diff --git a/CITATION.cff b/CITATION.cff index 25e22bb898..4491d3e9f9 100644 --- a/CITATION.cff +++ b/CITATION.cff @@ -12,11 +12,12 @@ repository-code: "https://github.com/CrocSwap/integer-mult-bounds" url: "https://github.com/CrocSwap/integer-mult-bounds" abstract: >- Exact rational certificates, source patches, and research notes for conditional - improvements to OpenAI's integer-multiplication manuscript. Direct nonadjacent - axis routing, shared finite circuits, auxiliary role sharing, an early-stopping - guard bound, and tighter Gaussian setup support kappa = 2^-59. An earlier - parameter-only witness gives kappa = 5.8e-33, near a certified ceiling under - the original cost accounting. The complete upstream algorithmic theorem is assumed. + improvements to OpenAI's integer-multiplication manuscript. Compact dirty-control + fields remove the simultaneous layer's spacing penalty. Together with complete + temporary reservations, local exceptional repair, a separately sized complex + network, and retained refinements, the supplied written proofs support + kappa = 83/10^12 > 2^-34. The complete upstream algorithmic theorem is assumed; + the new arguments are not independently reviewed or formally verified. keywords: - integer multiplication - computational complexity diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md index 0fd45db261..44f0a8fd53 100644 --- a/CONTRIBUTING.md +++ b/CONTRIBUTING.md @@ -6,7 +6,9 @@ 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. -The highest-value outstanding questions are listed in [the audit](docs/audit.md). +For the current result, start with the +[compact-control review guide](docs/research/compact-control-review.md). +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 @@ -19,7 +21,8 @@ make verify ``` Include regenerated certificates and patches in the same change. If the note -changes, also run `make note` and include the updated PDF. Review changes to +changes, also run its corresponding PDF target (`make compact-note` for the +current result) and include the updated PDF. 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. diff --git a/Makefile b/Makefile index 3cd0e24fe4..bb60f53aef 100644 --- a/Makefile +++ b/Makefile @@ -1,4 +1,4 @@ -.PHONY: verify note audit-note tuned-note reuse-note incidence-note dag-note shared-point-note paired-note fetch +.PHONY: verify note audit-note tuned-note reuse-note incidence-note dag-note shared-point-note paired-note compact-note complex-note fast-gaussian-note aligned-bit-note fetch verify: python3 scripts/certify.py @@ -22,6 +22,39 @@ verify: python3 scripts/make_shared_point_patch.py python3 scripts/paired_network.py python3 scripts/make_paired_patch.py + python3 scripts/prepare_layers.py + python3 scripts/audit_sparse_fusion.py + python3 scripts/audit_fused_block.py + python3 scripts/audit_short_guards.py + python3 scripts/audit_gather_schedules.py + python3 scripts/audit_coded_carries.py + python3 scripts/audit_cancellation.py + python3 scripts/audit_joint_frames.py + python3 scripts/audit_compact_controls.py + python3 scripts/compact_control_layer.py + python3 scripts/make_compact_control_patch.py + python3 scripts/complex_network.py + python3 scripts/make_complex_circuit_patch.py + python3 scripts/fast_gaussian.py + python3 scripts/make_fast_gaussian_patch.py + python3 scripts/aligned_bit_network.py + python3 scripts/make_aligned_bit_patch.py + python3 scripts/prime_field_network.py + python3 scripts/make_prime_field_patch.py + python3 scripts/audit_kappa_targets.py + python3 scripts/experiments/bit_breakthrough_affine_centers.py + python3 scripts/experiments/label_breakthrough_polynomial_screen.py + python3 scripts/experiments/assembly_breakthrough_guard.py --output certificates/assembly-breakthrough-guard.json + python3 scripts/experiments/ternary_direction.py + python3 scripts/experiments/ternary_depth_exchange.py --output certificates/ternary-depth.json + python3 scripts/experiments/ternary_reuse_core2.py + python3 scripts/ternary_reuse_network.py + python3 scripts/ternary_fused_network.py + python3 scripts/batched_stream_network.py + python3 scripts/dimension30_stream_network.py + python3 scripts/source_frame_stream_network.py --producer certificates/dimension30-stream-producer.json + python3 scripts/experiments/complex_all_residuals.py --output certificates/complex-all-residuals.json + python3 scripts/experiments/batched_role_targets.py --output certificates/batched-role-targets.json python3 scripts/audit_scratch_pooling.py python3 scripts/reuse_network.py python3 scripts/make_reuse_patch.py @@ -42,6 +75,11 @@ verify: git apply --check --directory=upstream patches/h46-dag-63.patch git apply --check --directory=upstream patches/h46-shared-point.patch git apply --check --directory=upstream patches/h50-paired-59.patch + git apply --check --directory=upstream patches/compact-control-34.patch + git apply --check --directory=upstream patches/complex-circuit-31.patch + git apply --check --directory=upstream patches/fast-gaussian-30.patch + git apply --check --directory=upstream patches/aligned-bit-30.patch + git apply --check --directory=upstream patches/prime-field28.patch note: mkdir -p artifacts @@ -75,5 +113,21 @@ paired-note: mkdir -p artifacts tectonic --outdir artifacts notes/paired-note.tex +compact-note: + mkdir -p artifacts + tectonic --outdir artifacts notes/compact-control-note.tex + +complex-note: + mkdir -p artifacts + tectonic --outdir artifacts notes/complex-circuit-note.tex + +fast-gaussian-note: + mkdir -p artifacts + tectonic --outdir artifacts notes/fast-gaussian-note.tex + +aligned-bit-note: + mkdir -p artifacts + tectonic --outdir artifacts notes/aligned-bit-note.tex + fetch: python3 scripts/fetch_upstream.py diff --git a/NOTICE b/NOTICE index 71becbdd19..321d87f67a 100644 --- a/NOTICE +++ b/NOTICE @@ -14,3 +14,49 @@ as separate files under patches/; upstream source files remain unchanged. The new research note, scripts, certificates, patches, and documentation were prepared by Douglas Colkitt with assistance from OpenAI Codex. This project is not an official OpenAI release or an OpenAI endorsement. + +Ternary five-subset construction and common-four-point resynthesis +Copyright 2026 Zhihao Chen (jacklightChen) + +The prime-field28 construction, paired triple producer, exact support checker, +and associated tests and artifacts are imported unchanged from PR #7: +https://github.com/CrocSwap/integer-mult-bounds/pull/7 +Commit: 6725c6a17b17871a35353fd29157f4ed851bc114 +Research, proof development, code and verification were prepared with +substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by the +contributor. This is not independent review or an OpenAI endorsement. +Follow-on research retains this attribution and cites that PR and its note. + +The inherited complex interface and fast Gaussian resampling are eumemic's +Claude-assisted contributions in PRs #3 and #5. Aligned pairing also appears +in Bortlesboat's PR #2; retained totals and bank sharing in dleen's PR #4; +and cheaper retained centers in eumemic's PR #6. These precede and remain +separately credited from the new ternary follow-up work on this branch. + +Batched recursive widths and controlled projector bases +Copyright 2026 IceKylin (icekylinx) + +The unmodified reference files under references/pr10 are from PR #10: +https://github.com/CrocSwap/integer-mult-bounds/pull/10 +Commit: 62691e395a0458ce089a1c7b5d89e74291e95e29 +Their original NOTICE and license are retained there. The local batching +experiments adapt that contribution's rank moments and assembly, including +its corrected Gaussian precision enclosure, to the smaller producer. + +Even-dimension bank matching and dimension-30 batching + +The unmodified references/pr12 sources are from Rohan Arun's Codex-assisted +continuation in PR #12, incorporating the matching extension from PR #11: +https://github.com/CrocSwap/integer-mult-bounds/pull/12 +Commit: 35d31e30f28bc5da0ae6a88e7b03d75ebc855534 +The original attribution and license remain in that reference directory. + +Auxiliary source frames and extended projector basis + +The unmodified references/pr13 sources are from eumemic's PR #13, prepared +with substantial assistance from Claude (Anthropic): +https://github.com/CrocSwap/integer-mult-bounds/pull/13 +Commit: 3ef246fa4f69c87ebfed78376418afa9ffcad145 +The source-frame composition on this branch uses that contribution's +auxiliary endpoint relocation and basis argument. Its original NOTICE and +license are retained with the pinned sources. diff --git a/README.md b/README.md index a98467cbd3..ff690ad51f 100644 --- a/README.md +++ b/README.md @@ -1,287 +1,214 @@ -# A sharper exponent for integer multiplication - -**Research draft by Douglas Colkitt — conditional on the underlying manuscript.** - -This repository improves the parameters, axis routing, and finite bit network in OpenAI's -[*Integer multiplication below n log n*](https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026) -(result family #109). Subject to its algorithmic interfaces and the construction -changes audited here, the strongest supplied bound is - -$$ -T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad -\boxed{\kappa=2^{-59}\approx1.7347\times10^{-18}}. -$$ - -The computational model is the manuscript's fixed finite-alphabet Turing machine -with a fixed finite number of one-dimensional tapes. The original manuscript -uses `kappa = 2^-182`. The latest witness improves the preceding `13*2^-66` -bound by a factor of 128/13 (about 9.85), giving a 2^123-fold increase in -exponent saving over the original. -These are asymptotic exponent comparisons, not measured practical speedups. - -**[Read the 2^-59 proof note (PDF)](artifacts/paired-note.pdf)** · -[Review the latest patch](patches/h50-paired-59.patch) · -[Inspect the routing audit](docs/nonadjacent-axis-audit.md) · -[Reproduce the checks](docs/reproducibility.md) - -## Latest improvement: paired sums and tighter downstream estimates - -Three changes combine to reach `2^-59`: - -1. A paired-block circuit keeps internal edge sums as vertex weights through - the recursion. At `h=50`, cross-group sharing gives **509,194 side roles**, - down from 694,495 for the preceding circuit at the same ground size. -2. Using the actual early stopping depth proves a **quadratic guard width**, - replacing the conservative twentieth power. -3. The smaller Gaussian width `alpha=ceil((32*d*b)^(1/4))` meets the retained - resampling interface and permits a larger transform dimension. - -The exact minimum margin is - -$$ -G=\frac{272158569}{156250000000000000000000000}>2^{-59}. -$$ - -The [construction and dependency audit](docs/research/paired-network.md) -explains all three changes and the [certificate](certificates/paired-network.json) -checks their arithmetic, circuit coefficients, and both frame directions. -The strict margin is about 0.4%. Both motifs now use `h=50`; the complex motif -retains its original construction. The upstream theorem remains an assumption. - -## Preserved cross-group sharing refinement - -Equal sums can now be shared across different common-point groups. The reverse -computation uses orthogonal complements of the forward source spans, so it no -longer needs a separate positive-definiteness proof for reachable-target spans. -Every retained source span still has a common point and is positive definite. - -This removes 45,706 additions and reduces side roles from **577,576 to 531,870**. -The rank deficit is unchanged. Exact checks support `a_b=203/10^11` and -`G=17661/10^23 > 13*2^-66`, with the complex saving unchanged. See the -[construction and next-search budget](docs/research/cross-point-sharing.md) -and [certificate](certificates/shared-point-network.json). With its then-retained guard and assembly bounds, that circuit fell short of -`2^-62`. The latest witness also changes those downstream bounds. - -## Preserved shared-computation improvement - -A cancellation-free computation graph shares intermediate sums across -outputs. Its reversible embedding uses **one role per output plus one per -binary addition**. Source supports provide nested forward frames; reachable -target supports provide nested frames for the reversed second stage. -All these spans are nondegenerate because each local circuit's triples share -a common point. Thus the new computation introduces no extra backward rank. - -At `h=46`, side roles per invocation fall from **2,394,438 to 577,576**, -preserving the absolute rank deficit. The certified bit saving is -`a_b=187/10^11`; the complex saving remains `a_c=9/500000000000`. The -existing parameter recipe yields - -$$ -G=\frac{3a_{\rm b}^2}{70}\approx1.4987\times10^{-19}>2^{-63}. -$$ +# Experiments to raise κ -The [construction audit](docs/research/shared-computation.md) proves the -reversible embedding and both frame assignments. The -[exact certificate](certificates/dag-network.json) checks every output -coefficient at the target size, all compiled support inclusions, and the -downstream inequalities. The compiler is reusable for other compatible -cancellation-free circuits; the latest witness builds on this transfer. +The current reproduced conditional witness is **κ = 1.076678e-6 > 2^-20**: +663× the starting aligned-bit branch, 39.8% above +[PR #13](https://github.com/CrocSwap/integer-mult-bounds/pull/13), and 18.6% +above [PR #14](https://github.com/CrocSwap/integer-mult-bounds/pull/14). +These compare asymptotic exponent savings, not practical runtimes. -## Preserved rectangle-circuit improvement +The construction combines a smaller h30 producer and bilateral wire reuse +(13,056,812 roles), PR #13's auxiliary source frames, and batching of every +complex residual. PR #10 supplies controlled batching and mixed-width +transfer; PR #12 supplies the even-dimension bank matching. -Neighboring triple pairs are partitioned into rectangles. A rectangle with -`a` sources and `b` targets uses an invertible mixer on `a+b-1` scratch roles, -instead of one role per pair. A common nondegenerate frame lets the mixer -share intermediate computations without adding backward rank. All auxiliary -roles, including centers, can then be shared between the first and third stages. +Reproduce with `python3 scripts/source_frame_stream_network.py`. +Python 3, a C++17 compiler, and several GB of memory are required. Exact +checks cover output supports, nondegenerate frames, physical allocation, +matching, recursive moments, and all assembly inequalities. The general +transfer arguments and inherited multiplication interfaces remain +mathematical dependencies; this is not formal verification. -At `h=46`, this reduces side roles per invocation from **41,122,620 to -2,394,438** while preserving the absolute rank deficit. The certified bit -saving is `a_b=46/10^11`; the complex saving remains `a_c=9/500000000000`. -The existing parameter recipe gives +[Working status](docs/research/current-status.md) · +[Construction](docs/research/source-frame-stream.md) · +[Certificate](certificates/source-frame-stream-witness.json) · +[Reproduction](docs/reproducibility.md) -$$ -G=\frac{3a_{\rm b}^2}{70}\approx9.0686\times10^{-21}>2^{-67}. -$$ +## Inherited project overview -The [construction and frame audit](docs/research/incidence-network.md) cover -dirty-scratch restoration, the reversed second stage, and every new frame -transition. The [exact certificate](certificates/incidence-network.json) -includes an exhaustive check of all 893,970 ordered disjoint pair instances -underlying the rectangle partition. This changes the scalar circuit and its -frames, so the bounds on pooling unchanged trajectories do not apply. +The overview below is preserved from the starting branch. Its authorship, +results, and publication artifacts describe that earlier work. -## Preserved stage-sharing improvement +# A sharper exponent for integer multiplication -An explicit matching lets the first and third stages share their side scratch -roles. Each invocation restores arbitrary initial scratch values. The matching -also makes the connecting frame labels nested, so sharing introduces no extra -rank loss. This removes one third of the side roles while preserving the -absolute rank deficit. The complex network is unchanged. +**Research draft by Douglas Colkitt — conditional on the underlying manuscript +and the written extensions supplied here.** -That witness uses bit saving `a_b = 27/10^12`, while the retained complex saving is -`a_c = 9/500000000000`. Set `tau = 1-a_b`, `sigma = 1-a_c`, and use the tuning -recipe below with `a = a_b`. The exact minimum margin is +This draft improves OpenAI's +[*Integer multiplication below n log n*](https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026) +(result family #109). In its fixed finite-alphabet Turing-machine model with a +fixed number of one-dimensional tapes, the strongest supplied witness is $$ -G=\frac{3a_{\rm b}^2}{70}\approx3.1243\times10^{-23}>2^{-75}. +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\boxed{\kappa=\frac{1624}{10^{12}}=1.624\times10^{-9}>2^{-30}}. $$ -The [proof](notes/stage-reuse-note.tex) accounts for scalar restoration, the new -edge ranks, every role's endpoint identity, and the separate primitive -exponents. The [certificate](certificates/stage-reuse.json) checks counts and -all downstream parameter inequalities. Finite tests check the full h=46 -matching, small rational projectors, and a complete h=6 shared-scratch circuit. -The earlier fixed-network ceiling does not apply to these changed role counts. - -## Preserved parameter improvement - -The new witness combines direct routing with the previously audited variable -stopping threshold. With `a = 9/500000000000`, set +The simpler **`kappa = 2^-30`** is a corollary; the witness remains below +`2^-29`. It is **1624/1479 ≈ 1.10 times** the fast-resampling witness +`1479/10^12`, **2.75 times** the compressed-complex witness `59/10^11`, and +about **19.6 times** the compact-control witness `83/10^12`. The original +manuscript uses `2^-182`. These compare asymptotic exponents, not practical +runtimes. + +**[Read the aligned-bit proof note (PDF)](artifacts/aligned-bit-note.pdf)** · +[fast-resampling note](artifacts/fast-gaussian-note.pdf) · +[Review the combined source patch](patches/aligned-bit-30.patch) · +[Inspect the exact certificate](certificates/aligned-bit-network.json) · +[Construction summary](docs/research/aligned-bit.md) + +This is a research claim supported by written proofs and reproducible checks. +The complete upstream theorem is assumed; the new arguments have not received +independent mathematical review or formal verification. + +## Latest improvement: an aligned bit circuit with cheaper centers + +With fast resampling, `kappa < a_b/2`, so only the bit network matters. Two +changes raise its saving from `296/10^11` to **`325/10^11`**: + +- **Cheaper centers.** Each center wire takes its value from the group total + computed inside the side circuit. Its label then rises only to the star span + `H_i` (dimension `h-1`) before the scatter, so it loses `h-1` instead of `h` + dimensions per invocation, in both stage directions. The deficit numerator + goes from `v - 6h^2` to `v - 6h(h-1)`. +- **Aligned side circuit.** Blocks `{2k, 2k+1}` are fixed across all + common-point groups, and each top-level pair-star chain is computed once for + the two groups that use it: **494,196 roles** per invocation including the + center wires, against 509,244. + +## Preserved fast Gaussian resampling + +With the compressed complex network the remaining cap was the Gaussian +resampling step. Its line maps cost `O(t p^(3/2+delta) alpha)`, which forces +the dimension exponent `epsilon < 1/5` and hence `kappa < a_b/5`. Two changes +remove that cap: + +- **Chirped correlations.** The identity + `(sigma a - b)^2 = sigma theta a^2 + sigma (a-b)^2 - theta b^2` + turns every block of Gaussian sums along a line into one correlation, + evaluated with the established integer multiplier in `O(p^(1+delta))` per + output, independently of the Gaussian width `alpha`. +- **Faster Neumann series.** A potential-function argument shows that the + powers of the correction `E = N - I` decay like `exp(-pi alpha^2 sigma n)` + after a burn-in of order `alpha^2/theta` bits. The Neumann series then needs + `O(p/alpha^2 + 1/theta)` terms instead of `p/(alpha^2 theta)`. + +With `alpha = floor(sqrt(b/(8d)))` the Gaussian cost per bit becomes +`O(d^2 p^delta)`, so `epsilon` may approach `1/2`. The compressed complex +network supplies the room for the larger guard parameter `beta = 19/25` this +requires. The new scoped ceiling for these networks is `kappa < a_b/2 < 2^-29`. + +## Preserved compressed complex network + +After compact control, the complex network was the binding motif. It still +used the original side wires, one per ordered neighbor pair: **3,693,800 side +wires per invocation** at `h=25`. A shared-sum circuit computes the same side +correction with **108,195 reversible roles**, raising the certified complex +saving from `418/10^12` to **`14/10^9`**. + +The complex labels are binary, and every frame residual needs an orthonormal +basis. Coordinate labels for disjoint sums and pair-star spans for +intersection-two sums satisfy this, with one necessary rule: an injected +disjoint piece must leave a point outside its target uncovered. Otherwise the +injection residual is an alternating hyperbolic plane with no orthonormal +basis. Exact checks cover every coefficient, label, inclusion and compiled +role in both stage directions. + +With the complex saving no longer binding, the compact-control recurrence has +internal exponent `chi = tau`, and the paired bit network sets the bound: +`kappa < a_b/5 < 2^-30` for the retained Gaussian margin. See the +[construction summary](docs/research/complex-circuit.md) for the remaining +ceiling and next targets. + +## Preserved compact-control movement + +The compact-control construction moves **compact control fields instead of entire spaced +windows**. For `f` selected axes, it replaces the layer's movement cost +`O(V*((f*K)^tau+1))` by $$ -\epsilon=\frac1{21},\quad \beta=\frac9{10},\quad c=\frac{9a}{10},\quad -\lambda=1-\frac{19a^2}{20},\quad \lambda'=1-\frac{9a^2}{10}. +O\!