difference_update({0}) + need(binary_rank(changed) != 9, "single corrupted pairing bit is detected") + normal = frame[0] + flat = [root for root in ports if dot(root, normal) == 0] + outside = next(root for root in ports if dot(root, normal) != 0) + need(prime_rank(flat) == 7 and prime_rank(flat + [outside]) == 8, + "inserting an outside root destroys the deficient flat") + wrong_normal = frame[1] + need(any(dot(root, wrong_normal) != 0 for root in flat), + "a substituted null normal is rejected") + return ["corrupted pairing bit", "outside-root insertion", "wrong exact normal"] + + +def subset_determinant(matrix): + """Exact determinant by subset dynamic programming, not Bareiss/pivots.""" + size = len(matrix) + values = {0: 1} + for row in matrix: + following = {} + for mask, coefficient in values.items(): + for column in range(size): + if mask >> column & 1: + continue + inversions = (mask >> (column + 1)).bit_count() + value = coefficient * row[column] * (-1 if inversions % 2 else 1) + following[mask | (1 << column)] = following.get(mask | (1 << column), 0) + value + values = following + return values[(1 << size) - 1] + + +def bind_receipt(receipt, independent): + """Bind all finite hyperplanes/minors to RESULT, without primary imports. + + Cross-call fixtures and the prose theorem are deliberately outside this + geometry auditor's coverage; the primary verifier checks its fixtures. + """ + lines, expected_frame, ports = direct_geometry() + geometry = receipt["geometry"] + finite = receipt["finite_premises"] + frame = [tuple(row) for row in geometry["standard_frame"]] + need(len(frame) == 8 and set(frame) == set(expected_frame), "receipt frame") + images = {} + for line in lines: + numerators = [dot(line, normal) for normal in frame] + need(all(value % 4 == 0 for value in numerators), "frame-map integrality") + image = tuple(value // 4 for value in numerators) + first = next(value for value in image if value) + if first < 0: + image = tuple(-value for value in image) + need(image not in images, "injective frame-coordinate map") + images[image] = line + hamming = [tuple(row) for row in geometry["hamming_roots"]] + need(len(hamming) == 120 and set(hamming) == set(images), "receipt root geometry") + source_lines = [images[row] for row in hamming] + deleted = geometry["deleted_indices"] + need(len(deleted) == 8 and len(set(deleted)) == 8 + and {source_lines[i] for i in deleted} == set(frame), "deleted frame indices") + live = [i for i in range(120) if i not in deleted] + port_index = {root: i for i, root in enumerate(ports)} + source_to_port = {i: port_index[source_lines[i]] for i in live} + labels = finite["labels_in_dual_coordinates"] + need(len(labels) == 120 and all(type(label) is int and 0 < label < 512 + for label in labels), "nine-bit labels") + expected_patterns = {tuple(sorted(pattern)) + for pattern in all_nonzero_patterns(pairing_rows(ports))} + deficient = {tuple(row["support"]): tuple(row["normal_doubled"]) + for row in independent["deficient"]} + records = finite["hyperplanes"] + need(len(records) == 511 and {row["functional"] for row in records} == set(range(1, 512)), + "all 511 functionals exactly once") + seen_patterns = set() + exceptional = [] + exceptional_supports = {} + for row in records: + functional = row["functional"] + outside = [i for i in live if (labels[i] & functional).bit_count() % 2] + pattern = tuple(sorted(source_to_port[i] for i in outside)) + need(pattern in expected_patterns and pattern not in seen_patterns, + "receipt functional space equals full pairing row space") + seen_patterns.add(pattern) + rank = 7 if pattern in deficient else 8 + need(row["outside_count"] == len(outside) and row["rank"] == rank, + "receipt hyperplane count/rank") + minor_rows, columns = row["minor_rows"], row["minor_columns"] + need(len(minor_rows) == rank and len(set(minor_rows)) == rank + and set(minor_rows) <= set(outside), "minor rows in hyperplane complement") + need(len(columns) == rank and len(set(columns)) == rank + and set(columns) <= set(range(8)), "minor columns") + determinant = subset_determinant([[hamming[i][j] for j in columns] + for i in minor_rows]) + need(determinant != 0 and determinant == row["minor_determinant"], + "exact minor