\left(V\bigl((f\log p)^\tau+1\bigr)\right). $$ -The network and routing schedule stay the same; the dimension and recursion -stopping parameters change. Exact checks give `G = 3a^2/70 > 2^-76`. -The guard remains sublinear: `epsilon C1 = 20/21 < 1`. -The [tuning note](artifacts/routing-tuned-note.pdf) audits composition, precision, -and setup, and gives a transferable recipe for `0 < a < 1/16` with compatible -primitive interfaces and `C1 = 20`. - -A simple necessary bound under those fixed primitive exponents and layer -constraints is `kappa < a^2/(20(1-a)) < 2^-75`. This is not an exact optimum -or a ceiling for stronger primitives or improved cost accounting. - -## Further search and its boundaries - -The [research log](docs/research/network-search-log.md) records the bounded -investigation that found stage sharing. The [scoped bounds](docs/research/network-screens.md) -exclude improvement from thinning the original triple family or using unequal -factor sizes, and limit one shared binary aggregation hierarchy. These are -family-specific results. The [next-step roadmap](docs/research/next-steps.md) -distinguishes concrete search targets from speculative stronger primitives. -The [block-label feasibility audit](docs/research/block-label-feasibility.md) -rules out several simple models for the proposed 2^-59 target; it does not -establish a new construction or a stronger multiplication bound. -The [additive-label obstruction](docs/research/additive-label-obstruction.md) -and [exact small control](docs/research/small-block-control.md) further narrow -the search; the small control does not have a positive network deficit. -The [quick 67 screen](docs/research/quick-67-screen.md) also excludes that -milestone from further pooling of the unchanged scratch trajectories. - -The [packed movement audit](docs/packed-movement-audit.md) reduces a constant -swap count and sharpens the layer recurrence, but finds no stronger exponent -certificate from those changes. The root cost still forces `c = O(a)` under -the audited estimates. It identifies the stronger sparse-bit movement bound -that would be needed to remove this restriction. The 2^-75 result comes from -the stronger network, not from that recurrence refinement. - -## The routing improvement and first witness - -The new routing schedules use direct nonadjacent swaps already supported by -the manuscript. CRT axis reversal needs floor(d/2) field interchanges; exposing -and restoring all resampling axes needs at most 2(d-1). This replaces the -original O(d^2) count of adjacent moves by O(d) full-field interchanges. -The finite network and numerical operations remain the same as in our earlier -h = 46 construction. The exact sequence of data-movement operations changes. - -The normalized layout cost improves from `d^2(1 + ell^tau)` to -`d(1 + ell^tau)`. Its exponent margin therefore becomes +The proof reserves temporary fields from existing address coordinates, +allows arbitrary initial temporary values, restores them exactly, and charges +exceptional-address repair at every recursion node. The temporary ranges +remain complete through padding and recursive row splitting. -$$ -g_4=(1-\tau)(1-\epsilon), -$$ +Removing `K^tau` removes the restriction responsible for the preceding +quadratic dependence on the finite-network saving. The bit network stays at +`h=50`. The original complex network is separately instantiated at `h=25`, +and a generalized stopping-depth guard completes the new parameter witness. +This is a change to the movement construction and its proof, beyond parameter +tuning of the preceding algorithm. -which permits a fixed dimension exponent `epsilon = 1/40`. With -`a = 9/500000000000`, the other choices are +The exact minimum assembly margin is $$ -\tau=\sigma=1-a,\quad \beta=\frac12,\quad c=\frac a2,\quad -\lambda=1-\frac{3a^2}{4},\quad \lambda'=1-\frac{a^2}{2},\quad -\delta=\frac1{16},\quad C_1=20. +G_* = \frac{333833}{4\cdot10^{15}} + = 8.345825\times10^{-11} > \kappa. $$ -Exact rational arithmetic gives +The remaining bottleneck at that stage was the complex layer's saving, which +the compressed complex network above removes. With the **fixed +`h=25` complex motif and retained Gaussian/leaf inequalities**, the scoped +ceiling is below `8.369598075e-11`, hence below `2^-33`. This is not a ceiling +for other networks or integer multiplication in general. -$$ -\min_i g_i=\frac{a^2}{80}=4.05\times10^{-24}>2^{-78}. -$$ +## Evidence and scope -The [routing audit](docs/nonadjacent-axis-audit.md) checks coordinate order, -padding, coefficient records, fixed tapes, descriptor costs, precision, and -all remaining parameter dependencies. The routing change itself uses primitives -already present in the source. With OpenAI's original network and recurrence -exponents unchanged, it also supports the simpler bound **kappa = 2^-107**. - -## The earlier parameter-only result and its ceiling - -Our [first note](artifacts/parameter-note.pdf) gives `kappa = 5.8e-33` under -the original layout cost accounting. Within the stated network-counting -family and those cost inequalities, it establishes `kappa < 5.838e-33`, even -allowing variable `beta` and unequal `tau` and `sigma`. The exact search covers -every admissible integer h, using a decreasing bound for the infinite tail. - -That ceiling remains valid for its stated assumptions. The original layout -cost forced `epsilon < a`, yielding a cubic constraint `kappa < a^3`. -The new schedules improve that cost and remove the restriction. With a fixed -epsilon, the supplied saving instead scales quadratically in a. We do not -claim that quadratic dependence is unavoidable or that the latest bound is optimal for -the revised accounting. - -## What has been checked - -| Component | Status | +| Component | Evidence | | --- | --- | -| Parameter inequalities and logarithm bounds | Exact rational certificates | -| Network-counting search and its infinite tail | Exact rational certificates | -| Nonadjacent routing, parameter dependencies, and earlier variable-beta extension | Written mathematical audits | -| Selected finite identities | Exhaustive small cases and deterministic sampled tests | -| Stage-sharing construction | Written frame proof, complete matching and finite scalar checks | -| Source patch application | All twelve alternatives checked | -| Full upstream multiplication theorem | Assumed; not independently established here | - -The scripts do not constitute a formal proof of the complete algorithm. This -repository contains no full multiplication-machine implementation or Lean -formalization of the claimed bound. Independent review is welcome, especially -of the finite-network interfaces and fixed-tape recurrence. See -[the audit's remaining work](docs/audit.md#what-remains-before-an-unconditional-claim). +| Parameters, logarithm enclosures, final margins | Exact rational certificate | +| Dirty-control identities, inverses and repair | Finite exhaustive cases and seeded tests | +| Wider-control tape bound, reservations and recursion | Written general proofs | +| Separate complex arity and precision guard | Written proofs and exact accounting | +| Compressed complex side circuit and binary frames | Exact full-size coefficient, label and role checks; written transfer proof | +| Source integration | Combined patch, reference checks and manuscript build | +| Full upstream multiplication theorem | Assumed | +| Independent review / full formalization | Not supplied | + +The [review guide](docs/research/compact-control-review.md) identifies the +compact-control proof obligations and their tests; the +[complex-circuit summary](docs/research/complex-circuit.md) lists the new ones. [Current research status](docs/research/current-status.md) +is authoritative when older notes describe superseded barriers or hypothetical +witnesses. The earlier artifacts remain available and unchanged. ## Reproduce -From the repository root, with Python 3.11 or newer, Git, and Make: +With Python 3.11 or newer, Git and Make, run from the repository root: ```sh make verify +git diff --exit-code -- certificates patches ``` -The mathematical checks use the Python standard library and bundled source; -no package installation or network access is needed. The command regenerates -the certificates and patches, runs the tests, verifies upstream hashes, and -checks each patch against the pinned manuscript. On a clean checkout, generated -files should have no changes: +No third-party Python packages or network access are needed for these checks. +They regenerate the certificates and patches, run the tests, verify upstream +hashes, and check each patch against the pinned manuscript. The second command +checks exact regeneration on a clean checkout. + +With Tectonic installed, rebuild the latest note using: ```sh -git diff --exit-code -- certificates patches +make complex-note ``` -To rebuild the latest PDF, install Tectonic and run `make dag-note`; -`make incidence-note` rebuilds the preceding rectangle-circuit note, -`make reuse-note` rebuilds the preceding stage-sharing note, -`make tuned-note` rebuilds the parameter tuning note, -`make audit-note` rebuilds the routing proof, and -`make note` rebuilds the earlier parameter-only note. The first run may -download TeX resources. See [reproducibility details](docs/reproducibility.md) -for individual commands and how to apply a patch in a disposable copy. -[GitHub Actions](.github/workflows/verify.yml) runs the numerical and patch checks. +The output is `artifacts/complex-circuit-note.pdf`; `make compact-note` +rebuilds the preceding compact-control note. The first PDF build may +download TeX resources. See [reproducibility instructions](docs/reproducibility.md) +for applying the combined patch in a disposable copy and building older notes. +[GitHub Actions](.github/workflows/verify.yml) runs the arithmetic and patch checks. +Passing tests does not establish the complete multiplication theorem; this +repository contains no full multiplication-machine implementation. -## Alternative patches +## Earlier witnesses and independent patches -Each patch applies independently to the unmodified pinned source. +Each patch applies independently to the **unmodified** pinned source; they are +alternatives, not a sequence to apply together. The +[result history](docs/research/result-history.md) records the earlier mechanisms +and scoped ceilings. -| Patch | Certified conditional saving | Scope | +| Patch | Conditional saving | Scope | | --- | --- | --- | | [frozen-154](patches/frozen-154.patch) | `2^-154` | Original network and recurrence exponents | | [balanced-153](patches/balanced-153.patch) | `2^-153` | Balanced assembly parameters | @@ -296,30 +223,34 @@ Each patch applies independently to the unmodified pinned source. | [h46-nonadjacent-76](patches/h46-nonadjacent-76.patch) | `2^-76` | Direct routing with tuned dimension and stopping parameters | | [h46-shared-side-75](patches/h46-shared-side-75.patch) | `2^-75` | Stage-1/stage-3 side-role sharing, routing, and parameter tuning | | [h46-incidence-67](patches/h46-incidence-67.patch) | `2^-67` | Rectangle incidence circuits, full auxiliary sharing, routing, and parameter tuning | -| **[h46-dag-63](patches/h46-dag-63.patch)** | **`2^-63`** | **Shared intermediate sums, reversible role allocation, full auxiliary sharing, routing, and parameter tuning** | - -Machine-readable results are in [parameters.json](certificates/parameters.json), [network-search.json](certificates/network-search.json), -[nonadjacent-axis.json](certificates/nonadjacent-axis.json), and -[routing-tuned.json](certificates/routing-tuned.json), and -[stage-reuse.json](certificates/stage-reuse.json), and -[incidence-network.json](certificates/incidence-network.json), and -[dag-network.json](certificates/dag-network.json). The first note also gives -a closed-form supremum for the earlier fixed-`beta` parameter problem. +| [h46-dag-63](patches/h46-dag-63.patch) | `2^-63` | Shared intermediate sums and reversible role allocation | +| [h46-shared-point](patches/h46-shared-point.patch) | `13*2^-66` | Cross-group sharing | +| [h50-paired-59](patches/h50-paired-59.patch) | `2^-59` | Paired sums, stopped guard and tighter Gaussian setup | +| [compact-control-34](patches/compact-control-34.patch) | `83/10^12 > 2^-34` | Compact controls, complete reservations, local repair and separate complex arity | +| [complex-circuit-31](patches/complex-circuit-31.patch) | `59/10^11 > 2^-31` | Compressed complex side circuit with binary frames, on top of compact control | +| [fast-gaussian-30](patches/fast-gaussian-30.patch) | `1479/10^12 > 2^-30` | Chirped-correlation Gaussian maps and a sharper Neumann count, on top of the compressed complex network | +| **[aligned-bit-30](patches/aligned-bit-30.patch)** | **`1624/10^12 > 2^-30`** | **Aligned bit side circuit and centers fed by group totals, on top of fast resampling** | ## Attribution, citation, and license -Author: **Douglas Colkitt**. The research, implementation, and drafting were -performed with assistance from OpenAI Codex. This assistance is not independent -review or endorsement by OpenAI. No priority claim is made. - -The original manuscript is by OpenAI. Its source is pinned at commit -`adc7f1241b42e322a6451854ab7e4b4c146bf78a`, with URLs and SHA-256 hashes in -[upstream/manifest.json](upstream/manifest.json). The files under `upstream/` -are preserved unchanged; modifications are distributed as separate patches. - -Use [CITATION.cff](CITATION.cff) for this project and also cite the -[upstream manuscript](upstream/README.md) when discussing the multiplication -bound. Until a release is archived, include the repository commit you used. - -Licensed under [Apache-2.0](LICENSE). See [NOTICE](NOTICE) for attribution and -[CONTRIBUTING.md](CONTRIBUTING.md) for review and contribution guidance. +Author: **Douglas Colkitt**. Research, implementation and drafting were performed +with assistance from OpenAI Codex. The compact-control proposal originated +with a separate research agent; the supplied note develops its tape, layout, +repair and assembly arguments. AI assistance is not independent review or +endorsement by OpenAI. No priority or unrestricted optimality claim is made. + +The compressed complex network (`complex-circuit-31`), the faster Gaussian +resampling (`fast-gaussian-30`) and the aligned bit circuit (`aligned-bit-30`) +were contributed by **eumemic**, prepared with +assistance from Claude (Anthropic); this is likewise not independent review or +endorsement by Anthropic. + +The original manuscript is by OpenAI, pinned at commit +`adc7f1241b42e322a6451854ab7e4b4c146bf78a`. Source URLs and SHA-256 hashes are in +[upstream/manifest.json](upstream/manifest.json). Files under `upstream/` remain +unchanged; modifications are supplied as separate patches. + +Use [CITATION.cff](CITATION.cff) and also cite the +[upstream manuscript](upstream/README.md). Until a release is archived, include +the repository commit used. Licensed under [Apache-2.0](LICENSE); see +[NOTICE](NOTICE) and [CONTRIBUTING.md](CONTRIBUTING.md). diff --git a/artifacts/aligned-bit-note.pdf b/artifacts/aligned-bit-note.pdf new file mode 100644 index 0000000000..45651783a1 Binary files /dev/null and b/artifacts/aligned-bit-note.pdf differ diff --git a/artifacts/compact-control-note.pdf b/artifacts/compact-control-note.pdf new file mode 100644 index 0000000000..12b3faf7a6 Binary files /dev/null and b/artifacts/compact-control-note.pdf differ diff --git a/artifacts/complex-circuit-note.pdf b/artifacts/complex-circuit-note.pdf new file mode 100644 index 0000000000..966dce2030 Binary files /dev/null and b/artifacts/complex-circuit-note.pdf differ diff --git a/artifacts/fast-gaussian-note.pdf b/artifacts/fast-gaussian-note.pdf new file mode 100644 index 0000000000..4a55ab6c37 Binary files /dev/null and b/artifacts/fast-gaussian-note.pdf differ diff --git a/certificates/affine-center-targets.json b/certificates/affine-center-targets.json new file mode 100644 index 0000000000..29e360ab0b --- /dev/null +++ b/certificates/affine-center-targets.json @@ -0,0 +1,25 @@ +{ + "ceiling": { + "auxiliary_role_floor_at_best": 17296, + "center_loss_floor_at_best": 2208, + "finite_candidates": 162, + "improvement_factor_upper": "1627/116", + "kappa_upper": "11389/500000000000", + "limitations": "Nonlinear triple predicates, changed side correction, interleaved centers, changed frames or tensor stages are outside this screen.", + "logarithmic_kappa_upper": "168381227328274558211047878253653634603865373147706507888055781644523265609004265552682304561983849079992056799500667272264515793358543707651777790160409319545/7392296962692181509341504884373707020863330157788787607027523873690953086571895657494476441778682819202755977654478405870314155164521356665791685748890530525569089536", + "optimistic_best_h": 48, + "optimistic_eta": "11/20791296", + "primitive_saving_lower": "30756571167769381972912504906892623327509167660765859468168678796588096610136188466730258022287293367736944711049940226724738590229999654235439479360/675139714901427643261136241894024905279653990649718289512373294805934029454426008731872679921252942351454633090474075480029960913270034272374385791305159973", + "primitive_saving_upper": "168381227328274558211047878253653634603865373147706507888055781644523265609004265552682304561983849079992056799500667272264515793358543707651777790160409319545/3696148481346090754670752442186853510431665078894393803513761936845476543285947828747238220889341409601377988827239202935157077582260678332895842874445265262784544768", + "scope": "Canonical B=Q Q^t over F2; affine point-incidence gather and scatter supports; one gather/scatter passage per center; source-span/complement frames; whole invocation factors through R auxiliary roles; unchanged three-stage counts and fast-Gaussian kappa 1-lambprime with beta>3/4.", + "upper": "13/8000000000" + }, + "status": "CONDITIONAL 1624/10^12 WITNESS; ALIGNED BIT CIRCUIT AND CHEAPER CENTERS; NOT FORMAL VERIFICATION", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a", + "witness": { + "absorption_gap": "46751/100000000000000000", + "constraint_slacks": { + "K_dominates_log_p": "499940001/1000000000", + "K_smaller_than_ell": "69999/1000000000", + "alpha_squared_theta_growth": "1/50000", + "beta_below_one": "6/25", + "beta_positive": "19/25", + "c_positive": "9999/10000", + "crt_layout": "650013/400000000000000", + "delta_below_one_eighth": "124999/1000000", + "delta_positive": "1/1000000", + "epsilon_positive": "49999/100000", + "gaussian_cost": "19/1000000", + "guard_width": "19969601/1000000000", + "kappa_positive": "203/125000000000", + "lambda_above_sigma": "21501/2000000000000", + "lambda_above_tau": "1/2000000000000", + "lambda_below_one": "6499/2000000000000", + "lambda_prime_above_lambda": "1/2000000000000", + "lambda_prime_below_one": "3249/1000000000000", + "leaf_cost": "111/1000000000000", + "packed_overhead": "1/2000000000000", + "prefix_cost": "69999/1000000000", + "prime_interval_growth": "1/50000", + "r_superpolynomial": "50001/100000", + "reserved_axes": "999899996751/1000000000000", + "scalar_cost": "500009/1000000", + "sigma_below_one": "7/500000000", + "sigma_positive": "499999993/500000000", + "tau_below_one": "13/4000000000", + "tau_positive": "3999999987/4000000000" + }, + "guard": { + "B": 338201706233372774319109678801287590064, + "C0": 321727027889740762489223803495350305328695729120143793284991524019954162292886152000000, + "C1": "19601/10000", + "E": 338201706233372774319082463160146840064, + "beta": "19/25", + "m": 15625, + "s": 27215641140750000, + "zeta": "1/10000" + }, + "limiting_margins": [ + "g3" + ], + "margins": { + "g1": "69999/1000000000", + "g2": "6499220013/4000000000000000000", + "g3": "162446751/100000000000000000", + "g4": "650013/400000000000000", + "g5": "19/1000000", + "g6": "500009/1000000", + "g7": "49999/100000" + }, + "minimum_margin": "162446751/100000000000000000", + "parameters": { + "C1": "19601/10000", + "beta": "19/25", + "c": "9999/10000", + "delta": "1/1000000", + "epsilon": "49999/100000", + "kappa": "203/125000000000", + "lam": "1999999993501/2000000000000", + "lamp": "999999996751/1000000000000", + "sigma": "499999993/500000000", + "tau": "3999999987/4000000000" + }, + "recurrence": { + "internal": "3999999987/4000000000", + "layer": "3999999987/4000000000", + "leaf": "6249999979/6250000000", + "preprocessing": "1/10000" + } + } +} diff --git a/certificates/assembly-breakthrough-guard.json b/certificates/assembly-breakthrough-guard.json new file mode 100644 index 0000000000..3412d8cdee --- /dev/null +++ b/certificates/assembly-breakthrough-guard.json @@ -0,0 +1,274 @@ +{ + "alternate_baseline_witness": { + "absorption_gap": "46751/100000000000000000", + "constraint_slacks": { + "K_dominates_log_p": "499940001/1000000000", + "K_smaller_than_ell": "69999/1000000000", + "alpha_squared_theta_growth": "1/50000", + "beta_below_one": "999/1000", + "beta_positive": "1/1000", + "c_positive": "9999/10000", + "crt_layout": "650013/400000000000000", + "delta_below_one_eighth": "124999/1000000", + "delta_positive": "1/1000000", + "epsilon_positive": "49999/100000", + "gaussian_cost": "19/1000000", + "guard_width": "49946051099/100000000000", + "kappa_positive": "203/125000000000", + "lambda_above_sigma": "21501/2000000000000", + "lambda_above_tau": "1/2000000000000", + "lambda_below_one": "6499/2000000000000", + "lambda_prime_above_lambda": "1/2000000000000", + "lambda_prime_below_one": "3249/1000000000000", + "leaf_cost": "10737/1000000000000", + "packed_overhead": "1/2000000000000", + "prefix_cost": "69999/1000000000", + "prime_interval_growth": "1/50000", + "r_superpolynomial": "50001/100000", + "reserved_axes": "999899996751/1000000000000", + "scalar_cost": "500009/1000000", + "sigma_below_one": "7/500000000", + "sigma_positive": "499999993/500000000", + "tau_below_one": "13/4000000000", + "tau_positive": "3999999987/4000000000" + }, + "guard": { + "B": 338201706233372774319109678801287590064, + "C0": 321727027889740762489223803495350305328695729120143793284991524019954162292886152000000, + "C1": "1001099/1000000", + "E": 338201706233372774319082463160146840064, + "W": 1741801270000, + "beta": "1/1000", + "constant_slacks": { + "exponent_above_one": "1099/1000000", + "one_piece_constant": 1029423546892481191846331538031317839932167687392682209855973070619839801581064, + "total_call_domination": 27215641140734225, + "whole_layer_constant": "160863513944870381244611901747675152664347864560071896642495762009977081146443075999982" + }, + "h": 25, + "m": 15625, + "path_branching": 15775, + "rho": "1001/1000", + "s": 27215641140750000, + "zeta": "1/10000" + }, + "limiting_margins": [ + "g3" + ], + "margins": { + "g1": "69999/1000000000", + "g2": "6499220013/4000000000000000000", + "g3": "162446751/100000000000000000", + "g4": "650013/400000000000000", + "g5": "19/1000000", + "g6": "500009/1000000", + "g7": "49999/100000" + }, + "minimum_margin": "162446751/100000000000000000", + "parameters": { + "C1": "1001099/1000000", + "beta": "1/1000", + "c": "9999/10000", + "delta": "1/1000000", + "epsilon": "49999/100000", + "kappa": "203/125000000000", + "lam": "1999999993501/2000000000000", + "lamp": "999999996751/1000000000000", + "sigma": "499999993/500000000", + "tau": "3999999987/4000000000" + }, + "recurrence": { + "internal": "3999999987/4000000000", + "layer": "3999999987/4000000000", + "leaf": "499999993007/500000000000", + "preprocessing": "1/10000" + }, + "scope": "Same aligned-bit kappa, additionally conditional on the written dependency-path guard proof.", + "status": "ALTERNATE BASELINE WITNESS; NO HEADLINE IMPROVEMENT" + }, + "assumption": "Existing nested residual frames and saved-input scalar gates", + "date": "2026-10-07", + "invocation_controls": [ + { + "ambient_dimension": 512, + "h": 8, + "invocations": [ + { + "conservative_gate_count": 2499, + "decreasing_edge_rank": 8, + "decreasing_edges": 9, + "inverse": false, + "maximum_path_descent": 8, + "maximum_path_rank": 528, + "roles": 999, + "stage": 1, + "upper_path_rank": 528 + }, + { + "conservative_gate_count": 2499, + "decreasing_edge_rank": 8, + "decreasing_edges": 9, + "inverse": true, + "maximum_path_descent": 8, + "maximum_path_rank": 528, + "roles": 999, + "stage": 2, + "upper_path_rank": 528 + }, + { + "conservative_gate_count": 2499, + "decreasing_edge_rank": 8, + "decreasing_edges": 9, + "inverse": false, + "maximum_path_descent": 8, + "maximum_path_rank": 528, + "roles": 999, + "stage": 3, + "upper_path_rank": 528 + } + ], + "maximum_three_stage_path_descent": 24, + "three_stage_path_rank_upper": 560 + }, + { + "ambient_dimension": 1000, + "h": 10, + "invocations": [ + { + "conservative_gate_count": 7875, + "decreasing_edge_rank": 10, + "decreasing_edges": 11, + "inverse": false, + "maximum_path_descent": 10, + "maximum_path_rank": 1020, + "roles": 2719, + "stage": 1, + "upper_path_rank": 1020 + }, + { + "conservative_gate_count": 7875, + "decreasing_edge_rank": 10, + "decreasing_edges": 11, + "inverse": true, + "maximum_path_descent": 10, + "maximum_path_rank": 1020, + "roles": 2719, + "stage": 2, + "upper_path_rank": 1020 + }, + { + "conservative_gate_count": 7875, + "decreasing_edge_rank": 10, + "decreasing_edges": 11, + "inverse": false, + "maximum_path_descent": 10, + "maximum_path_rank": 1020, + "roles": 2719, + "stage": 3, + "upper_path_rank": 1020 + } + ], + "maximum_three_stage_path_descent": 30, + "three_stage_path_rank_upper": 1060 + }, + { + "ambient_dimension": 1728, + "h": 12, + "invocations": [ + { + "conservative_gate_count": 18757, + "decreasing_edge_rank": 12, + "decreasing_edges": 13, + "inverse": false, + "maximum_path_descent": 12, + "maximum_path_rank": 1752, + "roles": 5873, + "stage": 1, + "upper_path_rank": 1752 + }, + { + "conservative_gate_count": 18757, + "decreasing_edge_rank": 12, + "decreasing_edges": 13, + "inverse": true, + "maximum_path_descent": 12, + "maximum_path_rank": 1752, + "roles": 5873, + "stage": 2, + "upper_path_rank": 1752 + }, + { + "conservative_gate_count": 18757, + "decreasing_edge_rank": 12, + "decreasing_edges": 13, + "inverse": false, + "maximum_path_descent": 12, + "maximum_path_rank": 1752, + "roles": 5873, + "stage": 3, + "upper_path_rank": 1752 + } + ], + "maximum_three_stage_path_descent": 36, + "three_stage_path_rank_upper": 1800 + }, + { + "ambient_dimension": 15625, + "h": 25, + "invocations": [ + { + "conservative_gate_count": 453605, + "decreasing_edge_rank": 25, + "decreasing_edges": 26, + "inverse": false, + "maximum_path_descent": 25, + "maximum_path_rank": 15675, + "roles": 112821, + "stage": 1, + "upper_path_rank": 15675 + }, + { + "conservative_gate_count": 453605, + "decreasing_edge_rank": 25, + "decreasing_edges": 26, + "inverse": true, + "maximum_path_descent": 25, + "maximum_path_rank": 15675, + "roles": 112821, + "stage": 2, + "upper_path_rank": 