determinant") + if rank == 7: + coordinate = row["normal_coordinate"] + need(type(coordinate) is int and 0 <= coordinate < 8 + and frame[coordinate] == deficient[pattern], "exceptional exact normal") + support = [i for i in live if hamming[i][coordinate] != 0] + need(row["support"] == support, "exceptional support") + label_rank = binary_rank([{j for j in range(9) if labels[i] >> j & 1} + for i in support]) + need(label_rank == row["support_label_rank"] == 8, "exceptional label rank") + exceptional.append(functional) + exceptional_supports[functional] = {source_to_port[i] for i in support} + need(seen_patterns == expected_patterns, "complete receipt support coverage") + need(finite["exceptional_functionals"] == exceptional, "exceptional functional order") + rows = pairing_rows(ports) + ranks = [[binary_rank([rows[i] & exceptional_supports[right] + for i in exceptional_supports[left]]) + for right in exceptional] for left in exceptional] + need(ranks == finite["exceptional_pairing_ranks"] + and all(rank == 8 for row in ranks for rank in row), "receipt exceptional pairings") + need(finite["hyperplane_count"] == finite["checked_integer_minors"] == 511, + "receipt finite coverage counts") + need(finite["exceptional_pairings_checked"] == 64 + and finite["complement_rank_histogram"] == {"7": 8, "8": 503}, "receipt rank summary") + return {"hyperplanes": 511, "exact_subset_determinants": 511, "pairings": 64} + + +def receipt_mutation_controls(receipt, independent): + changes = [ + ("missing functional", lambda value: value["finite_premises"]["hyperplanes"].pop()), + ("false rational rank", lambda value: value["finite_premises"]["hyperplanes"][0].__setitem__("rank", 8)), + ("false minor determinant", lambda value: value["finite_premises"]["hyperplanes"][0].__setitem__("minor_determinant", 1)), + ("false exceptional pairing", lambda value: value["finite_premises"]["exceptional_pairing_ranks"][0].__setitem__(0, 7)), + ] + for name, mutate in changes: + changed = copy.deepcopy(receipt) + mutate(changed) + try: + bind_receipt(changed, independent) + except ValueError: + continue + raise ValueError("accepted receipt mutation: " + name) + return [name for name, _ in changes] + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--json", action="store_true", help="print the independent receipt") + parser.add_argument("--result", type=Path, default=Path(__file__).with_name("RESULT.json"), + help="primary finite receipt to independently bind") + args = parser.parse_args() + result = audit_geometry() + result["mutation_controls"] = mutation_controls() + receipt = json.loads(args.result.read_text(encoding="utf-8")) + result["receipt_binding"] = bind_receipt(receipt, result) + result["mutation_controls"] += receipt_mutation_controls(receipt, result) + if args.json: + print(json.dumps(result, indent=2, sort_keys=True)) + else: + print("PASS_P112_BATCH_CENTRE_FLOOR_INDEPENDENT " + "functionals=511 rank7=8 rank8=503 support_pairs=64 " + "exact_minors=511 method=mod101-plus-integer-normals controls=7") + + +if __name__ == "__main__": + main() diff --git a/research/p112-batch-centre-floor/test_package.py b/research/p112-batch-centre-floor/test_package.py new file mode 100644 index 0000000000..09d4080ff3 --- /dev/null +++ b/research/p112-batch-centre-floor/test_package.py @@ -0,0 +1,81 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""CLI corruption checks on disposable copies; never edit predecessor inputs.""" +import hashlib +import json +from pathlib import Path +import shutil +import subprocess +import sys +import tempfile + +HERE = Path(__file__).resolve().parent + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def hashes(directory): + return {p.name: hashlib.sha256(p.read_bytes()).hexdigest() + for p in directory.iterdir() if p.is_file()} + + +def run(directory, script, expected_error=None): + result = subprocess.run([sys.executable, "-I", "-O", "-B", str(directory / script)], + capture_output=True, text=True, encoding="utf-8", timeout=45) + if expected_error is None: + need(result.returncode == 0, script + " positive failed: " + result.stderr) + else: + need(result.returncode != 0 and expected_error in result.stderr, + script + " mutation did not fail as intended: " + result.stdout + result.stderr) + + +def main(): + before = hashes(HERE) + # A sibling preserves repository-relative source lookup. The temporary + # directory is created here, validated here, then cleaned by its owner. + with tempfile.TemporaryDirectory(prefix=".p112-floor-test-", dir=HERE.parent) as name: + sandbox = Path(name).resolve() + need(sandbox.parent == HERE.parent.resolve() and sandbox != HERE, + "temporary copy must be a new package sibling") + for path in HERE.iterdir(): + if path.is_file(): + shutil.copy2(path, sandbox / path.name) + copied = hashes(sandbox) + run(sandbox, "verify.py") + run(sandbox, "audit.py") + need(hashes(sandbox) == copied, "positive default replay wrote files") + + result_path = sandbox / "RESULT.json" + source_path = sandbox / "SOURCES.json" + result_original, source_original = result_path.read_bytes(), source_path.read_bytes() + result = json.loads(result_original) + result["finite_premises"]["hyperplanes"][0]["minor_determinant"] += 1 + result_path.write_text(json.dumps(result), encoding="utf-8") + run(sandbox, "verify.py", "complete typed RESULT.json mismatch") + run(sandbox, "audit.py", "exact minor determinant") + result_path.write_bytes(result_original) + + sources = json.loads(source_original) + next(row for row in sources["files"] if row["usage"] == "input")["git_blob"] = "0" * 40 + source_path.write_text(json.dumps(sources), encoding="utf-8") + run(sandbox, "verify.py", "Git blob mismatch") + source_path.write_bytes(source_original) + + result = json.loads(result_original) + result["rectangular_coupling_fixture"]["cost"] -= 1 + result_path.write_text(json.dumps(result), encoding="utf-8") + run(sandbox, "verify.py", "complete typed RESULT.json mismatch") + result_path.write_bytes(result_original) + + text = result_original.decode("utf-8") + result_path.write_text('{"schema_version": 1,' + text[1:], encoding="utf-8") + run(sandbox, "verify.py", "duplicate JSON key") + need(hashes(HERE) == before, "package source bytes changed") + print("PASS_P112_BATCH_PACKAGE positive_relocation=2 mutation_groups=4 rejected_runs=5 read_only=1") + + +if __name__ == "__main__": + main() diff --git a/research/p112-batch-centre-floor/verify.py b/research/p112-batch-centre-floor/verify.py new file mode 100644 index 0000000000..3b9ecbe17e --- /dev/null +++ b/research/p112-batch-centre-floor/verify.py @@ -0,0 +1,544 @@ +#!/usr/bin/env python3 +# SPDX-License-Identifier: Apache-2.0 +"""Exact P112 batch-floor premises. Standard library; default read-only; -O safe. + +Predecessor code is hash-bound as provenance but never imported or executed. +--write regenerates only this package's complete deterministic RESULT.json. +""" +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +import itertools +import json +from pathlib import Path +import subprocess + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, indent=2, ensure_ascii=True) + "\n" + + +def digest(value): + return hashlib.sha256(canonical(value).encode()).hexdigest() + + +def unique_object(pairs): + result = {} + for key, value in pairs: + require(key not in result, "duplicate JSON key: " + key) + result[key] = value + return result + + +def read_json(path): + return json.loads(path.read_text(encoding="utf-8"), object_pairs_hook=unique_object, + parse_constant=lambda x: (_ for _ in ()).throw(ValueError(x))) + + +def source_pins(): + manifest = read_json(HERE / "SOURCES.json") + require(manifest.get("schema_version") == 1, "source schema") + rows, seen = [], set() + for item in manifest["files"]: + name = item["path"] + require(name not in seen, "duplicate source") + seen.add(name) + path = (ROOT / name).resolve() + require(path.is_relative_to(ROOT.resolve()), "external source") + if item["usage"] == "provenance": + command = ["git", "-C", str(ROOT), "show", manifest["base_commit"] + ":" + name] + process = subprocess.run(command, capture_output=True, check=False) + require(process.returncode == 0, "cannot read pinned historical provenance: " + name) + payload = process.stdout + binding = "historical Git blob at pinned base; current working file is not read" + else: + require(item["usage"] == "input" and path.is_file(), "missing/invalid input") + payload = path.read_bytes() + binding = "current