15675 + }, + { + "conservative_gate_count": 453605, + "decreasing_edge_rank": 25, + "decreasing_edges": 26, + "inverse": false, + "maximum_path_descent": 25, + "maximum_path_rank": 15675, + "roles": 112821, + "stage": 3, + "upper_path_rank": 15675 + } + ], + "maximum_three_stage_path_descent": 75, + "three_stage_path_rank_upper": 15775 + } + ], + "recurrence": { + "beta": "1/1000", + "complex_leaf_saving": "6993/500000000000", + "epsilon": "49999/100000", + "exact_power_comparison": "15775^1000 < 15625^1001", + "finite_recurrence_comparisons": 81, + "guard_exponent": "1001099/1000000", + "guard_slack": "49946051099/100000000000", + "h": 25, + "m": 15625, + "old_beta_19_over_25_leaf_saving": "21/6250000000", + "path_branching": 15775, + "rho": "1001/1000", + "zeta": "1/10000" + }, + "scope": "Conservative finite DAG controls support the written proof; not formal verification.", + "status": "RESEARCH GUARD REFINEMENT; NO NEW MULTIPLICATION EXPONENT" +} diff --git a/certificates/batched-role-targets.json b/certificates/batched-role-targets.json new file mode 100644 index 0000000000..52277b810d --- /dev/null +++ b/certificates/batched-role-targets.json @@ -0,0 +1,374 @@ +{ + "cases": { + "26": { + "100": { + "certified_role_budget": 5027844, + "favorable_bit_saving": "203/625000000", + "first_excluded_roles": 5027858, + "h": 26, + "lower_zero_role_gap": "-3099396416074427880954052653/39062500000000", + "output_floor": 66105, + "source_frames": false, + "target_kappa": "203/1250000000", + "upper_zero_role_gap": "-6198772356401780225433659929965821955773909888000/78124749868001195710239873077802147" + }, + "1000": { + "certified_role_budget": 344109, + "favorable_bit_saving": "203/62500000", + "first_excluded_roles": 344123, + "h": 26, + "lower_zero_role_gap": "-212132451668177880954052653/3906250000000", + "output_floor": 66105, + "source_frames": false, + "target_kappa": "203/125000000", + "upper_zero_role_gap": "-42424503061476198092262022665821955773909888000/781224986806146610239873077802147" + }, + "10000": { + "certified_role_budget": -1, + "favorable_bit_saving": "203/6250000", + "first_excluded_roles": 0, + "h": 26, + "lower_zero_role_gap": "76593944772447119045947347/390625000000", + "output_floor": 66105, + "source_frames": false, + "target_kappa": "203/12500000", + "upper_zero_role_gap": "1532017320116704641877768064178044226090112000/7809998686641700239873077802147" + } + }, + "28": { + "100": { + "certified_role_budget": 8910172, + "favorable_bit_saving": "203/625000000", + "first_excluded_roles": 8910195, + "h": 28, + "lower_zero_role_gap": "-56267265595367503231643907/152587890625", + "output_floor": 98658, + "source_frames": false, + "target_kappa": "203/1250000000", + "upper_zero_role_gap": "-230839288367053358067990069667345793760632832000/625999554392124528763411411146479" + }, + "1000": { + "certified_role_budget": 652655, + "favorable_bit_saving": 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"6235444996649406441541009344/762939453125", + "output_floor": 201872, + "source_frames": false, + "target_kappa": "203/12500000", + "upper_zero_role_gap": "31929987124056582828435705971100766041786023936/3904922412477207523705654634099" + } + } + }, + "scope": "Fixed three-stage geometry, unchanged center losses, four controlled recursive block classes (or PR #13 source relocation), and kappa=3, n selected-bit flips require at least n whole-slot affine movement gates, even with arbitrary address-dependent linear mixing among fixed complete-volume banks.", + "ordered_extension": "The same path-cover argument applies separately to fixed-order, whole-slot prefix-controlled affine updates, including nonlinear offsets from earlier fields. It does not allow swapping field order between those updates.", + "status": "BOUNDED CODED-CARRY ATTEMPT: EXACT BUT LINEAR; NO NEW KAPPA", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a" +} diff --git a/certificates/compact-control-audit.json b/certificates/compact-control-audit.json new file mode 100644 index 0000000000..4c9bbc30f8 --- /dev/null +++ b/certificates/compact-control-audit.json @@ -0,0 +1,26 @@ +{ + "bad_fraction_bound": "min(1, n*(2^(-G)+8*2^(G-K))) early; replace 2^(-G) by 2*2^(-G) late", + "candidate_cost": "O(V*((f*log(p))^tau+1)) if complete front/back compact fields exist and rotations and repair receive full tape accounting", + "controls": { + "early_swaps": 4, + "integer_cases": 2480, + "late_swaps": 12, + "packed_cases": { + "bad": 4932, + "early": 3600, + "good": 2268, + "late": 3600 + }, + "scope": "Exact integer arithmetic and deterministic samples of full modular address maps; not a tape-runtime proof." + }, + "cost_proof": "notes/compact-control-note.tex; arithmetic in certificates/compact-control-layer.json", + "outside_this_checker": [ + "Extend controlled rotation to polynomial-in-p offset work paid by large records.", + "Reserve complete compact fields outside the active axes and preserve them through row splitting, padding and every child.", + "Charge individually processed reserved axes and recompute the full layer recurrence and guard/assembly constraints.", + "Write the fixed-tape exceptional repair and inverse-descriptor argument for the full operation." + ], + "scope": "The highest selected bit is omitted here and supplied by the retained elementary two-bit XOR. This checker verifies addresses only; see the separate cost proof and layer certificate.", + "status": "ADDRESS-ONLY COMPACT-CONTROL CHECK; NO EXPONENT CLAIM BY THIS CHECKER", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a" +} diff --git a/certificates/compact-control-layer.json b/certificates/compact-control-layer.json new file mode 100644 index 0000000000..a087e47b8b --- /dev/null +++ b/certificates/compact-control-layer.json @@ -0,0 +1,339 @@ +{ + "allocation_controls": [ + { + "D": 1, + "G": 22, + "K": 1, + "active_chunks": 0, + "back_chunks": 22, + "back_slack_bits": 0, + "compact_capacity": 22, + "d": 1, + "front_chunks": 44, + "front_slack_bits": 0, + "mode": "individual", + "preprocessed_chunks": 1, + "recursion_depth_cap": 0, + "reserved_chunks": 112, + "row_bits": 46, + "row_chunks": 46 + }, + { + "D": 100, + "G": 142, + "K": 2, + "active_chunks": 0, + "back_chunks": 7100, + "back_slack_bits": 0, + "compact_capacity": 14200, + "d": 100, + "front_chunks": 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22, + "K": 48, + "late_bad_fraction_upper": "5/524288", + "n": 16, + "p": 16, + "uniform_upper": "5/524288" + }, + { + "G": 34, + "K": 72, + "late_bad_fraction_upper": "125/8589934592", + "n": 100, + "p": 100, + "uniform_upper": "1/25600000" + }, + { + "G": 86, + "K": 176, + "late_bad_fraction_upper": "78125/2417851639229258349412352", + "n": 1000000, + "p": 1000000, + "uniform_upper": "1/25600000000000000000" + } + ], + "scope": "Conditional on the pinned upstream interfaces, published paired-bit and Gaussian refinements, and the new compact-control movement, reservation, repair, independent-complex and generalized-guard proofs. Exact arithmetic is not formal verification.", + "scoped_ceiling": { + "below_next_dyadic": true, + "scope": "Unchanged complex h=25 motif and retained Gaussian/leaf assembly inequalities; not a bound on new complex networks or algorithms.", + "upper": "20746540553460608247112929262864045345085076105741608893520610937226467771276028699915418386370445044824659127166650635895871722137717597479546857376527708719255663717453838247976227897533268532668858228042014619339505681211015630361605588188325457596307994168644705454015569/247879771154821854878830500699773260764360935854381442829251991515018712527129563362382500245253233252357075435598151081543662393578612953585247357965410441040580391962056220774007355086170004496116271962271152918875464845809694379665906644415581567011045178785707969849087514050560000", + "witness_fraction_of_upper": 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In each provider, 38 entering width-8 helpers end at a certified terminal hyperplane, reducing their residual width to 7 and pairing them with width-9 owner slots. The change is mass-neutral and leaves the rank deficit unchanged while improving the paid profile.", + "pinned_sources": { + "multicold": "PR568 commit 71c99ac935a33bb7b59a1b4a7af8725a939b0b90", + "larger_gain": "PR569 source files pinned by the PR592 SPEC", + "structural_gain": "PR571 commit b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3", + "joint_prefix": "PR575 commit eac6f7e378ab622dac50a2fd4b7ef6eab014de44", + "frame_coalescing": "PR583 commit 89be768f0157f5a3d8e4db69cf1748768605eac9" + }, + "provenance": "PR592 credits chafreaky, jon314159, DominikScholz, jeremykothe2, eumemic, rohanarun, Dugongue, icekylinx, DreamingOfClouds and the original OpenAI #109 framework. PR592 discloses Claude Opus 5.5 (Anthropic) in its NOTICE and body; this independent replay was performed with OpenAI Codex assistance.", + "scope": "Conditional finite exact-complex witness. The all-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape/exact recovery, bit H1-H5 assumptions and analytic transfer remain assumptions. No unconditional theorem or full Lean certification is claimed.", + "conservative_complex_lower": "1615324489404801288267/1000000000000000000000000" +} diff --git a/certificates/fused-block-audit.json b/certificates/fused-block-audit.json new file mode 100644 index 0000000000..efad46873f --- /dev/null +++ b/certificates/fused-block-audit.json @@ -0,0 +1,73 @@ +{ + "block": { + "coordinate_layer_calls": 3, + "direct_equals_basis_sandwich_equals_three_layer_block": true, + "exact_gaussian_integer_arithmetic": true, + "exhaustive_basis_inputs": 168, + "layout_permutations": 0, + "minimum_coordinate_calls_from_cut_budget": 2, + "scope": "Finite exact identity; phase/scalar overhead has a separate written streaming audit. No faster recursive algorithm follows." + }, + "cut_controls": [ + { + "axes": 1, + "bit": 0, + "one_role_child_rank": 1, + "pointwise_rank": 0, + "target_rank": 2 + }, + { + "axes": 2, + "bit": 0, + "one_role_child_rank": 2, + "pointwise_rank": 0, + "target_rank": 4 + }, + { + "axes": 2, + "bit": 1, + "one_role_child_rank": 2, + "pointwise_rank": 0, + "target_rank": 4 + }, + { + "axes": 3, + "bit": 0, + "one_role_child_rank": 4, + "pointwise_rank": 0, + "target_rank": 8 + }, + { + "axes": 3, + "bit": 1, + "one_role_child_rank": 4, + "pointwise_rank": 0, + "target_rank": 8 + }, + { + "axes": 3, + "bit": 2, + "one_role_child_rank": 4, + "pointwise_rank": 0, + "target_rank": 8 + } + ], + "next_step": "Any surviving design must include a costed noncoordinate operation: sparse address movement or a direct directional butterfly not built solely from coordinate children and pointwise gates.", + "recursion": { + "conclusion": "Coordinate-butterfly children and address-pointwise gates alone cannot meet the strict contraction requirement, even after arbitrary legal fusion.", + "contraction_required": "S=2g." +} diff --git a/certificates/joint-frame-audit.json b/certificates/joint-frame-audit.json new file mode 100644 index 0000000000..e09158a229 --- /dev/null +++ b/certificates/joint-frame-audit.json @@ -0,0 +1,93 @@ +{ + "next_step": "Optimize a region crossing invocation boundaries or split the grouped central gates; optimizing only internal matrices of the present invocation is exhausted in this model.", + "physical_edge_scores": { + "X": 100, + "Y": 100, + "center": 108, + "side": 984 + }, + "physical_edges": 2042, + "published_invocation": { + "center_rank": 7500, + "center_return_loss": 2500, + "data_rank": 1920800, + "density": "392", + "density_minus_center_roles": "342", + "h": 50, + "side_rank": 25459700, + "side_roles": 509194, + "total_rank": 27388000, + "v": 19600 + }, + "rational_controls": { + "baseline_central_score": 2460, + "central_controls": [ + { + "lower_excess": "260", + "score": 3147 + }, + { + "lower_excess": "220", + "score": 7285 + }, + { + "lower_excess": "256", + "score": 3532 + }, + { + "lower_excess": "224", + "score": 3432 + }, + { + "lower_excess": "366", + "score": 5726 + }, + { + "lower_excess": "292", + "score": 3830 + }, + { + "lower_excess": "220", + "score": 5582 + }, + { + "lower_excess": "362", + "score": 3997 + }, + { + "lower_excess": "288", + "score": 4078 + }, + { + "lower_excess": "256", + "score": 3674 + }, + { + "lower_excess": "292", + "score": 3000 + }, + { + "lower_excess": "252", + "score": 5600 + } + ], + "h": 10, + "matrix_controls": 12 + }, + "scope": "Allows joint copy, injection, side and central frame changes. Retains active-factor embedding and all boundary matrices. Does not exclude new cross-invocation boundaries, changed central gate grouping, fewer roles or new scalar topology.", + "small_physical_graph": { + "center_rank": 108, + "center_return_loss": 36, + "data_rank": 200, + "density": "10/3", + "density_minus_center_roles": "-8/3", + "h": 6, + "side_rank": 984, + "side_roles": 164, + "total_rank": 1292, + "v": 20 + }, + "status": "FIXED-BOUNDARY BIT INVOCATION RANK OPTIMALITY; NO NEW KAPPA", + "theorem": "For h>=10, arbitrary rational internal active-factor matrices cannot lower the complete rank score with fixed invocation boundaries, side-role count, and the four grouped central gates.", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a" +} diff --git a/certificates/kappa-targets.json b/certificates/kappa-targets.json new file mode 100644 index 0000000000..fe996afd96 --- /dev/null +++ b/certificates/kappa-targets.json @@ -0,0 +1,61 @@ +{ + "baseline_kappa": "203/125000000000", + "distinction": "Optimistic ceilings discard real costs. They neither supply circuits nor exclude different terminal frames, scalar topologies or assemblies.", + "free_scratch": { + "bit_saving_lower": "3873294121340764213332539781850143772937883344336791872894247873388052681013864717907046223936787769703380475909620435869441790572951911759143110443762139858653798260855181314231487373567811440197934385521921757805327575259443217104702351207850518876979200000/45457224399796647658461137097812973169122529675193012374087564537106183608458383161531348308675742654384651519076560789376013393099480488843220453398914241193890741953997119154293588541184219155231281487856229285382873778844352312318548990446481564585837993314974623", + "bit_saving_upper": "12968618709846308749997343019587534954033091554699085560328836572615715689501399658205670244742540948448517621950615152228601709142037853656858085616863974175385156330635500208168064477666397370890859228241202740028023968710896182992740894791670001698524867/152200528124319132785026088762406442178514992768944757773762787950228051949992503411383509686109368907429976298338268645465021964999579909288893421130088691322795907318408734988132688478205555752509659382643752427423439986617266307072501947503010789928451041280000", + "eta_at_best": "1/1000000", + "factor_over_baseline_upper": "174685058052886701912679728173323477290316427191528630931153509868207377283154629016778963425950174413369041243946863580665432504607190916714144472209913445250338851436361802372953454709946085276008340897645511045636098716472200740742738345792968772878837109375/6658773105438962059344891383355281845310030933641333152602121972822477272812172024248028548767284889700061463052299253239094710968731621031389087174441380245372320945180382155730805120921493064172297597990664168699775499414505400934421960203256722059369733056", + "finite_candidates": 161, + "kappa_upper": "12968618709846308749997343019587534954033091554699085560328836572615715689501399658205670244742540948448517621950615152228601709142037853656858085616863974175385156330635500208168064477666397370890859228241202740028023968710896182992740894791670001698524867/304401056248638265570052177524812884357029985537889515547525575900456103899985006822767019372218737814859952596676537290930043929999159818577786842260177382645591814636817469976265376956411111505019318765287504854846879973234532614145003895006021579856902082560000", + "keep_designated_outputs": false, + "optimistic_best_h": 50, + "scope": "Three tensor stages, full triple family, aligned central loss, the retained negative-source charge and assembly. Every auxiliary role is optimistically free.", + "tail_bit_saving_upper": "27833142274242700215271088832669115753705207512506200661116895341181340108251606224021027707791928923512619941379564138696491167669799858823604024371133505222760202644845265858387466267130427995435823750848469693775/7078502969300212729822094800491960620668387726773175983267744615384987648449796790972336005495236810978783850779729191266357296777002324224996188122595699950478075100691587038050291082574501810343070693701501386284490854144", + "tail_start": 200 + }, + "retained_output_compiler": { + "bit_saving_lower": "4432183404945921257448454276479171639314273689115359711698626403142907079759413496243903290442341043771065384876996768876730822928519973685671081452686817049531167964782273832392969481697104012453433133899838920226979800349151194009093626746524048085770368028598890627147449958400000/208198310249216890880402088771065039824174432336840138115714400517700501152718860091477141212549790723334494775416172936195115115503507291577332899001394905331276342857180305823474423914596693274716186468388247997455949312414004958418840259949123380756809996933976243837151853368993747329871", + "bit_saving_upper": "17297479842763191861998063833202389746741178112039799830495154039873504852728479037308072189002681633771672458251513045425575780621374744752251340083396842724639172611366814745482245447284387078749275913858824058287014564993861882443278883544275006777217347767211772874606893036640909806077/812535435878950798292726337126155231763235896310953724966134337602703164608753202328004019415987097328120313195962513519535155957926043898861178231417254813760382481879525743883645379354329228891041622665132214780404434722751956280240136943563069714573736813207797010685813694701340677999061760000", + "eta_at_best": "49/196125000", + "factor_over_baseline_upper": "33284867308272767591591101895786618201087550246382003984173248999140826764024936571174707876005775927054480561598508785070766203474012362900729949359984688124690526114853014827359616393327407402149930560842872649105246622909986688814806964948189283361332642717079304330755259076048548733984375/5078346474243442489329539607038470198520224351943460781038339610016894778804707514550025121349919358300751957474765709497094724737037774367882363946357842586002390511747035899272783620964557680569010141657076342377527717017199726751500855897269185716085855082548731316786335591883379237494136", + "finite_candidates": 161, + "kappa_upper": "17297479842763191861998063833202389746741178112039799830495154039873504852728479037308072189002681633771672458251513045425575780621374744752251340083396842724639172611366814745482245447284387078749275913858824058287014564993861882443278883544275006777217347767211772874606893036640909806077/1625070871757901596585452674252310463526471792621907449932268675205406329217506404656008038831974194656240626391925027039070311915852087797722356462834509627520764963759051487767290758708658457782083245330264429560808869445503912560480273887126139429147473626415594021371627389402681355998123520000", + "keep_designated_outputs": true, + "optimistic_best_h": 50, + "scope": "Three tensor stages, full triple family, aligned central loss, the retained negative-source charge and assembly. Also retains q=3v+h designated outputs and the c+q compiler.", + "tail_bit_saving_upper": "27833142274242700215271088832669115753705207512506200661116895341181340108251606224021027707791928923512619941379564138696491167669799858823604024371133505222760202644845265858387466267130427995435823750848469693775/7078502969300212729822094800491960620668387726773175983267744615384987648449796790972336005495236810978783850779729191266357296777002324224996188122595699950478075100691587038050291082574501810343070693701501386284490854144", + "tail_start": 200 + }, + "status": "SCOPED UPPER BOUNDS AND NECESSARY TARGETS; NO NEW KAPPA", + "targets": { + "10000x": { + "first_excluded_m": 2023, + "necessary_m_at_most": 2022, + "scope": "Necessary only; even the optimistic smaller arities need a new valid network.", + "target_kappa": "203/12500000", + "upper_at_first_excluded": 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"17/2464463232", + "optimistic_saving_lower": "390248331122052612971871248875177207344243366090268476162068463038373874035413515308904833553740544378323225603217379669739099293402629758981379611597455642957770003590800039344565105691080756127579708498967557440793700469244857815474401222338805817599386132394435091273499089575471428697445212461601425292473829875918854817395401519631512798764728320/938460552997400085987449655210607212710677313290629204452356350061425639461916143895239656521620390245889718835938528234960831698532433246499460582149865152015771856598278578592808225518681840017666789789520708148362512357756563624599539228465425159220250457253564792865892719433505746019246094660893769271555794807519593919240387777395978838841805764196645219", + "optimistic_saving_upper": "1690580841970497081967632168578292809952507921803751434049256999900813828750419982748019350098376915393837843700506661533248810204217097903419095759262872863332353964425628541791356759553199534285729367412876990487868086599669930337181202002418931878920426805775844448512378465784547951854140797620108321894792352406504335278222006463172247367285057603462915/4065471407092940336520537909151325592505474557492476249784579158248664834633533009926313102851831241181377513614760450627224470206770052550759325914688054772702110616052838189451814058420185482197505077226024744812595292776054005868150275359282114304655100761631546696255030781564615963118887912462794746987441953971550195022953595066048514567823161620466172714729472", + "positive_deficit_possible": true, + "star_rank_lower": 230, + "vertices": 42504 + }, + { + "ambient_rank_lower": 273, + "deficit_numerator_upper": -14040, + "h": 14, + "k": 7, + "positive_deficit_possible": false, + "star_rank_lower": 208, + "vertices": 3432 + } + ], + "kappa_strict_upper": "1/1330999968", + "optimistic_relaxation": "All auxiliary roles are free; center loss uses only a lower bound for its source span.", + "scope": "Full k-subsets, odd k>=5, h>=2k; canonical degree-(k-1)/2 rational intersection polynomial; three tensor stages; h point centers returned through zero; arbitrary side circuit.", + "status": "SCOPED OBSTRUCTION; NO NEW MULTIPLICATION WITNESS" +} diff --git a/certificates/pr592-independent-replay/README.md b/certificates/pr592-independent-replay/README.md new file mode 100644 index 0000000000..06548df845 --- /dev/null +++ b/certificates/pr592-independent-replay/README.md @@ -0,0 +1,40 @@ +# Independent replay of PR592 + +**Conditional κ = 0.001612719427075359** (conservative value +`0.001612719424224312`). This is an independent clean replay of +[CrocSwap/integer-mult-bounds#592](https://github.com/CrocSwap/integer-mult-bounds/pull/592), +head `9092320f7932dbe61a78f15b6c5fd6a0cbd41307`. + +The replay rebuilt the frozen PR558 baseline and both final provider words +from the pinned PR568/569/571/575/583 source packages. It completed 32 +top-level stages plus the nine nested baseline stages in 338.45477745798416 +seconds, from a fresh baseline with no reused local or baseline outputs. The +result is `PASS_HPEQ4_NATIVE_COMPLEX_AND_CITED_BIT`. + +The replay checked the actual scalar words, arbitrary dirty restoration, both +reflected ledgers, phase edges, normalized wrappers, physical owner/profile +data, a fresh invoice, exact assembly, two exact moment engines, 47 outer +constraints, 147 assembly controls, timing controls, and recompression/rebound +controls. The resulting profile is +`{1:182818,2:34193,3:22560,4:14584,5:4032,6:4032,8:576,12:1,48:1969}` +with dispatch `ck=53`, `atm3=19`. + +PR592 changes only the owner dispatch from PR591: the hyperplane-ended helper +chain is retained, with 38 width-8 helpers per call ending on the certified +terminal hyperplane and becoming width 7. The changed literal owner map is +mass-neutral and leaves the rank deficit unchanged while improving the paid +profile. The independent replay's receipt SHA256 is +`13db1006088ecb880815fb116e6cc464b2e18c91426bdf3a9efdad481427beae`; its +result SHA256 is +`f36956b317d8dddbf12b956f76e50347e20d696c6168724b71eee990babd8867`. + +The replay used only the standard-library verifier in PR592, with immutable +source pins for PR568, PR569, PR571, PR575 and PR583. PR592's own provenance +and AI disclosure are retained: Claude Opus 5.5 (Anthropic) performed the +submitted construction, while this replay and certificate were prepared with +OpenAI Codex assistance. + +This remains conditional on the inherited all-size compiler, common-label / +mutable-source realization, whole-call hosting, precision/tape/exact +recovery, and analytic transfer. It is not an unconditional theorem or full +Lean certification. diff --git a/certificates/pr592-independent-replay/verify_receipt_hashes.py b/certificates/pr592-independent-replay/verify_receipt_hashes.py new file mode 100644 index 0000000000..a6d5bacc03 --- /dev/null +++ b/certificates/pr592-independent-replay/verify_receipt_hashes.py @@ -0,0 +1,27 @@ +"""Verify PR592 replay receipt, log and artifact hashes from an extracted work directory.""" +import argparse, hashlib, json +from pathlib import Path + +def sha(path): + return hashlib.sha256(path.read_bytes()).hexdigest() + +def main(): + ap=argparse.ArgumentParser() + ap.add_argument('work', type=Path) + a=ap.parse_args() + receipts=sorted(a.work.rglob('RECEIPT.json')) + assert receipts, 'no RECEIPT.json found' + checked=[] + for path in receipts: + rec=json.loads(path.read_text()) + for step in rec['steps']: + assert step['exit_code']==0, step + log=path.parent/(step['name']+'.log') + assert sha(log)==step['log_sha256'], (path, step['name']) + for rel,digest in rec['artifacts'].items(): + assert sha(path.parent/rel)==digest, (path, rel) + checked.append({'receipt':str(path.relative_to(a.work)), 'steps':len(rec['steps']), 'artifacts':len(rec['artifacts']), 'status':rec['status']}) + print(json.dumps({'status':'PASS_RECEIPT_LOG_ARTIFACT_HASHES','receipts':checked}, indent=2)) + +if __name__=='__main__': + main() diff --git a/certificates/prime-field28.json b/certificates/prime-field28.json new file mode 100644 index 0000000000..0c505954e2 --- /dev/null +++ b/certificates/prime-field28.json @@ -0,0 +1,241 @@ +{ + "complex_checks": { + "labels": { + "active_nodes": 64298, + "all_labels_nondegenerate": true, + "all_residuals_have_odd_vectors": true, + "checked_inclusions": 154860 + }, + "map": { + "coefficients": "+1/2 disjoint, -1/2 intersection two", + "nonzero_coefficients": 7780500, + "side_map_exact": true, + "targets": 3276 + }, + "roles": { + "forward_frames_nested": true, + "reverse_complement_frames_nested": true, + "roles": 93838 + }, + "stats": { + "additions": 61022, + "disjoint_additions": 43634, + "h": 28, + "injections": 32816, + "pair_star_additions": 17388, + "roles": 93838, + "roles_per_target": 28.644, + "v": 3276 + } + }, + "matching": { + "distinct_images": 98280, + "domain": 98280, + "every_intersection_two": true, + "h": 28 + }, + "producer": { + "global_before_replacement": { + "all_inputs_active": true, + "exact_support_keys": true, + "h": 28, + "local_inputs": 2600, + "outputs": 983178, + "pruned_additions": 11240978, + "removed_after_merging": 0, + "roles": 12224156, + "topological_order_checked": true, + "unique_additions": 11240978 + }, + "independent_template_check": { + "every_boundary_sum_verified": true, + "every_template_addition_disjoint": true, + "new_star_additions": 2623060, + "old_star_additions": 3006276, + "optimized_additions": 10857762, + "optimized_stars": 20475, + "role_upper_bound": 11840940, + "verified_templates": 365 + }, + "local": { + "additions": 41427, + "additions_by_core": { + "0": 8511, + "1": 18616, + "2": 14300 + }, + "all_disjoint_output_coefficients_exact": true, + "base": 4, + "binary_sha256": "460b55adf31b6128710d162b63ea17843e94786727a7208990b4ef1533b0ebaa", + "inputs": 2600, + "n": 26, + "outputs": 2600, + "retained_additions": 41439, + "retained_total_exact": true, + "roles": 44027, + "roles_per_output": 16.93346153846154, + "status": "SCALAR PRODUCER ONLY; NO COMPLETE NETWORK CLAIM" + }, + "replacements": { + "new_additions": 2623060, + "old_additions": 3006276, + "saved": 383216, + "sha256": "36d4b4bdfc89af922cb39a9f0469e4aa9c18d145ea6a6a542ddb8ddced55a136", + "stars": 20475, + "templates": 365 + } + }, + "proof_sha256": { + "notes/prime-field28-construction.tex": "1f850c13ce37ca15c38ea393dbecced241e666c5f4c11ceb3cb137810772c15c", + "scripts/paired_complex.py": "4f4c28102bfc528e264896e6c32bbd0e66da0973237e96e4044d8ccf7e78a1de", + "scripts/paired_triple_circuit.py": "8f5f2de32fdbd8e0005154638660ff416e5a8cd59623bd0dc03f6f7f97ce2fb1", + "scripts/prime_field_checks.py": "be21a411cdca5fa331fb9f36352ce98c055cb5e973553255e4530d07e99ae5c4", + "scripts/prime_field_circuit.py": "50485ac5a65b2707aab2b870477fbf503f4776699f6dd5ce38fb2006cc95d0f6", + "scripts/prime_field_network.py": "16629edd6aa8357b3fabcd5e1a70027860864a64475e8c0e7c822ee81dcd1ab9", + "scripts/prime_field_supports.cpp": "927dbb2ac2a411f06d8d9e0c440548f7ab965311fd83f4a573668d94d0cff1fe" + }, + "scope": "Full h28 formal support counts and replacement boundary sums; full complex h28 coefficients, binary labels and compiled role frames; all scalar basis inputs and rational physical frames at h8. General tensor-stage, finite-alphabet and multiplication transfer rely on the written proof and retained upstream interfaces.", + "small_controls": { + "all_dirty_basis": { + "all_basis_inputs": 1156, + "corrected_three_stage_exchange": true, + "dirty_scratch_restored": true, + "forward_inverse_exact": true, + "h": 8 + }, + "coefficients": { + "every_node_has_common_pair": true, + "h": 8, + "inputs": 56, + "retained_coefficients_exact": true, + "roles": 1044, + "side_coefficients_exact": true + }, + "physical_frames": { + "frames": [ + { + "every_edge_nested_and_nondegenerate": true, + "loss": 168, + "reverse": false, + "total_rank": 9472 + }, + { + "every_edge_nested_and_nondegenerate": true, + "loss": 168, + "reverse": true, + "total_rank": 9472 + } + ], + "h": 8 + } + }, + "status": "CONDITIONAL 373/10^11 > 2^-28 WITNESS; NOT FULL FORMAL VERIFICATION", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a", + "witness": { + "absorption_gap": "2313/250000000000000", + "bit": { + "D": 379712972620800, + "L": 284784729465600, + "N": 949282431552000, + "W": 230640858616896000, + "eta": "39/520019360", + "h": 28, + "m": 21952, + "retained_totals": 378, + "roles": 11840940, + "s": 5063027748645128371200, + "v": 98280 + }, + "complex": { + "D": 18030055680, + "I": 32196528, + "L": 26143580736, + "N": 35158608576, + "W": 2085111546336, + "eta": "5/12693352", + "h": 28, + "m": 21952, + "roles": 93838, + "s": 45772350635112192, + "scalar_gates": 8702721518400, + "v": 3276 + }, + "constraints": { + "K_dominates_log_p": "49985001/100000000", + "K_smaller_than_ell": "24999/100000000", + "alpha_squared_theta_growth": "1/5000", + "beta_below_one": "6/25", + "beta_positive": "19/25", + "c_positive": "9999/10000", + "crt_layout": "15003/4000000000000", + "delta_below_one_eighth": "124999/1000000", + "delta_positive": "1/1000000", + "epsilon_positive": "4999/10000", + "gaussian_cost": "199/1000000", + "guard_width": "2014601/100000000", + "kappa_positive": "373/100000000000", + "lambda_above_sigma": "3151/100000000000", + "lambda_above_tau": "1/100000000000", + "lambda_below_one": "749/100000000000", + "lambda_prime_above_lambda": "1/100000000000", + "lambda_prime_below_one": "187/25000000000", + "leaf_cost": "47/25000000000", + "packed_overhead": "1/100000000000", + "prefix_cost": "24999/100000000", + "prime_interval_growth": "1/5000", + "r_superpolynomial": "5001/10000", + "reserved_axes": "24997499813/25000000000", + "scalar_cost": "500099/1000000", + "sigma_below_one": "39/1000000000", + "sigma_positive": "999999961/1000000000", + "tau_below_one": "3/400000000", + "tau_positive": "399999997/400000000" + }, + "deficit_slacks": { + "bit": "25621089/1300048400000000000", + "complex": "619909/158666900000000" + }, + "guard": { + "B": 580186831374453739191529048436710092608, + "C0": 1330227007428750171975003167066483053874539349448633868671027791264533942584236249186304, + "C1": "19601/10000", + "E": 580186831374453739191483276086074980416, + "beta": "19/25", + "m": 21952, + "s": 45772350635112192, + "zeta": "1/10000" + }, + "log_upper": { + "bit": "9997/1000", + "complex": "10" + }, + "margins": { + "g1": "24999/100000000", + "g2": "149955003/40000000000000000", + "g3": "934813/250000000000000", + "g4": "15003/4000000000000", + "g5": "199/1000000", + "g6": "500099/1000000", + "g7": "4999/10000" + }, + "minimum_margin": "934813/250000000000000", + "parameters": { + "C1": "19601/10000", + "beta": "19/25", + "c": "9999/10000", + "delta": "1/1000000", + "epsilon": "4999/10000", + "kappa": "373/100000000000", + "lam": "99999999251/100000000000", + "lamp": "24999999813/25000000000", + "sigma": "999999961/1000000000", + "tau": "399999997/400000000" + }, + "recurrence": { + "internal": "399999997/400000000", + "layer": "399999997/400000000", + "leaf": "12499999883/12500000000", + "preprocessing": "1/10000" + } + } +} diff --git a/certificates/short-guard-audit.json b/certificates/short-guard-audit.json new file mode 100644 index 0000000000..9ef159d0e7 --- /dev/null +++ b/certificates/short-guard-audit.json @@ -0,0 +1,84 @@ +{ + "conditional_targets": { + "hypothetical_34": { + "exponents": { + "compact_gadget": "624999999421/625000000000", + "gathering": "312499999803/312500000000", + "layer": "624999999791/625000000000", + "leaf": "624999999791/625000000000", + "status": "IMPLICATION OF AN UNPROVED GATHERING BOUND" + }, + "missing_hypothesis": "Uniform gather and inverse in O(V f^tau K^(1-tau) polylog(p)), with complete descriptor, padding and cleanup costs.", + "target": "1/17179869184" + }, + "hypothetical_39": { + "exponents": { + "compact_gadget": "1999999999921/2000000000000", + "gathering": "9999999999753/10000000000000", + "layer": "9999999999901/10000000000000", + "leaf": "9999999999901/10000000000000", + "status": "IMPLICATION OF AN UNPROVED GATHERING BOUND" + }, + "missing_hypothesis": "Uniform gather and inverse in O(V f^tau K^(1-tau) polylog(p)), with complete descriptor, padding and cleanup costs.", + "target": "1/549755813888" + } + }, + "guard_formula": "4*ceil(log2(p))+6", + "naive_cost": "Separate window moves or binary-key sorting give only a linear-in-f certified bound, insufficient for a sublinear layer.", + "next_step": "Search a gatherer or an operation on strided windows directly; gathering is the remaining theorem, not a free preprocessing step.", + "reduction_cost": "O(G_gather + G_inverse + V*(f*log(p))^tau + V), with a fixed number of gather/inverse calls", + "repair_controls": [ + { + "bad_fraction_upper": "5/32", + "f": 2, + "g": 10, + "normalized_repair_at_A_equal_p_R_at_least_p": "5/8", + "p": 2 + }, + { + "bad_fraction_upper": "35/16384", + "f": 7, + "g": 18, + "normalized_repair_at_A_equal_p_R_at_least_p": "245/8192", + "p": 7 + }, + { + "bad_fraction_upper": "5/16384", + "f": 16, + "g": 22, + "normalized_repair_at_A_equal_p_R_at_least_p": "5/512", + "p": 16 + }, + { + "bad_fraction_upper": "125/268435456", + "f": 100, + "g": 34, + "normalized_repair_at_A_equal_p_R_at_least_p": "3125/33554432", + "p": 100 + }, + { + "bad_fraction_upper": "5/4294967296", + "f": 1024, + "g": 46, + "normalized_repair_at_A_equal_p_R_at_least_p": "5/2097152", + "p": 1024 + }, + { + "bad_fraction_upper": "78125/75557863725914323419136", + "f": 1000000, + "g": 86, + "normalized_repair_at_A_equal_p_R_at_least_p": "1220703125/590295810358705651712", + "p": 1000000 + } + ], + "status": "SHORT-GUARD REDUCTION; NO FAST GATHERER AND NO NEW KAPPA", + "stream_control": { + "active_bits": 12, + "guard_bits": 8, + "max_per_monotone_stream": 4351, + "records": 1048576, + "required_global_monotone_streams_at_least": 241, + "scope": "One-shot uncoded global monotone split/merge only; not all linear-time tape routines." + }, + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a" +} diff --git a/certificates/source-frame-stream-witness.json b/certificates/source-frame-stream-witness.json new file mode 100644 index 0000000000..dfc0da44a9 --- /dev/null +++ b/certificates/source-frame-stream-witness.json @@ -0,0 +1,1094 @@ +{ + "assembly": { + "absorption_gap": "170276524048839/500000000000000000000000000", + "constraints": { + "K_dominates_log_p": "49999946161/100000000000", + "K_smaller_than_ell": "53839/50000000000", + "alpha_squared_theta_growth": "53839/50000000000", + "beta_below_one": "999/1000", + "beta_positive": "1/1000", + "c_positive": "1", + "crt_layout": "107668065934695201/100000000000000000000000", + "delta_below_one_eighth": "1249999999/10000000000", + "delta_positive": "1/10000000000", + "epsilon_positive": "49999946161/100000000000", + "gaussian_cost": "26917/25000000000", + "guard_width": "62550201842411/250000000000000", + "kappa_positive": "538339/500000000000", + "lambda_above_sigma": "20381280001/10000000000000000", + "lambda_above_tau": "1/10000000000000000", + "lambda_below_one": "21533589999/10000000000000000", + "lambda_prime_above_lambda": 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Original tests were run in the full pinned snapshot; the retained test module references the excluded patch generator and is not independently runnable from this limited reference tree.", + "sha256": { + "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4", + "NOTICE": "68f78fa47282d16b8a54162d7097e3890c7911de79df5ba671305600119afffa", + "certificates/source-frame-network.json": "bd5bca02075fb8cb3ba427ac8e1e07261fbd8177ab419c289a9fa15d3752a5a1", + "notes/source-frame-assembly.tex": "12ad61dc0d7e549fe82891626ddfcde14bc4fab6a916023a3b16f81a0e79ea82", + "notes/source-frame-bit.tex": "c6f69e073c66b1bda8c94f245a92481a78239fb2bffd40caaf92513d28e7844c", + "notes/source-frame-complex.tex": "e6bc3bb5038d05cc84272443a419a05968dd1cb9dea4326ca67200a4c64ec7a9", + "scripts/batched_network.py": "45d918d5a064339299c21a12f02462ac3fe7e18380fa6a5a6e2542566865b830", + "scripts/source_frame_network.py": "e9b9992daf325d74edc5d3803a3e03fe1ffb00c3b9b9f0a4f604c8f1472966a2", + "tests/test_source_frame_network.py": "630861c4c243a081ee9229c597e6e00d8a858175f330f3f671afcb15d0e703f7" + }, + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/13" + } + }, + "scope": "Exact h30 producer, matching, frames, bilateral allocation, rank moments and assembly arithmetic. General source-frame, simultaneous-basis, nested-wire, all-complex batching, precision, and inherited multiplication arguments remain mathematical dependencies.", + "source_frame_basis_control": { + "common_basis": { + "h": 3, + "m": 27, + "new_class": "A4", + "new_profile": { + "corner_width": 0, + "middle_width": 21, + "nullity": 3, + "rank": 24, + "singleton_corner_pivots": 3 + }, + "new_projectors_checked": 9, + "old_class_counts": { + "A1": 27, + "A2": 3, + "A3": 27 + }, + "old_projectors_checked": 57, + "one_common_integer_parameter_tuple": true, + "parameter_sha256": "7c52f4532d5ec97319bb882d54b8a32a378611995583ee0134ec1664a36fc6c1", + "parameters": { + "G": [ + [ + [ + -45, + -17, + 7 + ], + [ + -27, + 3, + 45 + ], + [ + -40, + -26, + -4 + ] + ], + [ + [ + -13, + -27, + -35 + ], + [ + 24, + 46, + 17 + ], + [ + 43, + -8, + -1 + ] + ], + [ + [ + 10, + -38, + -39 + ], + [ + 2, + 39, + 48 + ], + [ + 47, + 28, + -38 + ] + ], + [ + [ + 2, + 1, + -23 + ], + [ + -3, + 4, + -34 + ], + [ + 22, + -11, + -8 + ] + ], + [ + [ + -36, + -49, + 32 + ], + [ + -48, + 31, + -10 + ], + [ + 5, + -12, + 26 + ] + ], + [ + [ + -4, + 50, + 31 + ], + [ + 9, + 9, + 9 + ], + [ + 17, + -42, + -1 + ] + ], + [ + [ + -21, + -7, + -28 + ], + [ + -32, + -27, + 24 + ], + [ + -35, + 21, + -7 + ] + ], + [ + [ + -38, + 42, + -15 + ], + [ + -29, + -7, + 20 + ], + [ + -18, + 47, + -43 + ] + ], + [ + [ + 44, + -32, + 17 + ], + [ + -13, + 31, + 29 + ], + [ + -15, + 11, + 25 + ] + ] + ], + "K": [ + [ + 40, + 18, + 9, + -15, + -14, + 5, + 45, + 28, + -47 + ], + [ + 27, + 38, + 34, + -40, + -41, + 17, + -4, + -46, + -20 + ], + [ + 7, + -39, + 36, + 44, + -29, + 48, + -34, + 35, + 45 + ], + [ + 9, + -13, + -2, + -24, + -8, + 25, + -27, + 45, + 50 + ], + [ + -8, + 43, + -6, + -12, + -41, + -7, + 10, + -43, + 45 + ], + [ + 0, + 42, + -21, + 26, + -28, + -39, + -32, + 43, + -10 + ], + [ + -44, + -36, + 30, + 31, + -37, + 12, + 37, + -37, + 39 + ], + [ + -25, + 20, + 47, + 28, + 48, + 38, + -11, + 37, + -23 + ], + [ + -13, + 1, + -17, + -3, + -31, + 12, + 45, + -38, + -23 + ] + ] + }, + "rational_corner_nonsingularity_certified": true, + "total_projectors_checked": 66 + }, + "five_subset": { + "accepted_trial": 0, + "endpoint_difference_identity": true, + "exact_line_norm": "3", + "exit_idempotent_mod_prime": true, + "h": 5, + "label_form": "I-2J/25", + "m": 125, + "new_entrance_rank": 0, + "new_total_rank": 125, + "old_total_rank": 125, + "pivot_profile": { + "corner_width": 0, + "middle_width": 115, + "nullity": 5, + "rank": 120, + "singleton_corner_pivots": 5 + }, + "ranks": { + "final": 25, + "internal": 5, + "new_exit": 120, + "old_exit": 100, + "source": 20 + }, + "source_frame_rank": 20 + }, + "predecessor": { + "author": "eumemic", + "head": "3ef246fa4f69c87ebfed78376418afa9ffcad145", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/13" + }, + "prime": 1000003, + "scope": "One common h3 basis covers 57 old and nine new projectors; an actual h5 five-subset tests the shifted source and exit. Nonzero modular corners certify rational nonzero corners. General simultaneous h28/h30 basis existence, rational pivot zero patterns, physical endpoint assignment and multiplication assembly remain written proof dependencies.", + "status": "FINITE SOURCE-FRAME BASIS CONTROL; GENERAL PROOF SEPARATE" + }, + "status": "CONDITIONAL SOURCE-FRAME STREAM WITNESS; NOT FORMAL VERIFICATION" +} diff --git a/certificates/sparse-fusion-audit.json b/certificates/sparse-fusion-audit.json new file mode 100644 index 0000000000..584ea8d56b --- /dev/null +++ b/certificates/sparse-fusion-audit.json @@ -0,0 +1,68 @@ +{ + "batching": { + "existing_active_exponent": "12499999963/12500000000", + "necessary_c_upper": "37/12499999963", + "proof": "For positive batch sizes b_i summing to f, sum b_i^tau >= f^tau. Separate calls to the existing whole-slot bound cannot improve its exponent by batching alone.", + "scope": "Optimizing these certified separate-call upper bounds, not a lower bound on actual tape time.", + "spacing_exponent": "12499999963/12500000000" + }, + "candidate_budgets": { + "hypothetical_34": { + "alternative_spacing_exponent": "1/1000", + "beta": "1/5", + "c": "1/10", + "maximum_spacing_exponent_at_current_active_saving": "330684010559/52166656000000000000", + "necessary_active_saving_for_alternative": "52166634142592961/104333312000000000000", + "note": "Strict necessary internal-cost budgets for these fixed parameters, not a primitive or sufficient whole-algorithm certificate.", + "target": "1/17179869184" + }, + "hypothetical_39": { + "alternative_spacing_exponent": "1/1000", + "beta": "1/100", + "c": "1/200", + "maximum_spacing_exponent_at_current_active_saving": "101359728243/16693329920000000000", + "necessary_active_saving_for_alternative": "83466636923924417/166933299200000000000", + "note": "Strict necessary internal-cost budgets for these fixed parameters, not a primitive or sufficient whole-algorithm certificate.", + "target": "1/549755813888" + } + }, + "central_frame_screen": { + "baseline_incident_rank": 56500, + "center_roles": 26, + "density_minus_center_count": "66", + "h": 25, + "joint_baseline": 112350, + "joint_frame_pairs": 4096, + "joint_minimum_changed_rank": 112850, + "minimum_sampled_changed_rank": 57000, + "rank_density": "92", + "sampled_changed_frames": 63, + "scope": "All four central frames may vary, but side frames, topology and the tensor factor are retained. Does not exclude changing those ingredients or the tape primitive.", + "theorem": "The original four central frames jointly minimize the directional rank lower bound when side frames and the active h-factor are fixed.", + "triples": 2300 + }, + "finite": { + "all_seven_product_is_identity": true, + "collective_symmetry_columns": [ + 62, + 61, + 59, + 55, + 47, + 31 + ], + "collective_symmetry_dimension": 6, + "gl3_invertible_matrices": 168, + "gl3_phase_preserving_pairs": 102, + "maximum_minimum_direction_calls": 4, + "phase_group_dimension": 3, + "phase_group_size": 512, + "scope": "Exact operator identities and optimistic directional-call distances; no tape cost or legal cross-gate fusion is inferred.", + "six_direction_minimum_calls": 1, + "six_direction_product_equals_inverse_of_missing_direction": true, + "unequal_phase_shear_commutes": false + }, + "next_step": "Search coordinated side/central changes, a new gate topology, or a new sparse address primitive. Central-only relabeling is excluded in the audited model. Require the complete gate product and tape accounting.", + "status": "BOUNDED SPARSE/FUSION AUDIT; NO NEW MULTIPLICATION EXPONENT", + "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a" +} diff --git a/certificates/ternary-depth.json b/certificates/ternary-depth.json new file mode 100644 index 0000000000..9ea694bf62 --- /dev/null +++ b/certificates/ternary-depth.json @@ -0,0 +1,53 @@ +{ + "direct_exchange": [ + { + "all_basis_inputs": 8, + "dirty_reflection_involution": true, + "exact_exchange": true, + "field": 2, + "reflection_is_identity": true, + "scratch_restored": true + }, + { + "all_basis_inputs": 8, + "dirty_reflection_involution": true, + "exact_exchange": true, + "field": 3, + "reflection_is_identity": false, + "scratch_restored": true + }, + { + "all_basis_inputs": 8, + "dirty_reflection_involution": true, + "exact_exchange": true, + "field": 5, + "reflection_is_identity": false, + "scratch_restored": true + }, + { + "all_basis_inputs": 8, + "dirty_reflection_involution": true, + "exact_exchange": true, + "field": 7, + "reflection_is_identity": false, + "scratch_restored": true + } + ], + "merged_two_stage_identity_by_field": { + "2": false, + "3": true, + "5": false, + "7": false + }, + "one_pass_and_basis": { + "arbitrary_target_blocks_checked": 81, + "basis_can_zero_decoder_noise": true, + "basis_changes_source_extraction": true, + "basis_fixes_additive_embedding": true, + "early_basis_does_not_fix_post_swap_noise": true, + "field": 3, + "noise_rank": 2, + "source_dimension": 2 + }, + "status": "SCALAR IDENTITIES AND SCOPED OBSTRUCTIONS; NO NEW RANK-SAVING NETWORK" +} diff --git a/certificates/ternary-direction.json b/certificates/ternary-direction.json new file mode 100644 index 0000000000..e16f3ca56f --- /dev/null +++ b/certificates/ternary-direction.json @@ -0,0 +1,321 @@ +{ + "distinction": "Optimistic ceilings discard real circuit costs; finite controls and exact assembly do not replace the general written proofs.", + "fixed_h28_role_targets": { + "h": 28, + "scope": "Necessary only, at fixed h28 with unchanged center loss, bank sharing and kappa 2⁻³⁴ in O(n (log n)^(1−κ)), building on OpenAI #109. -T(n) = O(n (log n)^(1 − κ)), with κ = 2^-78. - -That's about 570 million times our previous exponent saving of 5.8e-33—not a practical speedup claim. +Our exact witness, κ = 8.3 × 10⁻¹¹, is ~48 million times our previous 2⁻⁵⁹ exponent saving. This compares exponents, not practical runtimes. **2/4** -The key: the manuscript already supports nonadjacent axis swaps. Using them directly reduces layout routing from O(d²) to O(d) swaps, while preserving the network and arithmetic. A small scheduling change unlocks much stronger parameters. +The key: move compact control bits instead of entire spaced windows. We construct the temporary fields from existing address space and restore them exactly. Removing the spacing penalty breaks the quadratic bottleneck in our previous analysis. **3/4** -Our earlier ceiling applied to the original routing estimates. Improving those estimates removes the constraint that made the exponent saving cubic in the network saving. The new witness scales quadratically; we haven't established a new ceiling. +This is a structural proof extension. The note covers tape costs, recursive layout, exceptional repair and precision. The upstream theorem remains assumed; our new arguments await independent review. Exact arithmetic and finite tests support the written proof. **4/4** -The bound remains conditional on the upstream algorithmic interfaces. Notes, exact certificates, source patches, and the routing audit are public. Developed with OpenAI Codex; independent review welcome. - +Proof note, certificates, source patch and a focused review guide: https://github.com/CrocSwap/integer-mult-bounds -## Claim boundaries +The complex finite network now limits this construction. Improving it is the next research target. Developed with OpenAI Codex; independent review