immutable input bytes" + blob = hashlib.sha1(b"blob " + str(len(payload)).encode() + b"\0" + payload).hexdigest() + require(blob == item["git_blob"], "Git blob mismatch: " + name) + sha = hashlib.sha256(payload).hexdigest() + require(sha == item["sha256"], "source hash: " + name) + rows.append({"path": name, "sha256": sha, "git_blob": blob, "binding": binding}) + require(len(rows) >= 3, "empty source coverage") + return {"base_commit": manifest["base_commit"], "files": rows, + "hash_domain": "SHA256 of exact bytes; historical provenance also bound to its Git blob"} + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def binary_basis(rows): + basis, selected = {}, [] + for index, value in enumerate(rows): + v = value + while v: + pivot = v.bit_length() - 1 + if pivot in basis: + v ^= basis[pivot] + else: + basis[pivot] = v + selected.append(index) + break + return selected + + +def binary_rank(rows): + return len(binary_basis(rows)) + + +def rational_basis(rows): + """Fraction elimination, returning indices of original witness rows.""" + pivots, selected = {}, [] + for index, row in enumerate(rows): + v = list(map(Fraction, row)) + for column, basis in sorted(pivots.items()): + if v[column]: + q = v[column] + v = [x - q * y for x, y in zip(v, basis)] + column = next((j for j, x in enumerate(v) if x), None) + if column is not None: + q = v[column] + pivots[column] = [x / q for x in v] + selected.append(index) + return selected, sorted(pivots) + + +def determinant(matrix): + """Independent integer Bareiss determinant, applied to original-row minors.""" + n = len(matrix) + require(all(len(row) == n for row in matrix), "nonsquare minor") + if not n: + return 1 + a, previous, sign = [row[:] for row in matrix], 1, 1 + for k in range(n - 1): + pivot = next((i for i in range(k, n) if a[i][k]), None) + if pivot is None: + return 0 + if pivot != k: + a[k], a[pivot] = a[pivot], a[k] + sign = -sign + value = a[k][k] + for i in range(k + 1, n): + for j in range(k + 1, n): + numerator = value * a[i][j] - a[i][k] * a[k][j] + require(numerator % previous == 0, "Bareiss nonexact division") + a[i][j] = numerator // previous + a[i][k] = 0 + previous = value + return sign * a[-1][-1] + + +def geometry(): + roots = [tuple(2 * (i == j) for i in range(8)) for j in range(8)] + planes = [p for p in itertools.combinations(range(8), 4) + if p[0] ^ p[1] ^ p[2] ^ p[3] == 0] + require(len(planes) == 14, "affine plane count") + for plane in planes: + for tail in itertools.product((-1, 1), repeat=3): + row = [0] * 8 + for j, sign in zip(plane, (1,) + tail): + row[j] = sign + roots.append(tuple(row)) + return roots, planes + + +def positive_line(row): + return tuple(row if next(x for x in row if x) > 0 else [-x for x in row]) + + +def check_geometry(roots): + require(len(roots) == 120 and len(set(roots)) == 120, "120 distinct roots") + require(all(len(row) == 8 and dot(row, row) == 4 for row in roots), "root norm") + require(all(dot(a, b) % 2 == 0 for a in roots for b in roots), "Gram integrality") + require({dot(a, b) // 2 for i, a in enumerate(roots) for j, b in enumerate(roots) + if i != j} == {-1, 0, 1}, "off-diagonal Gram") + nrows = [sum(((1 + dot(a, b) // 2) & 1) << j for j, b in enumerate(roots)) for a in roots] + require(binary_rank(nrows) == 9 and len(set(nrows)) == 120, "rank nine / distinct labels") + require(all((nrows[i] >> i) & 1 for i in range(120)), "label diagonal") + return nrows + + +def geometry_receipt(roots, planes, nrows): + # Independent standard-coordinate signed-root generation and explicit map. + signed = [] + for i, j in itertools.combinations(range(8), 2): + for a, b in itertools.product((-2, 2), repeat=2): + row = [0] * 8 + row[i], row[j] = a, b + signed.append(tuple(row)) + signed.extend(row for row in itertools.product((-1, 1), repeat=8) + if sum(x == -1 for x in row) % 2 == 0) + require(len(signed) == 240, "standard signed roots") + standard = sorted({positive_line(row) for row in signed}) + frame = [] + for a in range(0, 8, 2): + for sign in (1, -1): + row = [0] * 8 + row[a], row[a + 1] = 2, 2 * sign + frame.append(row) + require(all(dot(a, b) == 8 * (i == j) for i, a in enumerate(frame) + for j, b in enumerate(frame)), "standard frame") + images, raw_images = [], [] + for row in standard: + require(all(dot(row, f) % 4 == 0 for f in frame), "frame map integral") + raw = [dot(row, f) // 4 for f in frame] + raw_images.append(raw) + images.append(positive_line(raw)) + require(set(images) == set(roots), "two root constructions disagree") + for i, a in enumerate(standard): + for j, b in enumerate(standard): + require(2 * dot(raw_images[i], raw_images[j]) == dot(a, b), "exact scaled Gram identity") + require(((1 + dot(a, b) // 4) & 1) == ((1 + dot(images[i], images[j]) // 2) & 1), + "pairing isometry") + require(len(rational_basis(roots[8:])[0]) == 8, "live root rank") + require(all(sum(row[i] * row[j] for row in roots[8:]) == 56 * (i == j) + for i in range(8) for j in range(8)), "live Gram X^T X=56I") + return {"hamming_roots": [list(r) for r in roots], "affine_planes": [list(p) for p in planes], + "standard_roots_sha256": digest(standard), "standard_frame": frame, + "standard_to_hamming": "h_j=(q dot f_j)/4; normalize line sign", + "exact_scaled_Gram_identity": "2*(h(q) dot h(q')) = q dot q' before line-sign normalization", + "exact_scaled_Gram_entries_checked": 14400, + "label_pairing_entries_checked": 14400, "label_pairing_rank": 9, + "label_pairing_sha256": digest(nrows), "live_ports": 112, + "deleted_indices": list(range(8)), "live_root_rank": 8, + "live_coordinate_Gram": [[56 * int(i == j) for j in range(8)] for i in range(8)]} + + +def source_geometry(roots, geom, sources): + name = "research/p112-single-lift/reference-geometry.json" + require(any(row["path"] == name for row in sources["files"]), "PR845 geometry must be pinned") + data = read_json(ROOT / name) + require(canonical(data["frame"]) == canonical(geom["standard_frame"]), "PR845 frame mismatch") + require(len(data["roots"]) == len(data["X"]) == len(data["labels"]) == len(data["duals"]) == 112, + "PR845 geometry lengths") + require(all(type(x) is int for row in data["roots"] + data["X"] for x in row), "integer source roots") + require(all(type(x) is int and 0 < x < 512 for x in data["labels"] + data["duals"]), "source labels") + require(all(len(row) == 8 for row in data["roots"] + data["X"]), "source coordinate widths") + images = [] + for row, expected in zip(data["roots"], data["X"]): + require(all(dot(row, f) % 4 == 0 for f in data["frame"]), "source image integrality") + image = [dot(row, f) // 4 for f in data["frame"]] + require(image == expected, "PR845 X compatibility") + images.append(positive_line(image)) + require(len(set(images)) == 112 and set(images) == set(roots[8:]), "PR845 live-set compatibility") + for i, a in enumerate(data["roots"]): + for j, b in enumerate(data["roots"]): + inherited = (data["duals"][i] & data["labels"][j]).bit_count() & 1 + require(inherited == ((1 + dot(a, b) // 4) & 1), "PR845 labels/duals compatibility") + require(inherited == ((1 + dot(images[i], images[j]) // 2) & 1), "PR845 pairing compatibility") + return {"path": name, "live_roots_matched": 112, "X_entries_checked": 896, + "label_dual_pairing_entries_checked": 12544, + "imports_predecessor_code": False} + + +def pairing_rank(nrows, readers, sources): + mask = sum(1 << j for j in set(sources)) + return binary_rank([nrows[i] & mask for i in set(readers)]) + + +def check_minor(roots, indices, columns, expected): + value = determinant([[roots[i][j] for j in columns] for i in indices]) + require(value != 0 and value == expected, "nonzero exact minor") + + +def census(roots, nrows): + basis = binary_basis(nrows) + require(len(basis) == 9, "label basis count") + labels = [sum(((row >> index) & 1) << j for j, index in enumerate(basis)) for row in nrows] + require(binary_rank(labels) == 9, "label coordinate map injective") + records, exceptional = [], [] + for lam in range(1, 512): + outside = [i for i in range(8, 120) if (labels[i] & lam).bit_count() & 1] + selected, columns = rational_basis([roots[i] for i in outside]) + indices = [outside[j] for j in selected] + rank = len(indices) + require(rank in (7, 8), "complement rank") + value = determinant([[roots[i][j] for j in columns] for i in indices]) + check_minor(roots, indices, columns, value) + row = {"functional": lam, "outside_count": len(outside), "rank": rank, + "minor_rows": indices, "minor_columns": columns, "minor_determinant": value} + if