welcome. -- The headline is a conditional asymptotic bound, not an independently proved - upstream theorem or a measured runtime improvement. -- The finite network and numerical operations are retained from our h = 46 - construction; the sequence of data-movement operations changes. -- The earlier ceiling remains valid under its stated cost assumptions. It is - not a lower bound for integer multiplication or the revised routing schedule. -- No priority or optimality claim is made for the new witness. +## Claim boundaries -Author: Douglas Colkitt. See the [routing note](../artifacts/nonadjacent-axis-note.pdf) -and [audit](nonadjacent-axis-audit.md) for the precise statements. +- The strongest witness is `83/10^12`, strictly between `2^-34` and `2^-33`. +- The approximately 48-million-fold comparison uses that exact witness. + Comparing only `2^-34` to `2^-59` gives exactly `2^25`, about 33.55 million. +- The result is conditional on retained upstream interfaces and the new written + arguments. Tests and PDF builds are not independent mathematical review. +- The scoped ceiling applies only to the fixed `h=25` complex motif and retained + Gaussian/leaf inequalities. It is not a general limit on integer multiplication. +- No practical benchmark, best-known claim, priority claim or unrestricted + optimality claim is made. + +Author: Douglas Colkitt. See the [proof note](../artifacts/compact-control-note.pdf) +and [review guide](research/compact-control-review.md) for the precise scope. diff --git a/docs/packed-movement-audit.md b/docs/packed-movement-audit.md index cd3ae56431..52a0bce0ba 100644 --- a/docs/packed-movement-audit.md +++ b/docs/packed-movement-audit.md @@ -1,5 +1,12 @@ # Packed movement audit: constants improve, the exponent obstruction remains +**Follow-up:** the [preparation pass](research/layer-preparation.md) extends +the recurrence sum below to both orderings of tau and sigma. The historical +calculations here retain their original parameters and scope. +The subsequent [short-guard reduction](research/short-guard-audit.md) shows +that logarithmic working guards suffice after gathering. The gathering and +inverse costs are still missing; it does not certify a new kappa. + October 7, 2026. This is a scoped follow-up to the conditional 2^-76 witness. It audits the existing packed routine and its recurrence, not all possible implementations of the same mathematical operator. No stronger headline kappa diff --git a/docs/release-notes.md b/docs/release-notes.md index 56a7bfc791..2d3e0ed7d0 100644 --- a/docs/release-notes.md +++ b/docs/release-notes.md @@ -1,46 +1,68 @@ -# Draft routing-improvement release +# Draft compact-control release Repository: [CrocSwap/integer-mult-bounds](https://github.com/CrocSwap/integer-mult-bounds) -Suggested description: +Suggested repository description: -> Conditional exponent improvements for integer multiplication, with exact rational certificates, source patches, and proof dependency audits. +> Conditional integer-multiplication bound with κ > 2⁻³⁴: compact-control movement, written proofs, exact certificates, and reproducible source patches. -Suggested release title: **Direct axis routing: conditional exponent saving 2^-78** +Suggested release title: **Compact controls: conditional exponent saving κ > 2⁻³⁴** ## Release body -This research draft improves the axis-routing schedule in OpenAI's -*Integer multiplication below n log n*. Its strongest supplied witness is -`kappa = 2^-78` in `O(n (log n)^(1-kappa))`, for the manuscript's -fixed-alphabet, fixed-tape Turing-machine model. +This research draft gives the conditional bound -Direct nonadjacent swaps reduce the number of layout interchanges from O(d²) -to O(d). The revised cost permits a fixed dimension exponent, removing the -cubic constraint behind our earlier parameter-only ceiling. The finite -network and numerical operations are retained from our h = 46 construction. + T(n) = O(n (log n)^(1-kappa)), + kappa = 83/10^12 = 8.3e-11 > 2^-34, -New artifacts: +in the fixed finite-alphabet, fixed-tape Turing-machine model of OpenAI's +*Integer multiplication below n log n* (result family #109). -- A routing research note and dependency audit. -- An exact rational certificate for the 2^-78 witness, plus a 2^-107 witness - using the original network and recurrence exponents. -- Three independent source patches, including a routing-only alternative. -- Finite checks of coordinate permutations, padding, coefficient records, - CRT order, and parameter constraints. +The new witness increases the exponent saving by approximately **47.85 million +fold** over our preceding published `2^-59` result. The convenient dyadic +corollary `2^-34` is exactly `2^25` times `2^-59`. These are exponent +comparisons, not measured or predicted practical speedups. -The earlier parameter-only note, certificates, and seven patches remain -available. Its ceiling applies to the original cost accounting. No ceiling -or optimality claim is established for the revised accounting. +The structural change is to move **compact dirty-control fields instead of +entire spaced windows**. This removes the `K^tau` spacing penalty from the +simultaneous layer and the restriction behind the previous quadratic +conversion of finite-network savings into the multiplication exponent. -The full upstream multiplication theorem remains an assumption. The exponent -comparison does not imply a practical speedup. Independent review is welcome. +The proof supplies wider-control streaming with all offset work charged, +front/back temporaries carved from existing address space, complete ranges +through padding and recursion, exact temporary restoration, and deterministic +exceptional repair charged locally at every node. The bit network remains at +`h=50`; the original complex network is independently instantiated at `h=25`. +A generalized stopping-depth guard and exact assembly checks complete the +witness. -Author: Douglas Colkitt, with assistance from OpenAI Codex. Apache-2.0 licensed. +Artifacts: + +- [Proof note (PDF)](../artifacts/compact-control-note.pdf) and [TeX source](../notes/compact-control-note.tex). +- [Exact layer certificate](../certificates/compact-control-layer.json) and [address checks](../certificates/compact-control-audit.json). +- [Combined patch](../patches/compact-control-34.patch), applied directly to the unchanged pinned upstream source. +- [Review guide](research/compact-control-review.md), [current status](research/current-status.md), and [reproduction instructions](reproducibility.md). +- Preserved historical proofs, certificates, patches and bounded research audits. + +The complete upstream theorem remains an assumption. The new general claims +rest on written proofs, supported by finite tests and exact arithmetic; they +have not received independent mathematical review or formal verification. +The repository does not contain a full multiplication-machine implementation. + +The remaining scoped ceiling, below `8.369598075e-11`, concerns the fixed +`h=25` complex motif and retained Gaussian/leaf inequalities. It is not a +limit on other networks or on integer multiplication generally. + +Author: **Douglas Colkitt**, with assistance from OpenAI Codex. The initial +compact-control proposal came from a separate research agent. Apache-2.0 +licensed. No independent endorsement, priority or unrestricted optimality +claim is made. Independent mathematical review is welcome. ## Publication details -Attach `artifacts/nonadjacent-axis-note.pdf` if publishing a release. Add a -release version and date to citation metadata when a release is actually -published. This draft does not create a GitHub release or post an announcement. -Suggested thread: [announcement.md](announcement.md). +This file is prepared release copy, not a record of a GitHub release. +Attach `artifacts/compact-control-note.pdf` when publishing. For a GitHub +release body, replace the relative artifact links above with links under +`https://github.com/CrocSwap/integer-mult-bounds/blob//`. +Set release version/date metadata only when a release is actually published. +The accompanying [announcement](announcement.md) is draft copy for that time. diff --git a/docs/reproducibility.md b/docs/reproducibility.md index 8cdac287e9..3c8507b557 100644 --- a/docs/reproducibility.md +++ b/docs/reproducibility.md @@ -1,9 +1,10 @@ # Reproducing the result -The primary artifacts are the [paired-network note](../artifacts/paired-note.pdf), -[strongest patch](../patches/h50-paired-59.patch), and exact JSON certificates. -They are conditional on the algorithmic interfaces identified in the -[dependency audit](audit.md). +The primary artifacts are the [compact-control note](../artifacts/compact-control-note.pdf), +[combined patch](../patches/compact-control-34.patch), and +[exact layer certificate](../certificates/compact-control-layer.json). +They are conditional on the retained algorithmic interfaces and written +extensions identified in the [review guide](research/compact-control-review.md). ## Requirements @@ -50,6 +51,54 @@ This command performs these steps: h=50 paired-network certificate and independent patch for kappa=2^-59. They also check the stopped guard constants and revised Gaussian inequalities; these proof models are selected explicitly in the parameter checker. + `scripts/prepare_layers.py` records generalized guard and unequal-exponent + recurrence estimates, the independent complex-family search, and clearly + marked hypothetical sparse-primitive targets in `layer-preparation.json`. + This preparation pass does not change the published 2^-59 witness. + `scripts/audit_sparse_fusion.py` records finite phase identities, a bounded + central-frame screen, and the cost targets for future sparse primitives. + Its output is a research audit, not a stronger multiplication certificate. + `scripts/audit_fused_block.py` checks a complete parity-butterfly sandwich + and records the cut-rank obstruction for coordinate-child/pointwise-only + recursion. Its exact identity does not supply a faster multiplication bound. + `scripts/audit_short_guards.py` records logarithmic working guards and a + reduction to a still-unproved fast gathering routine. Exact address tests + check the reduction's maps, not a new tape-time or multiplication bound. + `scripts/audit_gather_schedules.py` checks a reversible merge tree, scoped + interval-schedule obstructions, and optimistic round-reuse accounting. + It does not rule out coded gatherers or certify a new multiplication bound. + `scripts/audit_coded_carries.py` checks an exact but linear carry code and + small controls for a scoped flat affine-circuit obstruction. The bounded + attempt supplies no sublinear recurrence or new kappa. + `scripts/audit_cancellation.py` checks the first cancellation circuit's + scalar map and dirty-scratch restoration, then records nonorthogonal-path + bounds rejecting two topologies under the retained data-stage frames. + `scripts/audit_joint_frames.py` reproduces a complete small bit invocation's + edge ranks and checks rational controls for the fixed-boundary local + optimality proof. It does not certify a stronger multiplication bound. + `scripts/audit_compact_controls.py` checks the compact dirty-control address + maps, inverses and exceptional repair. `scripts/compact_control_layer.py` + records the new conditional `83/10^12 > 2^-34` witness, with reservations, + the revised recurrence, independently sized complex motif, generalized + guards and all assembly margins. Its general tape claims depend on the + written [compact-control proof](../notes/compact-control-note.tex). + `scripts/make_compact_control_patch.py` generates the independent combined + patch, including the changed global exceptional-stream accounting. + `scripts/complex_network.py` builds the compressed complex side circuit of + `scripts/complex_circuit.py`, checks every side coefficient, binary label, + residual and compiled role in both stage directions, and records the + conditional `59/10^11 > 2^-31` witness. `scripts/make_complex_circuit_patch.py` + extends the compact-control patch with the new complex construction. + `scripts/fast_gaussian.py` checks the fast-resampling witness `1479/10^12`, + the exact step inequality behind the bound on powers of the resampling + correction, and the Neumann counts; `scripts/make_fast_gaussian_patch.py` + extends the compressed-complex patch with the chirped-correlation Gaussian + maps, the revised interfaces and the new assembly parameters. + `scripts/aligned_bit_network.py` builds the aligned bit circuit of + `scripts/bit_circuit.py`, checks every partial output, group total, + addition and role frame in both directions, and records the + `1624/10^12` witness; `scripts/make_aligned_bit_patch.py` extends the + fast-resampling patch with the new bit interface. `scripts/research_networks.py` and `scripts/search_network_variants.py` record scoped family bounds and clearly marked exploratory scores. 4. The standard-library unittest suite checks certificate boundaries, selected @@ -63,8 +112,20 @@ that the checked-in outputs reproduce exactly, run on a clean checkout: git diff --exit-code -- certificates patches ``` +The aligned-bit note builds with `make aligned-bit-note`, the +fast-resampling note with `make fast-gaussian-note`, the +compressed-complex note with `make complex-note`, and the compact-control +note with `make compact-note`. Their statements remain +conditional on the identified upstream interfaces and supplied written +extensions. See the +[current research status](research/current-status.md) before treating older +research pages or preparation certificates as current conclusions. + The theorem certificates and all-h bound comparisons use `fractions.Fraction`; -no floating-point result controls their acceptance. The separate exploratory +no floating-point result controls their acceptance. The fast-resampling tests +also compare explicit resampling matrices against the power bound at 160 +decimal digits; the certificate itself checks the underlying step inequality +exactly. The separate exploratory higher-subset screen uses floating point to select candidates, then encloses their scores exactly; it certifies neither a new construction nor optimality. Sampled finite identity tests use fixed random seeds. @@ -79,7 +140,7 @@ existing one. It leaves the bundled source untouched: mkdir -p build mkdir build/review cp -R upstream/build build/review/build -git apply --directory=build/review patches/h46-dag-63.patch +git apply --directory=build/review patches/compact-control-34.patch ``` Read `build/review/build/main.tex` and its included sections. The other patches @@ -90,18 +151,23 @@ are alternatives based on the same original manuscript, not successive commits. With Tectonic installed: ```sh -make note +make compact-note ``` -The result is `artifacts/parameter-note.pdf`. Build the separate routing audit +The result is `artifacts/compact-control-note.pdf`. Build the earlier +parameter-only note with `make note`, producing `artifacts/parameter-note.pdf`. +Build the separate routing audit with `make audit-note`, producing `artifacts/nonadjacent-axis-note.pdf`. Build the tuning note with `make tuned-note`, producing `artifacts/routing-tuned-note.pdf`. Build the preserved stage-sharing note with `make reuse-note`, producing `artifacts/stage-reuse-note.pdf`. Build the rectangle-circuit note with `make incidence-note`, producing `artifacts/incidence-note.pdf`. -Build the latest shared-computation note with `make dag-note`, producing +Build the shared-computation note with `make dag-note`, producing `artifacts/dag-note.pdf`. +Build the cross-group and paired notes with `make shared-point-note` and +`make paired-note`, producing `artifacts/shared-point-note.pdf` and +`artifacts/paired-note.pdf`, respectively. The first run may download fonts and TeX packages; the numerical verification does not use them. PDF builds may differ in metadata or typesetting across TeX environments. The exact-byte @@ -129,5 +195,60 @@ and [setup-python](https://github.com/actions/setup-python) actions, with read-only repository permissions. The PDF is supplied for readers and can be rebuilt locally using the command above. -Build the latest note with `make paired-note`. The earlier notes and +Build the latest note with `make compact-note`. The earlier notes and patches remain available as independent witnesses. + + +## Branch experiments (`codex/raise-kappa`) + +The PR #7 import is pinned to `6725c6a17b17871a35353fd29157f4ed851bc114`. +Its original source, attribution, tests, and certificate are retained. +The new branch work consists of code, certificates, and working derivations. + +```sh +python3 scripts/source_frame_stream_network.py --workdir /private/tmp/source-frame-stream +python3 scripts/batched_stream_network.py --workdir /private/tmp/batched-stream +python3 scripts/ternary_reuse_network.py +python3 scripts/ternary_fused_network.py +python3 -m unittest discover -s tests -v +``` + +The source-frame driver reproduces the current conditional +**κ = 1.076678e-6 > 2^-20** result. It rebuilds the h30 selected 16+14 +producer, stars, exact support/core audit, complete frame construction, and +bilateral physical compiler. It checks every partner permutation image and +combines PR #13's source relocation with the complete h28 complex residual +histogram and precision bound. It writes `source-frame-stream-witness.json`. +After reproducing the dimension-30 producer separately, pass +`--producer certificates/dimension30-stream-producer.json` to reuse it with +source-hash validation. Pinned PR #12 and #13 sources live in `references/`. + +The earlier batched driver reproduces conditional **κ = 1.64886e-7** +at h28. It regenerates the selected 16+12 plan, checks every support family, +resynthesizes the four-point stars, independently checks the rewritten DAG, +audits all rational frames, and materializes the full nested-wire register +schedule. It then checks the recursive moments and all assembly inequalities. +Its three outputs are `ternary-stream-producer.json`, +`ternary-stream-witness.json` (singleton recursion), and +`batched-stream-witness.json` under `certificates/`. Source and artifact hashes +bind the finite results to the working arguments. PR #10's unmodified +dependencies and pinned manifest live in `references/pr10/`. + +The fused driver regenerates the direct producer, direct assembly, frame +fusion, and fused assembly certificates together. It checks every output +support family, repairs and checks the rational frames, computes F3 boundary +ranks, and checks the exact assembly inequalities. Its source hashes include +the working arguments in `docs/research/`. It requires a C++17 compiler, +several GB of memory, and roughly two minutes on the development machine. +The reuse driver checks a separate improvement to PR #7; its savings must +not be added to the direct producer's savings. + +`make verify` includes these checks and all inherited checks. For a quick +arithmetic check without regenerating certificates, use +`python3 scripts/source_frame_stream_network.py --quick` (including the +complex histogram but without the large producer). Exact finite checks do +not replace the general compiler, controlled-basis, transfer, and tensor-stage +arguments or the inherited multiplication interfaces. The current composition +uses PR #10's corrected Gaussian shifted-error enclosure; the historical +source is preserved. Older role-budget ceilings assumed singleton recursion +and do not apply to the batched result. diff --git a/docs/research/aligned-bit.md b/docs/research/aligned-bit.md new file mode 100644 index 0000000000..d8bbad17ee --- /dev/null +++ b/docs/research/aligned-bit.md @@ -0,0 +1,62 @@ +# Aligned bit circuit with cheaper centers: conditional kappa = 1624/10^12 + +October 8, 2026. Contributed by eumemic, prepared with assistance from Claude +(Anthropic). Conditional on the pinned upstream interfaces, compact control, +the compressed complex network, fast Gaussian resampling and the written +center-frame argument. Not independently reviewed or formally verified. + +- [Proof note (PDF)](../../artifacts/aligned-bit-note.pdf) and + [source](../../notes/aligned-bit-note.tex) +- [Construction text](../../notes/aligned-bit-construction.tex) +- [Independent upstream patch](../../patches/aligned-bit-30.patch) +- [Exact certificate](../../certificates/aligned-bit-network.json) + +## Why the bit network + +After fast resampling, `g2, g3, g4 <= a_b min(epsilon, 1 - epsilon)`, so +`kappa < a_b/2`. The bit deficit per data element is + + eta_b = (v - 6 (central loss)) / (2 m (v + R)), + +with `v = 19600`, `m = 125000`, `R` the auxiliary roles per invocation and +central loss `h^2 = 2500` per invocation. One unit of central loss is worth +about 640 side roles. + +## Cheaper centers + +Each center wire `C_i` used to gather all sources at the frame `D_1` and +scatter at `D_0`, losing `P (x) F` (dimension `h`). Now `C_i` is the role +carrying the group total `z_i = sum_{T contains i} x_T` out of the side +circuit. Its frame there is `D_{H_i}` with `H_i` the star span, of dimension +`h - 1`. The scatter still runs at `D_0`, so the loss is `h - 1`. In stage 2 +the center rises to `D_1` at the scatter and falls to `D_{H_i^perp}` at the +reversed total, again losing `h - 1`. The central loss per invocation becomes +`h(h-1) = 2450`, and the numerator `v - 6h(h-1) = 4900` instead of 4600. + +## Aligned side circuit + +The paired recursion now uses the global blocks `{2k, 2k+1}` in every group, +with the partner of the common point as a singleton. At the top level, the +leave-one-block-out sums of the pair-star `{i, q}` are taken over the blocks +avoiding both `i` and `q`, by one prefix/suffix chain shared by groups `i` +and `q`, plus one group-specific triple. Side additions fall from 450,394 to +435,346, including the 50 group totals. With 58,800 partial outputs and 50 +center roles, `R = 494196`. + +## Result + + eta_b = 49/1284490000 > (325/10^11)(11737/1000), + kappa = 1624/10^12, minimum margin g3 = 162446751/10^17. + +The leaf needs `(1 - beta) a_c > 1 - lambda'` and the guard `beta > 3/4`; +`beta = 19/25` satisfies both with the compressed complex saving `14/10^9`. +A further 3% on `a_b` would close this window, so the next bit improvement +must come with a stronger complex saving or the sharper guard `C1 = 4 - 3 beta` +(available because `s_c < m_c^4`). + +## Bounded search notes + +Exact CP-SAT searches over center families of "triples inside Y" and +"star-in-Y" wires at `h = 8, 10` found nothing below `h(h-1)`. A rank +argument bounds any center loss below by about `h^2/6`, so the gap is open. +Block sizes 3 and 4 in the paired recursion were worse than 2. diff --git a/docs/research/assembly-breakthrough.md b/docs/research/assembly-breakthrough.md new file mode 100644 index 0000000000..2f062409a9 --- /dev/null +++ b/docs/research/assembly-breakthrough.md @@ -0,0 +1,209 @@ +# Dependency paths remove the large complex guard exponent + +October 7, 2026. Research refinement prepared with assistance from OpenAI +Codex. Conditional on the existing compressed complex circuit, its nested +binary frames and the retained recursive layer interface. This is not +independent review or formal verification, and it announces no new kappa. + +The current guard bounds coefficient dependency depth by concatenating all +`s_c = 27,215,641,140,750,000` recursive calls. The network's dependency +graph gives a much smaller factor: `q = m + 6h = 15,775` at `h = 25`. +Consequently the stopped guard may use + + C1 = beta + rho(1-beta) + zeta, rho = 1001/1000, + +in place of `5 - 4 beta + zeta`. The old large constants remain sufficient. +At `beta = 1/1000`, `zeta = 1/10000`, this gives `C1 = 1.001099`. +With `epsilon = 0.49999`, the guard condition has nearly one half of slack. + +This permits a small stopping exponent without weakening the guard. The +existing complex leaf saving becomes `(1-beta) a_c = 1.3986e-8`, versus +`3.36e-9` at `beta = 19/25`. That is useful room for a better bit circuit; +it does not itself change the current bit-limited multiplication exponent. + +## 1. Count recursive calls on a dependency path + +Consider one invocation of the finite complex network that reduces an +`e`-axis layer to children on `e/m` axes. Each edge has nested labels `U,V`. +Its operator is a product of exactly + + r(edge) = abs(dim V - dim U) + +recursive child kernels, with only the retained phase and coordinate +operations around them. The