rank == 7: + normals = [j for j in range(8) if all(roots[i][j] == 0 for i in outside)] + require(len(normals) == 1, "unique missing-coordinate normal") + normal = normals[0] + inside = [i for i in range(8, 120) if not ((labels[i] & lam).bit_count() & 1)] + require(inside == [i for i in range(8, 120) if roots[i][normal]], "exceptional support equality") + require(binary_rank([nrows[i] for i in inside]) == 8, "exceptional support dimension") + row.update({"normal_coordinate": normal, "support": inside, "support_label_rank": 8}) + exceptional.append(row) + records.append(row) + require(Counter(r["rank"] for r in records) == {8: 503, 7: 8}, "rank histogram") + require(sorted(r["normal_coordinate"] for r in exceptional) == list(range(8)), "missing normals") + pairings = [[pairing_rank(nrows, a["support"], b["support"]) for b in exceptional] for a in exceptional] + require(all(v == 8 for row in pairings for v in row), "64 exceptional pairings rank eight") + return {"basis_root_indices": basis, "labels_in_dual_coordinates": labels, + "hyperplanes": records, "exceptional_functionals": [r["functional"] for r in exceptional], + "exceptional_pairing_ranks": pairings, "hyperplane_count": 511, + "complement_rank_histogram": {"7": 8, "8": 503}, "checked_integer_minors": 511, + "exceptional_pairings_checked": 64} + + +def identity(n): + return [[int(i == j) for j in range(n)] for i in range(n)] + + +def shear_pair(a, b, source, target, coefficient): + for row in a: + row[target] += coefficient * row[source] + for row in b: + row[source] -= coefficient * row[target] + + +def root_matrix(roots, k): + return [[x if block == call else 0 for block in range(k) for x in root] + for call in range(k) for root in roots[8:]] + + +def multiply(a, b): + columns = list(zip(*b)) + return [[dot(row, col) for col in columns] for row in a] + + +def transpose(a): + return [list(row) for row in zip(*a)] + + +def check_product(d, t, target): + require(len(d) == len(target) and len(t) == len(target[0]), "factor dimensions") + require(d and t and len(d[0]) == len(t[0]), "factor width") + for i, row in enumerate(d): + for j, other in enumerate(t): + require(dot(row, other) == target[i][j], "full identity including cross-call entries") + + +def term_costs(d, t, nrows): + result = [] + for column in range(len(d[0])): + readers = {8 + i % 112 for i, row in enumerate(d) if row[column]} + sources = {8 + i % 112 for i, row in enumerate(t) if row[column]} + result.append(pairing_rank(nrows, readers, sources)) + return result + + +def aligned_columns(coefficients): + """A block-root-code column is aligned iff all blocks use one root axis.""" + return [len({row % 8 for row in range(len(coefficients)) if coefficients[row][column]}) == 1 + for column in range(len(coefficients[0]))] + + +def check_exact_cost_law(a, b, costs): + left, right = aligned_columns(a), aligned_columns(b) + predicted = [8 if x or y else 9 for x, y in zip(left, right)] + require(costs == predicted, "exact 8m plus two-sided mixing charge") + return {"reader_aligned": left, "source_aligned": right, + "two_sided_mixed_terms": sum(not (x or y) for x, y in zip(left, right))} + + +def fixture(roots, nrows, k, mode): + n = 8 * k + a, b, operations = identity(n), identity(n), [] + if mode.startswith("shared"): + require(k >= 2, "shared fixture requires independent calls") + for block in range(1, k): + for j in range(8): + for source, target, coefficient in ((j, 8 * block + j, 1), (8 * block + j, j, -1)): + shear_pair(a, b, source, target, coefficient) + operations.append([source, target, coefficient]) + if mode == "shared_cross_coordinate": + shear_pair(a, b, 0, 9, 2) + operations.append([0, 9, 2]) + if mode == "two_sided_coordinate_mixing": + # Only column one has full label span on BOTH sides. The other + # two changed columns each retain one aligned side: exact cost65. + for i, row in enumerate(((1, 1, 0), (0, 1, 1), (0, 0, 1))): + a[i][:3] = row + for i, row in enumerate(((1, 0, 0), (-1, 1, 0), (1, -1, 1))): + b[i][:3] = row + if mode == "overcomplete": + for i in range(n): + a[i].append(a[i][0]) + b[i].append(-b[i][0]) + b[i][0] *= 2 + require(multiply(a, transpose(b)) == identity(n), "coefficient factorization") + x = root_matrix(roots, k) + d, t = multiply(x, a), multiply(x, b) + target = multiply(x, transpose(x)) + check_product(d, t, target) + costs = term_costs(d, t, nrows) + exact_cost = check_exact_cost_law(a, b, costs) + if mode == "two_sided_coordinate_mixing": + require(sum(costs) == 64 * k + 1 and exact_cost["two_sided_mixed_terms"] == 1, + "one-unit strict mixing charge") + require(all(c >= 8 for c in costs) and sum(costs) >= 64 * k, "batch floor") + if mode in ("canonical", "shared_same_coordinate"): + require(costs == [8] * n, "attaining fixture") + mixed = [c for c in range(len(a[0])) if + len({i // 112 for i, row in enumerate(d) if row[c]}) > 1 or + len({i // 112 for i, row in enumerate(t) if row[c]}) > 1] + if mode.startswith("shared"): + require(bool(mixed), "fixture must mix independent calls") + return {"calls": k, "mode": mode, "terms": len(costs), "term_costs": costs, + "cost": sum(costs), "lower_bound": 64 * k, "coefficient_A": a, "coefficient_B": b, + "exact_cost_law": exact_cost, + "paired_shears": operations, "mixed_call_columns": mixed, + "matrix_entries_checked": len(d) * len(t), + "off_diagonal_call_entries_checked": k * (k - 1) * 112 * 112, + "factors_sha256": digest([d, t]), "target_sha256": digest(target)} + + +def rectangular_fixture(roots, nrows): + # Three output blocks, two independent input blocks, rank(C)=2. + # Compute the target directly from C and root dot products, independently + # of the attaining coefficient factors used to generate D and T. + coupling = [[1, 0], [0, 1], [1, -2]] + a = [[coupling[block][term // 8] * int(axis == term % 8) + for term in range(16)] for block in range(3) for axis in range(8)] + b = identity(16) + d = multiply(root_matrix(roots, 3), a) + t = multiply(root_matrix(roots, 2), b) + target = [[coupling[i // 112][j // 112] * dot(roots[8 + i % 112], roots[8 + j % 112]) + for j in range(224)] for i in range(336)] + check_product(d, t, target) + costs = term_costs(d, t, nrows) + require(costs == [8] * 16, "rectangular coupling attainment") + exact_cost = check_exact_cost_law(a, b, costs) + return {"C": coupling, "rank_C": len(rational_basis(coupling)[0]), + "input_blocks": 2, "output_blocks": 3, "terms": 16, + "cost": sum(costs), "term_costs": costs, "exact_cost_law": exact_cost, + "matrix_entries_checked": 336 * 224, "factors_sha256": digest([d, t])} + + +def expect_failure(action, label): + try: + action() + except ValueError: + return label + raise ValueError("control did not reject: " + label) + + +def controls(roots, nrows, counts): + results = [] + corrupted = list(roots) + corrupted[8] = tuple(2 * x for x in roots[8]) + results.append(expect_failure(lambda: check_geometry(corrupted), "root norm mutation")) + bad_pairing = nrows[:] + bad_pairing[0] ^= 1 + results.append(expect_failure(lambda: require(bad_pairing == check_geometry(roots), "pairing definition"), + "binary pairing mutation")) + results.append(expect_failure(lambda: require([r["functional"] for r in counts["hyperplanes"][:-1]] == + list(range(1, 512)), "hyperplane coverage"), "omitted hyperplane")) + first = counts["hyperplanes"][0] + bad_indices = first["minor_rows"][:] + bad_indices[-1] = bad_indices[0] + results.append(expect_failure(lambda: check_minor(roots, bad_indices, first["minor_columns"], + first["minor_determinant"]), "duplicate minor row")) + x = root_matrix(roots, 2) + target = multiply(x, transpose(x)) + results.append(expect_failure(lambda: check_product([r[1:] for r in x], [r[1:] for r in x], target), + "omitted rank-one summand")) + leaked = [row[:] for row in x] + for row in leaked: + row[0] += row[8] + results.append(expect_failure(lambda: check_product(leaked, x, target), "cross-call crosstalk")) + wrong = multiply(leaked, transpose(x)) + require(all(wrong[i][j] == target[i][j] for i in range(224) for j in range(224) + if i // 112 == j // 112), "crosstalk must pass diagonal-only check") + # Exact factorization outside the root code. Its final term has cost one. + d, t = [row[:] for row in x], [row[:] for row in x] + for i in range(224): + z = int(i == 0) + d[i][0] += z + d[i].append(-z) + t[i].append(t[i][0]) + check_product(d, t, target) + outside_costs = term_costs(d, t, nrows) + require(outside_costs[-1] == 1, "outside-code