direction changes whether a child is forward +or inverse, but does not change its coefficient-depth bound. + +Evaluate each fixed scalar gate from saved inputs, as in the existing guard +argument. A dependency path through that gate enters at an input and exits +at an output. All its incidences have the same frame. Therefore a path +through the network selects a directed path of labeled edges even when it +switches roles at a gate. Using all-to-all dependencies at each gate is a +safe overestimate; zero scalar coefficients need not be exploited. + +Write `D(path)` for the sum of the downward dimension changes along this +path. Cancellation of consecutive signed changes gives the exact identity + + sum r(edge) = dim(last frame) - dim(first frame) + 2 D(path). + +The same bound covers a path ending partway through an edge. Insert one +intermediate label for each orthonormal residual vector; these dimensions +are monotone between the original endpoints. It also covers intermediate +values inside a scalar gate because the gate adds no further child call. + +Every frame dimension is between zero and `m`. Thus, if `D(path) <= D_*` +for every dependency path, at most `m + 2 D_*` recursive child calls lie +on any one path. Let `A(e)` bound coefficient dependency depth. With the +same additive charge `E = 64(W+m+1)^3` for fixed scalar arithmetic and phase +corrections as in the current proof, we obtain + + A(e) <= (m + 2 D_*) A(e/m) + E. + +This changes the guard recurrence only. The *time* recurrence still uses +the total `s/W`; calls on different branches still take time. Copying, +parking, swaps, compact controls and exceptional repair permute whole +coefficient encodings, so they do not introduce coefficient arithmetic. + +## 2. The existing circuit has `D_* <= 3h` + +For a stage `j`, write `a = h^(j-1)` and suppress the future line factor. +The usual labels have + + dim D0 = h^j - h, dim D1 = h^j, + dim D_U = h^j - h + dim U. + +Each physical X data role moves from dimension `a` to `h^j`; each physical +Y role moves from `a-1` to `h^j-1`. Their intermediate labels are monotone. +Side roles use the monotone compiled labels in the forward stage and their +reverse complements in the inverse stage. Their initial and final +connections are monotone as well. These are the existing frame obligations, +not new assumptions that arbitrary side circuits automatically satisfy. + +The only downward edges are the `h+1` central returns. Every such edge goes +from `D1` to `D0`, losing `h` dimensions. They occur between the same two +central gate times. A directed dependency path can cross at most one of +them in an invocation: it cannot return to an earlier gate time, and it +chooses only one operand on entry to a later gate. The fact that there are +`h+1` central wires does not multiply the loss on one path. + +Invocations within a stage have disjoint data and scratch roles. A path +therefore visits at most one invocation in each stage. A data role can carry +the path to the next stage, while a scratch role has only its own invocation +and then its terminal connection. The compressed complex construction uses +separate scratch roles in the three stages. It follows that a whole-network +path crosses at most three central returns, so + + D_* <= 3h, q := m + 2 D_* <= m + 6h. + +This stage argument must be revisited if a future complex construction +shares scratch between stages or introduces a new decreasing frame edge. +The [ternary follow-up](ternary-direction.md) performs that check for PR #7: +its chronological stage-1/stage-3 joins are monotone and there is no sharing +within a stage, so the same bound holds. It does not automatically apply to +other sharing schedules. + +## 3. An explicit stopped guard + +At `h=25`, `m=15625` and `q=15775`. The integer comparison + + 15775^1000 < 15625^1001 + +certifies `q < m^rho` with `rho=1001/1000`, without floating-point logarithms. +Stop at `e < d^beta`, as before. The number of internal levels is at most +`j <= (1-beta) log_m d + 1`, and the leaf depth is at most `8 d^beta`. +Unrolling the new recurrence gives + + A(e) <= (8 d^beta + E) q^j + <= q(8+E) d^[beta + rho(1-beta)]. + +Keep the existing `B=s+E` and + + C0 = ceil(max(128 m B^2, 18 m B^2(1+1/zeta))). + +Since `q<=s`, the existing exact comparison +`s(8+E)<=9B^2` also bounds `q(8+E)`. There are at most +`m(1+1/zeta)d^zeta` base-m pieces. Reserved axes, outer phases and the final +denominator shift have the same `18d` allowance as before. Consequently + + A_layer <= 9mB^2(1+1/zeta) d^[beta+rho(1-beta)+zeta] + 18d + <= C0 d^C1, + +where `C1=beta+rho(1-beta)+zeta > 1`. The original magnitude and denominator +induction then applies unchanged: `Delta=ceil(C0 d^C1)` guard bits suffice, +and `epsilon C1 < 1` keeps the coefficient widths `O(p)`. + +In particular, no norm bound for intermediate circuits and no intermediate +rounding is being substituted for the existing exact arithmetic argument. + +## 4. Exact controls + +Run: + +```sh +python3 scripts/experiments/assembly_breakthrough_guard.py \ + --output certificates/assembly-breakthrough-guard.json +``` + +The checker uses the actual compiled side roles and label dimensions at +`h=8,10,12,25`. For each of the three stage directions it executes the full +early/middle/late frame schedule on the X, Y, side and central roles. Every +gate is conservatively treated as all-to-all. At `h=25`, each invocation has +112,821 roles. The checker finds exactly 26 decreasing edges, all central +returns of rank 25; the maximum downward weight of a dependency path is +25, and its maximum edge-rank weight is 15,675. The latter includes direct +scratch-terminal connections to the full ambient space. Composition of the +three stages uses the structural argument above, rather than materializing +the full network's trillions of wires. + +The certificate also checks the integer-power comparison and 81 finite +unrollings of the recurrence, including large additive charges. It computes +the existing `C0` and checks `q<=s`, `q(8+E)<=9B^2`, and the final whole-layer +constant comparison with exact integers and rationals. These +controls validate the finite frame accounting and arithmetic; they do not +mechanically verify the underlying subspace inclusions, the stream machine, +or the asymptotic proof. + +An alternate witness in the same research certificate retains the aligned +bit parameters, including `kappa=1624/10^12`, while setting `beta=1/1000` +and `C1=1001099/10^6`. It checks every `fast_constraints` slack, the complete +compact-control layer exponents, and all seven `fast_margins`. The minimum +remains `g3=162446751/10^17`, with absorption gap `46751/10^17`. This is a +concrete way to review the guard change without treating it as a new +headline or modifying the published certificates. It is additionally +conditional on this note's dependency-path argument. + +The focused regression checks are in `tests/test_dependency_guard.py`. +They distinguish a dependency path from chronological execution, check a +path ending partway through a decreasing edge, verify that sequential +decreases cannot be treated as parallel, and exercise the compiled forward +and inverse schedules, constant comparisons and alternate witness. + +## 5. What assembly and bootstrap alone cannot amplify + +With bit saving `a_b`, the present completed layer has exponent at least +`1-a_b`, giving `kappa < epsilon a_b`. Transform layout costs +`p K^(-a_b)` and the size constraint requires `K=o(ell)`, where +`ell=Theta(p^(1-epsilon))`; hence this layout also allows at most +`a_b(1-epsilon)`. The independent CRT/layout row has that same margin. +Together they imply `kappa < a_b/2` within this architecture. + +Recursively using an improved integer multiplier for the packed polynomial +products improves only the already generous product margin. If its saving +is `kappa_0`, then that row changes from margin `epsilon` to +`epsilon+(1-epsilon)kappa_0`. The layer and movement rows remain unchanged. +Thus that bootstrap does not amplify the bottleneck saving. + +Tensoring a completed primitive by ordinary recursive composition also +preserves its exponent: arity `m^t` with normalized branching `(s/W)^t` +has logarithmic exponent `log_m(s/W)`. It does not supply a stronger bit +primitive. Any many-order improvement must improve that primitive or prove +a different layout/whole-transform algorithm that bypasses these rows. + +The one-stage scalar identity `SWAP = I + [I;I][I I]` in characteristic two +is a possible different architecture. Its transparent evaluation can +restore dirty scratch because equal additions to both banks preserve +`X+Y`. However the obvious frame assignment makes an early X injection +raise a scratch role to its target line, followed by a Y source-copy visit +at the zero frame. This introduces downward losses for side roles that were +monotone in the three-stage construction. No valid low-loss one-stage +assignment is proved here; correct scalar algebra alone does not establish +the required rank deficit. diff --git a/docs/research/batched-stream.md b/docs/research/batched-stream.md new file mode 100644 index 0000000000..7efcbe1e58 --- /dev/null +++ b/docs/research/batched-stream.md @@ -0,0 +1,133 @@ +# Batching the smaller producer + +This working result combines the local producer/compiler experiments with +IceKylin's [PR #10](https://github.com/CrocSwap/integer-mult-bounds/pull/10), +pinned at `62691e395a0458ce089a1c7b5d89e74291e95e29`. The conditional assembly +check gives **κ = 164886/10^12 = 1.64886e-7**, or `82443/812 ≈ 101.531` times +the starting aligned-bit value. PR #10 itself gives `6149999/(5*10^13)`. +The inherited multiplication interfaces and the new transfer arguments remain +assumptions; the executable checks do not formally verify that whole theorem. + +## Producer and physical registers + +The selected top partition is 16+12. Rectangular transpositions and selected +local split substitutions give 9,763,505 additions. Resynthesizing fixed +four-point stars removes 107,097 additions. Every replacement uses disjoint +source supports and returns exactly the same boundary sums. Those source +vectors have Gram matrix `I+2J`, so every replacement source span is positive. + +The resulting producer has 9,656,408 additions and 98,658 output uses, +or 9,755,066 roles under the original compiler. Its full rational-frame audit +reassigns 724 nodes to source spans and leaves no unresolved frame. Nested +delivery reuse, including exact saturated-source and saturated-dual inclusions, +then removes 983,670 roles. There are 20,337 dual-to-dual continuations and +36,945 source-to-dual continuations. The physical allocator independently +counts **8,771,396** registers and checks every transition and protected output. +Each invocation still has center loss 9,828 and rank sum 250,925,864 in either +orientation. + +The [stream argument](ternary-stream.md), +[source saturation argument](ternary-stream-saturated.md), and +[dual saturation argument](ternary-stream-bilateral.md) explain the scalar map, +dirty-scratch restoration, ordering, and frame inclusions. Both invocation +directions retain every auxiliary role's low and high boundary frames. The +producer certificate includes a strict dual-frame control with frames of +dimensions 24 and 27: seven roles become six, and all 22 data/scratch basis +symbols pass the transparent wrapper. It also retains the full small direct +producer control. Thus the added rule is exercised by a role-saving example. + +## Why PR #10 applies + +Batching changes the recursive child widths, preserving the rank deficit. +Let `h=28`, `m=h^3`, `v=C(h,5)`, `N=v^3`, and `R=8,771,396`. Then + +``` +W = 2 v²(v+R) +L = 3 v² C(h,2)(h−2) +s = Wm−N+2L +``` + +Our compiler leaves the relevant boundary edges unchanged: + +| Edge class | Copies | Original rank | +| --- | ---: | ---: | +| Shared stage-one/stage-three auxiliary join | `v²R` | `m−2h` | +| Stage-two auxiliary sink | `v²R` | `m−h²` | +| Stage-three data entrance | `2N` | `(h²−1)(h−1)` | + +The controlled-basis argument depends on these tensor boundary forms, not +the internal addition graph. Its recursive block widths are `m−4h`, +`m−2h²`, `m−2h²−2h+2`, and the extra stage-two corner `h²`. Every remaining +rank unit stays a singleton child. An independent small basis control checks +the corner minors and the actual lower-triangular elimination pivot blocks. + +For each class let `r_i` be child width divided by `m`, and `w_i` its rank +mass divided by `Wm`. The exact checker verifies + +``` +sum(w_i) = s/(Wm) +sum(w_i / (1−a_b log_upper(1/r_i))) < 1 +a_b = 329773/10^12 +``` + +This bounds the recursive moment using `exp(x) <= 1/(1−x)`. Increasing +`a_b` by `10^-12` fails this conservative comparison. Omitting the controlled +stage-two corner also fails. These are negative controls on this certificate, +not global optimality claims. + +The complex circuit remains PR #7's circuit. PR #10's whole-residual batching +supports `a_c=7/10^7`. Its path guard has `q=22120`, `rho=6/5`, and +`C1=11999/10000` at `beta=1/1000`. The existing chronological stage-one to +stage-three sharing preserves that path argument. + +## Assembly and the Gaussian correction + +Use `epsilon=499999917/10^9`, `c=1`, `delta=10^-10`, and put lambda and +lambda-prime respectively `10^-16` and `2*10^-16` above `tau=1−a_b`. +All 29 constraints and seven absorption margins are strict. The minimum is +`824432362644205083/(5*10^24)`, giving gap +`2362644205083/(5*10^24)` above the selected κ. The exact grid check maximizes +the minimum margin on the `10^-9` epsilon grid with the other parameters fixed: +the intersection of the increasing third margin and decreasing Gaussian +margin lies between epsilon `0.499999917` and `0.499999918`, and the first wins. +The selected κ is the largest strict `10^-12` grid value below the minimum. +PR #10's original epsilon +would leave too little Gaussian/prefix margin for this larger κ. + +This result includes PR #10's correction to the inherited Gaussian input +error bound. The generator shifts an approximation by +`B=ceil(1.14 alpha²)`; absolute error is multiplied by `2^B`. Therefore use +`F=2^B`, not just the smaller exact-magnitude bound `exp(pi alpha²/4)`. +For integer `p>100` and `2<=alpha= 6`, their rational span has these dimensions. Write `a=|A|` and +let `1_A` be the indicator of `A`. + +| Mask size | Span dimension | A normal when the span is a hyperplane | +| --- | --- | --- | +| `a=0` | `0` | Predicate is zero | +| `a=1` or `a=2` | `h-1` | `3 1_A - 1` | +| `3 <= a <= h-3` | `h` | No normal | +| `a=h-2` | `h-1` | `e_i-e_j`, where the omitted pair is `{i,j}` | +| `a=h-1` | `h-1` | `e_i`, where `i` is omitted | +| `a=h` | `h` | All triples | + +For the middle case, differences of allowed triples span all coordinate +differences within `A` and within its complement. These provide `h-2` +independent directions. Allowed intersection sizes one and three give +independent vectors of block sums `(1,2)` and `(3,0)`, supplying the remaining +two dimensions. For `a=1,2`, only the first block-sum pattern occurs, leaving +exactly the displayed one relation. Complement masks reduce the last cases +to triples containing either zero or two points of the omitted pair, or +avoiding the omitted singleton. Those have exactly the displayed relation. +The all-triples family spans the whole space by coordinate differences and +one triple of nonzero coordinate sum. + +Thus **every nonzero affine predicate spans at least `h-1` dimensions**. +The script exhausts every mask for `h=6,7,8,9` and every mask-size orbit at +`h=10,11,24,38,48,50`. A full modular rank is a rational lower bound; the exact +integer normals, or ambient dimension, give the matching rational upper +bound. The general statement follows from the proof above, not from +extrapolating the finite checks. + +## Center loss is at least `h(h-2)` + +A useful center has nonzero gather and scatter predicates. Its source span +has dimension at least `h-1`; the orthogonal complement of its target span +has dimension at most one. The center must therefore lose at least `h-2` +dimensions along its gather-to-scatter passage. Allowing a nonnested path +cannot evade the bound: rank subadditivity bounds the entire passage through +its endpoint rank difference. Additional decreases only increase the charge. + +There must be at least `h` centers. Indeed, `Q` has full column rank over +`F2`: differences of triples span the even-weight subspace, and a triple has +odd weight. Hence `rank_F2(Q Q^t)=h`. Every center rectangle has scalar rank +at most one, so any factorization of `B` has at least `h` nonzero terms. +Consequently the loss per invocation obeys + + center_loss >= h(h-2). + +The aligned loss is `h(h-1)`. This theorem leaves room for one dimension per +center; it does not claim the current loss is exactly optimal. In particular, +complement-pair predicates have normals `e_i-e_j`, and some such hyperplanes +can have orthogonal normals. Assuming all parity masks span the full space, +or assuming every center necessarily costs `h-1`, would be incorrect. + +## Even an optimal side circuit in this class stays within a factor 15 + +The auxiliary-bank factorization supplies another useful lower bound: +`rank(J L V)=v`, where `v=C(h,3)`, so the number of auxiliary roles satisfies +`R >= v`. This is independent of the current binary-addition compiler. + +Give the candidate both optimistic floors, `R=v` and loss `h(h-2)`, whether +or not they can be simultaneously achieved. Under the retained network +counts, + + eta <= [v - 6h(h-2)] / [4 h^3 v], + a_b <= -log(1-eta) / log(h^3), + kappa < a_b/2. + +The optimistic deficit first becomes positive at `h=38`. Exact logarithm +enclosures isolate `h=48` as the maximum among `11 <= h < 200`. For all +`h >= 200`, the uniform bound + + a_b < 1 / (4*200^3 - 1) + +is below that maximum, because `eta < 1/(4h^3)`, +`-log(1-eta) < eta/(1-eta)`, and `log(h^3)>1`. +The resulting convenient exact ceiling is + + kappa < 22778/10^12 = 2.2778e-8, + +which is less than `14.027` times the current conditional witness. This is +an upper bound for the specified construction class, not an achievable +witness and not a general bound on integer multiplication. + +Thus merely changing the basis of point centers and making the current side +circuit arbitrarily efficient cannot provide multiple orders of magnitude. +A larger jump needs to escape at least one of the explicit restrictions: +nonlinear center supports or a changed scalar correction, smaller rational +label spaces, a different stage architecture, or a different transfer. + +Reproduce with: + +```sh +python3 scripts/experiments/bit_breakthrough_affine_centers.py +python3 -m unittest discover -s tests -p test_affine_center_targets.py -v +``` diff --git a/docs/research/cancellation-audit.md b/docs/research/cancellation-audit.md new file mode 100644 index 0000000000..073c05724b --- /dev/null +++ b/docs/research/cancellation-audit.md @@ -0,0 +1,279 @@ +# First cancellation circuits: scalar savings defeated by path rank costs + +**No new kappa.** The simplest cancellation circuit computes the correct +side map and restores arbitrary scratch. At h=50 it would reduce the side +role count from 509,194 to 137,150. However, with the retained data-copy and +injection frames it cannot have a positive rank deficit. The obstruction +allows arbitrary rational matrices at its internal gates, not just the old +source-span labels. + +A second prefix/suffix construction removes explicit self-corrections and +uses 234,900 side roles, but also fails a nonorthogonal-path rank bound. +These are scoped rejections of explicit topologies, not a lower bound on all +cancellation circuits. The published conditional +kappa=2^-59 witness remains unchanged. Code and exact controls are in +[audit_cancellation.py](../../scripts/audit_cancellation.py) and +[cancellation-audit.json](../../certificates/cancellation-audit.json). + +## 1. Use the correct scalar field + +The bit network's scalar gates operate over F2, although its frame matrices +are rational. These two fields must not be confused. + +Index inputs and outputs by triples of [h]. Let A1 be the matrix whose +entries are one exactly when the two triples intersect in one point. If +B1(S,T)=|S intersect T|, then over F2, + + A1 = B1 + I. + +For off-diagonal intersections 0,1,2 this has the desired values. On the +diagonal, 3+1=0 in F2. The integer identity involving pair-incidence terms +reduces to this formula; charging a coefficient -2 as a nonzero bit-network +operation would be incorrect. + +Define point aggregates and outputs by + + G_i = XOR_(S contains i) x_S, + z_T = XOR_(i in T) G_i XOR x_T. + +Each point aggregate is computed with its own balanced binary reduction +tree. Then each output uses two additions to combine its three point values +and a final addition of its own input. This fixed topology deliberately +retains cancelled paths rather than deleting them from the frame audit. + +With v=C(h,3), the point reductions use + + h (C(h-1,2)-1) = 3v-h + +additions. The output calculations use 3v, for c=6v-h additions in total. +The existing reversible gather/fanout embedding uses c+v=7v-h side roles. +At h=50: + + v=19600, c=117550, R_side=137150. + +This is approximately 3.71 times fewer side roles than the published circuit. +It is only a scalar-circuit comparison, not a multiplication improvement. + +## 2. Arbitrary scratch restoration still works + +The old reversible circuit compiler can be generalized to additions with +overlapping supports: its scalar invertibility does not require disjoint +supports. The source-span frame proof does require more, and is not reused. + +Track each node's actual coefficient support by XOR and its graph-reachable +input set by union. Every output z_T has coefficient zero on x_T, but x_T +is reachable through the final private input edge. This distinction is the +key to the failed frame transfer. + +Let L be the reversible mixer, V the input copy, and J the output injection. +The audited schedule + + L, J, inverse L, R, V, G, R, L, J, inverse L, G, V + +still produces side contribution JLVx and restores all initial scratch. +Here G,R are the retained central point gather/scatter operations. Since +JLV=A1 and RG=B1, the full invocation is y <- y+x. The three-stage schedule +still exchanges the banks and restores the shared auxiliary banks. + +Exact small input-basis checks include every dirty scratch input, in both +forward and reverse invocations. A complete small three-stage check includes +the first/third-stage auxiliary sharing. Thus scalar correctness is not the +reason this candidate fails. + +## 3. A rank-excess bound for a private diagonal path + +For any rational edge matrices M_tail,M_head, + + rank(M_head-M_tail) >= rank(M_head)-rank(M_tail). + +The sum of the quantities on the right telescopes over the network because +all incidences at a scalar gate share one frame matrix. With the original +source and sink matrices, this signed rank sum is Wm-2N: sources contribute +N and sinks Wm-N. Call the difference between an edge's rank charge and its +signed rank change its excess. Every edge has nonnegative excess. + +For a nondegenerate triple line t in F=Q^h with form I-J/9, let P_t be its +rational projection. At the first stage, the retained data-copy frame is +P_t and the late target-injection frame is I-P_t, within the active factor. +Their ranks are 1 and h-1, while + + rank((I-P_t)-P_t) = rank(I-2P_t) = h. + +The last equality follows because P_t is a projection: I-2P_t acts as -1 +on the line and +1 on its complement. On any edge path between these two +frames, rank subadditivity therefore gives total charge at least h. Its +signed rank change is h-2, so its total excess is at least **two**. + +No condition on the intermediate rational matrices is used. They may be +nonsymmetric, nonnested, or not projections. The future tensor factor at +stage one is a nondegenerate line, so tensoring with its projection leaves +these ranks unchanged. At later forward stages, the common B component +cancels in the difference and the same excess calculation applies. + +## 4. The explicit circuit contains enough disjoint paths to kill its saving + +For every T, the middle forward mixer has a path + + copy of x_T -> input fanout -> final private XOR for z_T -> injection into y_T. + +The first and last edges are unique to T. The direct x_T input to the final +XOR also has its own physical role. These paths are edge-disjoint, even +though other inputs to the output calculations come from shared point sums. +Changing a pivot role does not remove the graph path: all incidences at its +gate must still have the same matrix. + +There are v paths in each invocation and v^2 invocations in the first stage, +so already that stage contains N=v^3 edge-disjoint paths with excess at +least two each. Hence + + s >= (Wm-2N) + 2N = Wm. + +Strict contraction requires s=|U intersect T|-1. +Both have squared norm two. When c is nonzero, + + rank(I-P_t-P_u) = h. + +Indeed, any kernel vector must lie in span(u,t). On that span, the operator +sends alpha*u+beta*t to -(c/2)(beta*u+alpha*t), an invertible map. Its +endpoint signed rank change is still (h-1)-1=h-2. Consequently every path +from the retained copy frame for U to the injection frame for T has excess +at least two whenever the labels are nonorthogonal. This includes distinct +triples intersecting in two points, even without a diagonal path. + +### An explicit disjoint path packing + +Take even h>=6. Within each common-point module, order the pairs of the +remaining h-1 points by an Euler circuit in K_(h-1). Such a circuit exists +because every vertex has even degree. Consecutive pairs, including the last +and first, share one point. After adjoining the common point, consecutive +triples intersect in two points and hence have nonorthogonal labels. + +The prefix/suffix circuit contains edge-disjoint paths implementing the +cyclic assignment of source position k to target position k+1 modulo q: + +- For k= (Wm-2N)+2N = Wm. + +This rejects the explicit Euler-ordered prefix/suffix topology with retained +copy/injection frames, allowing arbitrary internal rational matrices. It +does not reject every ordering or implementation of leave-one-out sums. +Removing explicit self-corrections alone is insufficient. + +## 7. Research consequence + +Neither explicit circuit is a useful target for internal frame optimization: +the path bounds already permit arbitrary internal matrices and rule them +out under the retained data frames. + +The next cancellation attempt should first screen edge-disjoint paths +between **any nonorthogonal source and target labels**, not only diagonal +paths. A viable candidate must reduce that path packing or change the +data-stage frames themselves. Passing the screen is necessary, not +sufficient: scalar correctness, scratch restoration, physical role count, +and the complete edge-rank sum must still be checked. + +Shared corrections may help if they actually reduce the edge-disjoint path +packing. Mere sharing of algebraic expressions, or cancellation of final +coefficients, does not establish this. Conversely, reachability alone does +not prove rejection: shared edges cannot be charged repeatedly. + +## Verification + +Run `python3 scripts/audit_cancellation.py` and +`python3 -m unittest discover -s tests -p test_cancellation.py -v`, or +`make verify`. Ten tests cover small scalar maps, physical private edges, +arbitrary dirty inputs and the shared three-stage network for the first +candidate, exact rational projection ranks, arbitrary intermediate-matrix +controls, the local formula, and the h=50 count and loss budgets. They also +check the leave-one-out map, Euler orders, perfect matching and disjoint +paths at h=6,8,10, and distinct nonorthogonal endpoint ranks. The general +rejections follow from the written rank and matching arguments, not the +finite tests alone. Published certificates and patches are unchanged. diff --git a/docs/research/coded-carry-audit.md b/docs/research/coded-carry-audit.md new file mode 100644 index 0000000000..0ea862415a --- /dev/null +++ b/docs/research/coded-carry-audit.md @@ -0,0 +1,206 @@ +# Bounded coded-carry attempt and assessment + +**Outcome: exact gap restoration, but no sublinear recurrence and no new +kappa.** Stop this bounded attempt. The construction below works, yet its +best demonstrated cost is linear in the number of windows. A scoped bound +also excludes compressing it into a sublinear number of whole-slot affine +movements in a fixed-bank circuit. This does not exclude recursive coded +networks or general tape algorithms. + +The published result remains conditional kappa=2^-59. This follows the +[gather-schedule audit](gather-schedule-audit.md). The implementation is +[audit_coded_carries.py](../../scripts/audit_coded_carries.py), with exact +[controls and assessment](../../certificates/coded-carry-audit.json). + +## 1. Concrete target and stopping criterion + +An earlier control slot supplies n values x_j in [Q], Q=2^g. A target slot +has n periods of K bits, K>g, each containing a low g-bit value y_j and +arbitrary gap bits. Perform + + y_j <- y_j+x_j modulo Q + +without changing any gap bit, control value, spectator coordinate or payload. +This is the strided affine operation that the previous whole-slot substitute +failed to implement: its carries changed a positive-density set of gaps. + +For this bounded experiment, require an exact small identity and a cost +that could lead to a sublinear recurrence. A correct identity with a linear +number of full-stream stages is not sufficient to continue a large search. + +## 2. Exact code: split by carries, translate, recombine + +For one window define c_j=[y_j+x_j>=Q]. On the c_j branch, translate the +whole target slot by + + (x_j-Q c_j) 2^(jK). + +The new short-field value lies in [0,Q), so there is no net carry or borrow +into its gap. Let D_(j,c) be the address-dependent diagonal mask selecting +that branch, and P_b the full-slot cyclic translation by b. The exact +operator is + + T_j = P_(x_j 2^(jK)) D_(j,0) + + P_((x_j-Q) 2^(jK)) D_(j,1). + +This identity holds for arbitrary record values. The two masked streams +have disjoint destination support, so their sum, or bitwise XOR for bit +payloads, gives the required permutation. Selecting a branch can be done +by a full-stream scan with address arithmetic polynomial in p, absorbed by +the retained superpolynomial record width. Each translation is a legitimate +ordered-affine call because its offset uses only the earlier controls and +the fixed branch choice. Temporary streams use a fixed number of tapes and +are cleared after use. + +The different T_j commute as exact operators, because each updates a +different short field and restores all gaps. Their product implements the +whole target. The code tests the product and a direct address map on complete +arrays, including all arbitrary gap values. + +### Two implementations, neither with a saving + +Expanding the product gives 2^n masked translation terms, indexed by the +complete carry pattern c. Their offsets are + + sum_j (x_j-Q c_j) 2^(jK). + +When every x_j is nonzero, every carry pattern occurs, and the offsets are +distinct modulo 2^(nK). This flat expansion is exponential. + +Keeping the product factored uses two translations per window and O(n) +full-stream stages. It costs O(Vn) in the retained record regime. A balanced +description has normalized recurrence + + F(n) = 2F(n/2)+O(1), F(1)=O(1). + +Both subcomputations still act on the entire array. Creating two temporary +copies does not give two role streams of volume V/2. There is no analogue of +the original all-role network's s/W contraction. Restoring arbitrary gaps +is solved here; improving the recursive cost is not. + +## 3. Why a fixed invertible encoding cannot simply hide the carry + +Already for one window, full-slot unit translation on K bits has order 2^K, +whereas incrementing only the low g bits has order 2^g. After 2^g full-slot +increments, a gap bit changes; the desired short-field operator is identity. + +Conjugation by an invertible encoding preserves operator order. Thus no +fixed invertible encoding of the entire arbitrary-input state space turns +that full shift into the short-field increment. The statement also holds +for invertible linear encodings of full data banks. It does not exclude +redundant encodings followed by projection or a multi-operation construction +such as the exact code above. + +## 4. Flat affine coding still needs linearly many movements + +The next question was whether algebraic cancellation between coded streams +could combine the exponential carry branches into only a few movements. +There is a useful obstruction for a precisely specified class. + +### Circuit class + +Use a fixed number W of banks with the same complete address domain and +unchanging bank volume. Allow arbitrary address-dependent pointwise linear +mixing between banks, including masks, copies and cancellations. Movement +gates apply affine permutations to full equal-width address slots. They may +mix those slots affinely and temporarily change arbitrary auxiliary slots. +Their parameters may depend on read-only controls. Each movement is charged +as a separate complete-bank pass. There is no recursive row splitting, +smaller-volume child, or permutation of a proper subfield hidden in a gate. +Changing radix or padding to a different address domain is also outside this +fixed-domain model. + +Fix the read-only controls. For k movement gates, expanding any output/input +bank block into computation paths gives at most 2^k address maps: each path +uses a chronological subset of those gates. Pointwise masks and coding +change coefficients or cancel paths, but introduce no new address graph. +Each path map is affine. For a distinguished slot y, fixing all other input +slot values makes its output coordinate ay+b modulo 2^L, with L=nK. +The coefficient a need not be odd when slots are mixed. + +### An affine graph covers only a small part of the target + +Consider the required special case of flipping n separated selected bits: + + T(y)=y XOR M, M=sum_j 2^(rho+jK), 0<=rho<=K-3. + +At most 2^(L-n) values of y satisfy T(y)=ay+b for any a,b modulo 2^L. +Here is an elementary proof, valid for every n and K in the stated range. +Write y=sum_i 2^i y_i and let s_i be -1 at selected positions and +1 +elsewhere. Then + + T(y)-ay = M + sum_i (s_i-a) 2^i y_i modulo 2^L. + +If a is even, all s_i-a are odd. The summand weights have distinct +2-adic valuations 0,...,L-1. Their subset sums are distinct modulo 2^L: +in any difference, the least nonzero valuation cannot cancel. Thus every +right-hand side occurs at most once. + +If a=1 modulo 4, choose the n selected coordinates. Each coefficient +-1-a has valuation one, giving distinct weight valuations rho+jK+1= 2^n, so k>=n. + +This permits arbitrary coding coefficients and cancellations. A required +nonzero target entry must lie in the support of at least one path, regardless +of how the path coefficients combine. With fixed W and separate full-bank +passes, k>=n precludes a sublinear-in-n bound in this model. + +The one-bit target also occurs in the general short-field interface: choose +x_j=2^(g-1). Addition modulo 2^g then flips only the top short-field bit, +so rho=g-1; K>=g+2 supplies the two gap positions needed by the proof. +Thus this is not merely a small-g artifact. At K=2,rho=0 the clean bound +does have a boundary exception: the corresponding agreement maximum is +2^(L-n+1). The controls explicitly test that exception. + +### Fixed-order ordered-affine operations + +The same path argument applies, separately, to the manuscript's forward +whole-slot operations while field order stays fixed. These can add nonlinear +functions of earlier fields to a later target. Earlier fields never depend +on the initial value of y. Along each path, the output y is still affine in +its own initial value when the other initial coordinates are fixed. +Consequently the same agreement bound applies. + +This extension does **not** allow combining those nonlinear operations with +arbitrary field exchanges: moving a later field back into y can introduce +nonlinear dependence on the original y and escape the argument. Nor does the +argument cover recursion that splits banks into smaller volumes. These are +real escape routes, including mechanisms used in the upstream machinery. +The result is not a general lower bound for coded gathering or multiplication. + +## 5. Assessment and decision + +The bounded attempt produced an exact operation and a meaningful limitation +of its natural coded implementation. It produced **no candidate with a +sublinear recurrence**. Cheap exact carry cancellation alone is insufficient +when it still takes a full-stream stage per window. + +The evidence does not justify expanding this into a broad topology search +now. Return the main effort to stronger finite networks. Keep the short-guard +reduction and conditional linear targets as reusable infrastructure, but +reopen this direction only for a concrete construction that supplies smaller +child volumes or costed backward/nonlinear address movement and demonstrates +the complete all-role contraction. A suggestive identity or a new encoding +without that cost calculation is not enough. + +## Verification + +Run `python3 scripts/audit_coded_carries.py` and +`python3 -m unittest discover -s tests -p test_coded_carries.py -v`, or +`make verify`. Eight tests check complete small basis inputs, arbitrary gap +values, exact carry codes, translation support, exhaustive small affine +agreement, the distinct-valuation proof controls, encoding order, and the +scope boundary for nonlinear ordered operations. These finite checks support +the written arguments; they do not formally verify the upstream theorem. +Published certificates, source patches and the bundled manuscript are unchanged. diff --git a/docs/research/compact-control-audit.md b/docs/research/compact-control-audit.md new file mode 100644 index 0000000000..05ec3e6031 --- /dev/null +++ b/docs/research/compact-control-audit.md @@ -0,0 +1,159 @@ +# Compact dirty controls: a new movement candidate + +**Follow-up completed locally:** the wider-control, reservation, repair and +layer proofs are now supplied in the [compact-control note](../../notes/compact-control-note.tex), +with a conditional `kappa=83/10^12 > 2^-34` witness. See +[current status](current-status.md). The text below preserves the initial +address-only audit and its then-open obligations; it is historical. +An independent research agent proposed moving compact control fields instead +of gathering the spaced target windows. This pass checks that proposal with +arbitrary temporary values, including the case where the source slot follows +the target slot. It is now the priority movement candidate, ahead of the +previously proposed central-gate splitting experiment. + +The checked code is [audit_compact_controls.py](../../scripts/audit_compact_controls.py) +and its output is [compact-control-audit.json](../../certificates/compact-control-audit.json). +These are address-permutation controls, not a certified fixed-tape runtime +or a replacement manuscript proof. + +## 1. Why the dirty temporary cancels + +For a control bit x, target segment u, and arbitrary temporary digit t, +perform the integer operations + + u1 = u + 2*x*t, + t1 = t + (u1 mod 2), + u2 = u1 + x*(1-2*t1), + t2 = t1 - ((u2 mod 2) XOR x). + +The first operation preserves target parity a=u mod 2. Consequently + + u2 = u + x*(1-2*a), + t2 = t. + +Thus the target parity is XORed with x, its guard quotient is restored, and +the arbitrary temporary digit returns exactly to its original value. The +temporary is not assumed to be zero, and its load is modular addition, not +an overwrite or a free copy. + +For f selected bits, use n=f-1 such segments, omitting the highest selected +position as in the original packed-rectangle proof. Each segment begins at +j_i=rho+iK. Pack the n temporary digits into nG consecutive bits, with radix +B=2^G. The four simultaneous integer updates become whole-field modular +rotations. Supply the omitted highest selected bit separately through the +retained elementary two-bit XOR. + +## 2. Actual physical field order + +When x precedes y, assume complete compact fields t before x and b after y, +each of width nG. The physical layout may be written t,x,y,b, with arbitrary +spectator intervals. The first and third rotations target y and use x,t as +controls. To update t using y, swap t and b, rotate the back field using the +preceding y, then swap back. Apply the same pattern when unloading t. + +This uses four width-nG swaps and four rotations. The field b has arbitrary +initial contents, restored by the two swaps around each load or unload. + +When x follows y, use layout u,t,y,x,b, with three compact fields u,t,b. +First perform the earlier-source procedure controlled by the digit parities +of u. Load the selected bits of x into u by swapping u with b, rotating the +back field, and swapping back. Perform the earlier-source procedure again, +now controlled by the updated digit parities of u. Finally unload x from u. +Off temporary-digit overflow, the two control parities XOR to x. This uses +twelve compact swaps and ten rotations. One back field can be reused because +each load/unload restores its arbitrary contents. + +Every rotation in the checker has an explicit list of controls. It checks +that all controls physically precede that rotation's target. + +## 3. A sufficient invariant exceptional set + +Exclude any address with a temporary t digit equal to B-1. For each used +K-bit segment of y, write its initial value as 2g+a and require + + 2B <= g < 2^(K-1)-2B. + +This conservative interval keeps every intermediate target value within its +own segment. The temporary load also stays in its own radix-B digit. Hence +the packed modular operations agree with the independent integer identities +on the good set. They preserve all bits outside the used selected positions. + +For a late source, additionally exclude any u digit equal to B-1. The first +parity flip preserves the target guards; loading x into u cannot then carry +between temporary digits. The second flip and unload restore u,t,b. + +With complete independent field ranges, union bounds give + + delta_early <= min(1, n*(2^(-G)+8*2^(G-K))), + delta_late <= min(1, n*(2*2^(-G)+8*2^(G-K))). + +The ideal operation preserves the target guards and every temporary field, +so this bad set is invariant under it. Every actual primitive is a bijection, +and the complete operation agrees with the ideal map off the bad set. +The actual operation therefore preserves the same bad set. As in the +upstream repair argument, T S^-1 is supported on that set. The checker +constructs the inverse by reversing every rotation and swap and verifies +the repaired address map, including bad inputs. + +Taking G=4*ceil(log2 p)+6 makes the temporary-overflow term O(p^-3) when +n<=p. For K=d^c with fixed positive c and d=Theta(p^epsilon), the term +p*2^(G-K) is smaller than every inverse fixed power of p. This is the +right density scale for the retained sorting repair, but its complete +descriptor, inverse-computation and tape costs still need a written audit. + +## 4. Two substantive proof obligations + +**Wider controls.** The original ordered-affine lemma requires controls of +O(b) digits for a b-digit target. Here a y control can have nK bits while +the back target has only nG. A new bound must explicitly charge offset work. +The natural extension is O(V+M*poly(p)) for M records, because every control +and descriptor has O(p) bits and a fixed number of fields is used. Under +the existing superpolynomial record-width assumption, this would be O(V). +The two-piece rotation implementation in the original lemma suggests this +extension, but the old hypothesis cannot simply be omitted when citing it. + +**Complete temporary fields in every child.** The current layer contract +does not promise front and back scratch fields. They cannot be appended as +new independent address coordinates: that would multiply the record volume +by their ranges. + +A candidate allocation is to process selected axes in reserved leading and +trailing chunks individually, then retain their address bits as spectators. +Keep the front temporary region separate from the row index divided among +roles. Every row and recursive child must retain the complete temporary +ranges; padding and reassembly must preserve this property. + +Reserving O(dG) bits at each end needs O(dG/K+1) chunks, and their individual +processing would cost + + O(V*(d^(1-c)*log(p)+1)). + +This is a new top-level cost. It must be included in the layer exponent, +with strict slack above 1-c, and in the treatment of small D where the +reserved regions would overlap. It need not automatically fit below the +recursion's stopping exponent beta: what matters for this preprocessing +cost is the final layer bound. This allocation is a proposed route, not a +completed recursive invariant or an integrated proof patch. + +If both obligations and the repair are discharged, the candidate row-addition +cost is O(V*((f*log(p))^tau+1)), without the old K^tau factor. The full layer +must then be recomputed, including preprocessing, leaves, guards, complex +recursion and assembly margins. The earlier 2^-39 and 2^-34 targets remain +hypothetical until that work is complete. + +## Verification and next priority + +The address suite exhausts 2480 elementary integer cases and 24,576 small +packed cases, and checks 7200 deterministic samples with several segments, +extreme selected offsets, arbitrary scratch, inversion, bad-set invariance, +and exact exceptional repair. It also verifies the physical order of every +rotation's controls. These counts do not establish asymptotic tape costs. + +Run `python3 scripts/audit_compact_controls.py` and +`python3 -m unittest discover -s tests -p test_compact_controls.py -v`. + +The next bounded research goal should audit the wider-control streaming +bound and temporary-field allocation first. If either requires an overhead +that restores the old spacing penalty, record that obstruction promptly. +If they work, finish the repair and full recurrence before promoting any +new exponent. The proposed finite-network goal becomes the fallback. diff --git a/docs/research/compact-control-review.md b/docs/research/compact-control-review.md new file mode 100644 index 0000000000..f61a7465ab --- /dev/null +++ b/docs/research/compact-control-review.md @@ -0,0 +1,109 @@ +# Compact-control result: proof dependencies and review guide + +This guide accompanies the conditional witness `kappa = 83/10^12 > 2^-34`. +It records the supplied arguments and their verification limits. It is an +author-side publication review, not independent mathematical review. + +Start with the [nine-page note](../../artifacts/compact-control-note.pdf), then +the [combined patch](../../patches/compact-control-34.patch). The patch applies +directly to the pinned upstream source; it includes the retained improvements +and all four new construction sections. No earlier patch should be applied first. +The patched title page identifies Douglas Colkitt's modifications and states +their conditional status. Every changed source file has a modification notice; +the resulting manuscript is not presented as an OpenAI release. + +## Retained assumptions + +The upstream commit is `adc7f1241b42e322a6451854ab7e4b4c146bf78a`. Its complete +multiplication theorem is not proved or formally verified by this repository. +The present result retains its stream model, bit-transfer and complex-phase +interfaces, analytic estimates, resampling, prime selection, synthetic +transforms, and final rounding. It also retains the project's paired bit +network, nonadjacent routing and tighter Gaussian setup. + +The relevant source contracts are: + +- `02-streams.tex`, `lem:elementary-streams` and `lem:ordered-affine-streams`: + fixed-tape stream operations and ordered controlled rotations. +- `04-swap.tex`, `lem:chunk-swap`: arbitrary-width, nonadjacent binary chunk + interchange, including arbitrary spectators, padding and cleanup. +- `03-motifs.tex`, `lem:motif-residuals` and the complex phase interface: + scalar circuit, residual orthonormal bases and signed endpoints. +- `05-layers.tex` and `08-assembly.tex`: the retained layer interface and + final multiplication assembly, with the replacements identified below. + +Section labels refer to the bundled source or to the combined patched source, +as appropriate. The [reproduction instructions](../reproducibility.md) explain +how to materialize the latter without modifying `upstream/`. + +## New obligations and supplied arguments + +| Obligation | Argument supplied | What to challenge | +| --- | --- | --- | +| Controls wider than a rotation target | `lem:record-paid-rotation` in [movement](../../notes/compact-control-movement.tex): two-piece