cost-one counterexample") + require(len(rational_basis(roots[9:])[0]) == 8, "singleton outside-code premise") + results.append(expect_failure(lambda: require(all(c >= 8 for c in outside_costs), "root-code domain"), + "outside-root-code exact factorization")) + # Correlated two-input/two-output demand C=ones(2,2), rank(C)=1. + single = root_matrix(roots, 1) + repeated = single + single + correlated_target = multiply(repeated, transpose(repeated)) + check_product(repeated, repeated, correlated_target) + correlated_costs = term_costs(repeated, repeated, nrows) + require(correlated_costs == [8] * 8, "rank-one demand attaining fixture") + require(correlated_target != target, "correlated endpoint differs") + results.append(expect_failure(lambda: check_product(repeated, repeated, target), "correlated demand substituted")) + # Exact Z[i] unit rephasing: D_0*=i, T_0*=-i, with literal integer pairs. + phase_d = [[(0, x) if c == 0 else (x, 0) for c, x in enumerate(row)] for row in single] + phase_t = [[(0, -x) if c == 0 else (x, 0) for c, x in enumerate(row)] for row in single] + phase_entries = 0 + for i, a in enumerate(phase_d): + for j, b in enumerate(phase_t): + real = sum(ar*br - ai*bi for (ar, ai), (br, bi) in zip(a, b)) + imaginary = sum(ar*bi + ai*br for (ar, ai), (br, bi) in zip(a, b)) + require((real, imaginary) == (dot(single[i], single[j]), 0), "Gaussian ordinary-transpose phase") + phase_entries += 1 + results.append(expect_failure(lambda: require(all(dot(a, b) - 2*a[0]*b[0] == dot(a, b) + for a in single for b in single), "Hermitian mismatch"), + "conjugate transpose substituted")) + results.append(expect_failure(lambda: require(canonical({"count": True}) == canonical({"count": 1}), + "typed comparison"), "boolean replaces integer receipt")) + return {"rejected": results, "count": len(results), + "outside_code": {"entries_checked": 50176, "terms": len(outside_costs), + "term_costs": outside_costs, "cost": sum(outside_costs), + "new_term_cost": 1, "factors_sha256": digest([d, t]), + "not_a_cheaper_decoder_claim": True}, + "correlated_demand": {"C": [[1, 1], [1, 1]], "rank_C": 1, + "input_blocks": 2, "output_blocks": 2, "cost": 64, + "terms": 8, "entries_checked": 50176}, + "Gaussian_unit_phase_entries_checked": phase_entries, + "crosstalk_passes_diagonal_only": True} + + +def generate(): + sources = source_pins() + roots, planes = geometry() + nrows = check_geometry(roots) + geom = geometry_receipt(roots, planes, nrows) + source_match = source_geometry(roots, geom, sources) + counts = census(roots, nrows) + fixtures = [fixture(roots, nrows, k, "canonical") for k in (1, 2, 3)] + fixtures.extend(fixture(roots, nrows, k, "shared_same_coordinate") for k in (2, 3)) + fixtures.append(fixture(roots, nrows, 2, "shared_cross_coordinate")) + fixtures.append(fixture(roots, nrows, 2, "overcomplete")) + fixtures.append(fixture(roots, nrows, 1, "two_sided_coordinate_mixing")) + return {"schema_version": 1, "claim": "Common-chart block-root-code copy optimum 64 rank(C); cost 8m+b; independent-call floor 64k", + "scope": "all-pairs block-diagonal exactness; union support pricing; separate scalar totals", + "sources": sources, "geometry": geom, "source_geometry_compatibility": source_match, + "finite_premises": counts, + "fixtures": fixtures, "controls": controls(roots, nrows, counts), + "rectangular_coupling_fixture": rectangular_fixture(roots, nrows), + "fixture_entries_checked": sum(f["matrix_entries_checked"] for f in fixtures), + "fixture_off_diagonal_entries_checked": sum(f["off_diagonal_call_entries_checked"] for f in fixtures)} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write", action="store_true", help="regenerate only RESULT.json") + args = parser.parse_args() + result = generate() + path = HERE / "RESULT.json" + if args.write: + path.write_text(canonical(result), encoding="utf-8", newline="\n") + else: + require(canonical(read_json(path)) == canonical(result), "complete typed RESULT.json mismatch") + print("PASS_P112_BATCH_CENTRE_FLOOR: 511 exact hyperplanes; 64 exceptional pairings; " + + str(result["fixture_entries_checked"]) + " fixture entries; " + + str(result["controls"]["count"]) + " controls; charge >=64k") + + +if __name__ == "__main__": + main()