streaming, polynomial offset work per prefix fiber, at most one fiber per record | Whether every counter, reset and offset computation is paid by `O(V+M A^C)` without a descriptor scan per payload bit | +| Dirty compact controls | Earlier/later-source constructions in the same source; four-update cancellation, modular swaps and loads | Control order, arbitrary initial temporaries, digit overflow and both source orders | +| Deterministic exceptional repair | Ideal map preserves the bad set; actual bijection agrees off it; sort records under `T S^-1` | Bad-set invariance, inverse offsets on current controls, guard boundaries and exact restoration | +| Existing-coordinate temporaries | [Layout proof](../../notes/compact-control-layout.tex): two front ranges and one back range carved from reserved chunks | Full ranges in every child, no hidden extra volume, small-`D` fallback and reserved-axis cost | +| Recursive cost | Same source: `F(e) <= (s/W)F(e/m) + O((e log p)^tau+1)` plus leaf and reservation costs | Exactly `V/W` child volumes, spectator layout, stopping depth and uniform constants | +| Precision | [Generalized guard](../../notes/compact-control-guard.tex): additive node depth charge and actual stopping depth | New operations must only permute whole coefficient encodings; reserved kernels and all base-`m` pieces must be counted | +| Independent complex arity | [Complex interface](../../notes/independent-complex.tex): original motif at `h=25`, unchanged bit motif at `h=50` | Residual nondegeneracy and norm-one witnesses, scalar restoration, endpoints and absence of a hidden equal-arity requirement | +| Final assembly | Note and patched `08-assembly.tex`: seven margins and a strict absorption gap | Every changed cost must enter the completed layer estimate; retained numerical hypotheses must still hold | + +The old ordered-affine lemma alone does **not** establish the wider-control +bound: its original control-width hypothesis is narrower. The new proof pays +polynomial offset work using the superpolynomial record length. + +Likewise, the old global exceptional-sort estimate based on `2^-K` does **not** +apply. The new bad fraction is at most `5/(128 p^3)` beyond the common cutoff. +Sorting costs `O(V_call/p^2)` and extraction/reinsertion `O(V_call)` at each +node; these enter the volume-weighted recurrence. The patch replaces the +old global argument explicitly. + +The complete-field invariant concerns address ranges, not zero scratch values. +Each row contains all front, active and back coordinates. Padding adds whole +rows, and recursive role splitting divides only the separate row index. +Completed calls restore every temporary coordinate and preserve each original +row, so the padded zero rows can be removed. + +## Exact witness and scoped ceiling + +The [layer certificate](../../certificates/compact-control-layer.json) records +all parameter slacks and the hashes of the four construction sources. It gives + + min(g_1, ..., g_7) = 333833/(4*10^15) = 8.345825e-11, + kappa = 83/10^12 = 8.3e-11, + min(g_i) - kappa = 1833/(4*10^15) > 0. + +The strongest supplied witness is about 47.85 million times the previous +`2^-59` saving. If comparing the simpler dyadic corollaries alone, +`2^-34 / 2^-59 = 2^25 = 33,554,432`. Neither ratio is a runtime benchmark. + +The upper enclosure `8.369598075e-11` applies to the unchanged `h=25` complex +motif and the retained Gaussian/leaf inequalities. It is not an all-network +ceiling, an optimality theorem for integer multiplication, or a barrier to a +different assembly. The witness exceeds 99% of this scoped upper enclosure. + +## Reproducible checks and their limits + +Run the focused checks with: + +```sh +python3 scripts/audit_compact_controls.py +python3 scripts/compact_control_layer.py +python3 scripts/make_compact_control_patch.py +python3 -m unittest discover -s tests -p 'test_compact_control*.py' -v +git apply --check --directory=upstream patches/compact-control-34.patch +``` + +Run `make verify` for the full suite and `make compact-note` for the note. +The finite checks cover packed modular address maps and inverses, dirty +temporaries, repair, two-piece rotations, row padding/splitting, recurrence +orderings, rejected parameter choices, and source integration. They do not +simulate the complete fixed-tape multiplication machine or mechanically verify +the asymptotic proofs. PDF compilation checks typesetting and references. + +Publication preparation found no additional blocker in this author-side pass. +Independent review remains outstanding, especially for the wider-control tape +bound, recursive layout invariant and local repair summation. A substantive +failure in those arguments would revoke the new headline; passing arithmetic +tests alone would not rescue it. diff --git a/docs/research/complex-all-residuals.md b/docs/research/complex-all-residuals.md new file mode 100644 index 0000000000..ad18210d94 --- /dev/null +++ b/docs/research/complex-all-residuals.md @@ -0,0 +1,63 @@ +# Batching every complex residual + +The unchanged h28 paired complex network supports complex saving +`4191487/10^12`, compared with PR #10's `7/10^7` and PR #13's `18/10^7`. +This is headroom for a stronger bit producer, not a multiplication exponent +on its own. It uses IceKylin's whole-residual identity and mixed-width +recurrence from [PR #10](https://github.com/CrocSwap/integer-mult-bounds/pull/10). + +Each nonzero physical edge has an orthonormal residual basis and rank +`r < m = 21952`. The identity for mixed forward/inverse factors in PR #10 +replaces all `r` singleton calls by one child of width `r/m`. The sign passes, +phase, and coordinate changes stay in fixed local work. Applying that +identity to every edge leaves the network, rank sum, and deficit unchanged. + +The executable schedule counts 34 residual ranks. Stage one stops auxiliary +roles at dimension h; stage three begins those shared roles at dimension h, +so it counts the join exactly once. Stage two keeps its separate bank. +The summed ranks equal the inherited total `45772350635112192`; both original +bulk multiplicities also agree. Independent enumeration of the old three-bank +schedule, followed by the bank-sharing histogram substitution, agrees at +h8, h10, and h28. No assumption of a uniform edge rank enters the moment. + +For each rank r with multiplicity c, set `w=cr/(Wm)`. Exact outward logarithm +enclosures give + +``` +sum(w) = 1−eta +sum(w/(1−a_c log_upper(m/r))) < 1 +a_c = 4191487/10^12 +``` + +The strict gap exceeds `1.34e-14`; the next `10^-12` grid point fails this +conservative comparison. + +Batching additional residuals requires a new precision bound. The inherited +path argument still gives total rank at most `q=m+6h=22120`. The largest +edge is `A=m−2h=21896`. For `rho=3/2`, convexity shows that any partition +of at most q into parts at most A has normalized power sum at most + +``` +(A/m)^(3/2) + ((q−A)/m)^(3/2) + < 27921787004127/28000000000000 + < 999/1000. +``` + +The first inequality uses exact rational upper square-root bounds. It does +not assume a path visits only one selected edge. Every edge contributes at +most four wrapper operations, and there are at most s nonzero edges, so the +existing `4s` local overhead allowance suffices. The same induction with +`E=64(W+m+1)^3` yields `C1=3/2−beta/2+zeta=3749/2500` for +`beta=1/1000`, `zeta=1/10000`. This leaves room for epsilon near one half. + +The inherited coefficient/label/frame construction, PR #10 transfer, and +general path argument remain mathematical dependencies. The finite histogram +and rational inequalities are checked exactly; they are not formal proof of +the multiplication theorem. + +Reproduce with: + +```sh +python3 scripts/experiments/complex_all_residuals.py --output certificates/complex-all-residuals.json +python3 -m unittest discover -s tests -p test_complex_all_residuals.py -v +``` diff --git a/docs/research/complex-circuit.md b/docs/research/complex-circuit.md new file mode 100644 index 0000000000..8a56402823 --- /dev/null +++ b/docs/research/complex-circuit.md @@ -0,0 +1,122 @@ +# Compressed complex network: conditional kappa = 59/10^11 + +The compact-control witness `83/10^12` was limited by the complex network, +which still used the original side wires: one wire per ordered neighbor pair, +`v*z_c = 3,693,800` per invocation at `h=25`, against `v = 2,300` data values. +This contribution computes the same side correction with shared sums. + +- [Proof note (PDF)](../../artifacts/complex-circuit-note.pdf) +- [Construction source](../../notes/complex-circuit-construction.tex) +- [Independent source patch](../../patches/complex-circuit-31.patch) +- [Exact certificate](../../certificates/complex-network.json) + +| | Original complex motif | Compressed side circuit | +| --- | ---: | ---: | +| Side roles per invocation | 3,693,800 | 108,195 | +| `W_c` | 58,645,352,620,000 | 1,741,801,270,000 | +| Certified `a_c = 1-sigma` | `418/10^12` | `14/10^9` (33.5x) | +| Headline kappa | `83/10^12` | `59/10^11` (7.11x) | + +The bit network, compact-control movement, guard, Gaussian setup and +assembly are unchanged. Every new step is conditional in the same way as the +compact-control result: the upstream theorem's interfaces and the written +extensions are assumed, and nothing here is independently reviewed. + +## 1. The side map and the circuit + +Target `S` must receive `(1/2) * sum{x_T : T disjoint from S}` minus +`(1/2) * sum{x_T : |S cap T| = 2}`. Two cancellation-free families compute it. + +- **Disjoint sums.** A recursive halving of the 25 ordered points computes + triple sums avoiding up to three points. Cross terms between halves are + pair sums in the link of each point on the other side, followed by + leave-out sums over that point. Each target receives six pieces: the two + half sums and the two mixed families, each split over the halves of its + outer index. +- **Pair stars.** For each pair `{a,b}`, prefix and suffix sums of `x_abu` + over the other points. Target `{a,b,c}` receives the prefix before `c` and + the suffix after `c`. + +Disjoint pieces are injected with weight `+1/2` and pair-star pieces with +`-1/2`. At `h=25`: 80,595 additions, 27,600 injected pieces, and +`R = c + q = 108,195` reversible roles, using the bit network's role compiler +and its transparent twelve-operation schedule (with signs). + +## 2. The binary frame condition, and why pieces are split + +Complex labels live in `F2^25` with the dot product. Every edge residual must +have an orthonormal basis, which for a nondegenerate binary space means it +must contain a vector of odd weight. The labels are: + +- input `T`: the line ``; +- disjoint-sum node: the coordinate space of its covered points; +- pair-star node: the span of its `t_abu`, which is already orthonormal. + +The injection gate for `S` sits at the physical `Y` frame `t_S^perp`. If an +injected disjoint piece covered every point outside `S = {a,b,c}`, the +residual would be the plane ``: nondegenerate, but all its +vectors have even weight, so it has no orthonormal basis. No relabeling +helps, since a nondegenerate space between `F_{outside}` and `t_S^perp` is one +of those two spaces. The construction therefore injects only pieces that +leave an outside point `y` uncovered; then `e_y` lies in the residual. If a +piece would cover everything, its two summands are injected instead. At least +four pieces per target are needed for this label class; the circuit uses six +disjoint and six pair-star pieces. + +The reversed second stage uses orthogonal complements, as in the bit network. +Every reversed residual equals a forward one, so the same condition covers it. + +## 3. Rank, phases and depth + +Each side role runs monotonically from label `0` to the full space, so it +contributes exactly `m`; central and data roles are unchanged. Hence + + W_c = 2N + 3v^2 (R + h + 1) = 1,741,801,270,000 + s_c = W_c m - 2N + 2L_c = 27,215,641,140,750,000 + eta_c = 28/205,789,375, log m < 966/100, a_c = 14/10^9. + +The coefficient-depth bound needs only `total gates <= 12 W_c`: an invocation +has `4*82,895 + 4v + 4` gates, which gives `5.41e12 <= 2.09e13`. Gate +coefficients stay in `{0, +-1, +-1/2}`, so `E = 64(W_c+m+1)^3` and the +generalized guard apply with the new constants. Also `2 <= s_c < m^5`. + +## 4. Parameters + +With `sigma < tau` the compact-control internal exponent is `chi = tau`, so +the layer needs `lambda > tau` and `g3 = epsilon*(1-lambda') < epsilon*a_b`. +The witness uses + + tau = 1-296/10^11, sigma = 1-14/10^9, epsilon = 19999/100000, + c = 999/1000, beta = 1/1000, zeta = 1/10000, delta = 1/10^6, + C1 = 49961/10000, lambda = 1-2958/10^12, lambda' = 1-2956/10^12, + kappa = 59/10^11. + +The old `c = 1/5` would make `g2 = epsilon*c*a_b` binding. The spacing +exponent enters only through `epsilon(1+c) < 1`, `K/log p -> infinity`, and the +reserved-axis count `O(log d + d^(1-c) log p) = o(d)`. The minimum margin is +`g3 = 14779261/(25*10^15)`, above `kappa` by `29261/(25*10^15)`. + +## 5. The next ceiling + +With the published `h=50` paired bit network and the retained Gaussian margin +(`epsilon < 1/5`), `kappa < a_b/5 < 5.92e-10 < 2^-30`. The witness is above 99.6% +of that bound. The complex network now has spare saving (`a_c = 4.7 a_b`), so +the next improvement in this strategy must come from a stronger bit network +or a Gaussian step that permits larger `epsilon`. This is a scoped statement, +not a ceiling for other networks or algorithms. + +## Verification + +`python3 scripts/complex_network.py` regenerates the certificate. It checks +all 3,693,800 nonzero side coefficients and every zero, all 82,895 node labels +(nondegeneracy and odd-vector residuals on 188,790 inclusions), and all +108,195 compiled roles in both stage directions. The tests in +`tests/test_complex_network.py` cross-check the binary label algebra against +enumeration, reproduce the alternating-plane obstruction, run complete small +circuits (`h=8..12`) through dirty-scratch forward and inverse invocations and +the signed exchange, push every input basis vector through an `h=8` +invocation, and reject the witness with `c=1/5`, `epsilon=1/5`, or the old +complex saving. `scripts/make_complex_circuit_patch.py` produces the combined +patch; the patched manuscript compiles with Tectonic and no unresolved +references. These checks support the written arguments; they do not verify +the upstream theorem. diff --git a/docs/research/current-status.md b/docs/research/current-status.md new file mode 100644 index 0000000000..9092ba431d --- /dev/null +++ b/docs/research/current-status.md @@ -0,0 +1,213 @@ +# Working status: raising κ + +Updated October 8, 2026. Branch `codex/raise-kappa` starts at +`research/aligned-bit` (`5015011`). The current reproduced conditional +witness is **κ = 1.076678e-6 > 2^-20**, about **663×** the starting value. + +| Construction | κ | +| --- | --- | +| Starting aligned-bit branch | `1.624e-9` | +| Zhihao Chen's PR #7 | `3.73e-9` | +| IceKylin's PR #10 | `1.2299998e-7` | +| PR #12, dimension 30 | `1.2649e-7` | +| Eumemic's PR #13, auxiliary source frames | `7.699e-7` | +| PR #14, source frames and data corners | `9.0799e-7` | +| Smaller h30 producer + source frames + all complex residuals | **`1.076678e-6`** | + +The [selected construction](source-frame-stream.md) uses 13,056,812 +auxiliary roles per invocation. Selected splits and star resynthesis reduce +the producer; bilateral nested-wire reuse reduces physical registers without +changing center loss or auxiliary invocation boundaries. PR #13's source +relocation then removes the expensive stage-two auxiliary entrance calls. +The bit saving is `2153359/10^12`. + +Batching [every complex residual](complex-all-residuals.md) supports +`4191487/10^12`, beyond the complex saving in PR #13. Its convex path bound +allows precision constant `C1=3749/2500`. All 29 assembly constraints and +seven margins pass exactly. The [certificate](../../certificates/source-frame-stream-witness.json) +binds the complete producer, scalar/support/frame checks, physical allocation, +matching, independent small basis controls, rank moments, and source hashes. + +Original certificates from PR #7, #10, and #13 reproduced exactly. Their +attribution is preserved; selective unmodified dependencies are in +`references/`. The composition uses PR #10's corrected shifted Gaussian +error enclosure. General compiler, source-frame, basis, transfer, precision, +and inherited multiplication arguments remain mathematical dependencies. +PR #14's new data-corner blocks are not part of this witness. + +Further work continues on additional compatible bit blocks and scratch +reclamation. Those exploratory gains are excluded from the selected result. +Under the present source-frame recurrence, reaching 1000× at h30 would need +roughly 7.67 million roles instead of 13.06 million. Earlier singleton-only +role ceilings do not apply to batching; the newer exact budget screen is +`scripts/experiments/batched_role_targets.py`. + +Run `python3 scripts/source_frame_stream_network.py` for full reproduction. +`make verify` includes inherited checks. See the +[reproduction instructions](../reproducibility.md). + +## Inherited status snapshot + +The following snapshot, including its authorship and publication references, +is preserved from `research/aligned-bit`. “Current” below refers to that +starting branch, not to the experiments above. + +# Current contracts and research status + +Updated October 7, 2026. Author: Douglas Colkitt. All results remain conditional +on the pinned upstream algorithmic interfaces and the identified written +extensions. Nothing in this page asserts formal or independent verification. + +## Current and earlier results + +| State | Exponent saving kappa | Artifacts | +| --- | --- | --- | +| Published baseline | `2^-59` | [paired note](../../artifacts/paired-note.pdf), [certificate](../../certificates/paired-network.json), [patch](../../patches/h50-paired-59.patch) | +| Preceding conditional research draft | `83/10^12 = 8.3e-11 > 2^-34` | [proof note](../../artifacts/compact-control-note.pdf), [source](../../notes/compact-control-note.tex), [certificate](../../certificates/compact-control-layer.json), [patch](../../patches/compact-control-34.patch) | +| Preceding conditional research draft (contributed) | `59/10^11 = 5.9e-10 > 2^-31` | [proof note](../../artifacts/complex-circuit-note.pdf), [source](../../notes/complex-circuit-note.tex), [certificate](../../certificates/complex-network.json), [patch](../../patches/complex-circuit-31.patch), [summary](complex-circuit.md) | +| Preceding conditional research draft (contributed) | `1479/10^12 = 1.479e-9 > 2^-30` | [proof note](../../artifacts/fast-gaussian-note.pdf), [source](../../notes/fast-gaussian-note.tex), [certificate](../../certificates/fast-gaussian.json), [patch](../../patches/fast-gaussian-30.patch), [summary](fast-gaussian.md) | +| Current conditional research draft (contributed) | `1624/10^12 = 1.624e-9 > 2^-30` | [proof note](../../artifacts/aligned-bit-note.pdf), [source](../../notes/aligned-bit-note.tex), [certificate](../../certificates/aligned-bit-network.json), [patch](../../patches/aligned-bit-30.patch), [summary](aligned-bit.md) | + +**Update (aligned bit circuit).** With fast resampling, `kappa < a_b/2`. The +center wires now take their values from group totals computed inside the side +circuit, so each loses `h-1` rather than `h` dimensions per invocation, and the +side circuit aligns its blocks across groups and shares its top-level +pair-star chains: 494,196 roles per invocation against 509,244. The bit +saving rises from `296/10^11` to `325/10^11` and the witness to +`1624/10^12`. See [the summary](aligned-bit.md). + +**Update (fast Gaussian resampling).** The Gaussian line maps, whose cost +forced `epsilon < 1/5`, are replaced by chirped block correlations, and the +Neumann series for the resampling inverse uses a sharper bound on the powers +of its correction. The Gaussian cost per bit falls from +`p^(3/4+delta+5eps/4)` to `O(d^2 p^delta)`, so `epsilon` may approach `1/2`; +the compressed complex network supplies room for the larger guard parameter +`beta = 19/25`. The witness becomes `1479/10^12 > 2^-30`, a factor 1479/590 +above `59/10^11`. The scoped ceiling for these networks is now +`kappa < a_b/2 < 2^-29`. See [the summary](fast-gaussian.md). + +**Update (compressed complex network).** The complex motif's side wires are +replaced by a shared-sum circuit with binary coordinate and pair-star labels: +108,195 side roles per invocation instead of 3,693,800. The certified complex +saving rises from `418/10^12` to `14/10^9`, so the complex network no longer +binds and the witness becomes `59/10^11 > 2^-31`, a factor 590/83 above +`83/10^12`. The bit network now sets the scoped ceiling `kappa < a_b/5 < 2^-30` +under the retained Gaussian margin. See [the summary](complex-circuit.md); the +compact-control description below remains accurate for the movement, guard +and assembly, which are unchanged. + +The compact-control witness increased kappa by approximately 47,846,242 times over the +published `2^-59`. This compares asymptotic exponent savings, not practical +runtime. The dyadic statement `kappa=2^-34` is a weaker convenient corollary. +The witness remains below `2^-33`. The earlier published artifacts and pinned +source remain unchanged. + +## What changed + +The finite bit network is unchanged: `h_b=50`, `m_b=125000`, with +`a_b=1-tau=296/10^11`. The new movement construction swaps compact dirty +control fields, rather than the full spaced slots. For f selected axes it +costs `O(V*((f log p)^tau+1))`, including deterministic repair, under the +layer's record regime and the supplied reservation layout. + +Two front fields and one back field are carved out of already existing +address chunks. Their selected kernels are processed individually first. +They remain outside the row index divided among roles, so each child retains +the complete compact ranges and still receives exactly `V/W` volume. The +extra preprocessing is `O(V*(log d+d^(1-c) log p+1))` for the chosen `c<1`. +No independent address coordinates, zero-temporary assumption, or free +gathering operation are used. + +The complex network uses its original construction independently at `h_c=25`, +`m_c=15625`, supporting `a_c=1-sigma=418/10^12`. The separate-arity proof is +now integrated into the source patch. The generalized stopped-depth guard +has `C1=5-4 beta+zeta`. Compact operations only permute complete coefficient +encodings, so they do not increase the coefficient arithmetic depth. + +## Exact current parameters and bottleneck + +The certificate uses + + epsilon = 1999/10000, c = 1/5, + beta = 1/1000, zeta = 1/10000, delta = 1/10^6, + C1 = 49961/10000, + lambda = 1-1671/(4*10^12), + lambda' = 1-167/(4*10^11), + kappa = 83/10^12. + +The internal, leaf and reservation exponents are respectively + + chi = tau+(1-beta)*max(sigma-tau,0), + leaf = sigma+beta*(1-sigma), + reserve = max(1-c,0). + +All lie strictly below lambda' after the stated intermediate-lambda checks. +All seven final margins exceed kappa. Their minimum is + + G = g3 = 333833/(4*10^15), + G-kappa = 1833/(4*10^15) > 0. + +The former quadratic restriction from `K^tau` has been removed by this +construction. The remaining limiting margin is the completed layer's saving, +controlled here by the complex network's stopped leaves. The unchanged h=25 +complex motif and retained Gaussian/leaf inequalities have the scoped upper +bound `kappa < 8.369598075e-11 < 2^-33`. Our simple rational witness is above +99% of that upper enclosure. It is not an unrestricted algorithmic ceiling. + +## Verification boundary + +The general proofs are in the four included construction sources: + +- [movement and deterministic repair](../../notes/compact-control-movement.tex); +- [reservations, row splitting and recurrence](../../notes/compact-control-layout.tex); +- [generalized guard](../../notes/compact-control-guard.tex); +- [independent complex interface](../../notes/independent-complex.tex). + +The [publication review guide](compact-control-review.md) maps each new +obligation to its proof and tests. Release copy and citation metadata have +been updated for this draft. This preparation is not independent review. + +The certificate stores their SHA-256 hashes. The standalone note supplies +the complete assembly comparison and scope. The combined patch also updates +the old global exceptional-stream paragraph: a `p^-3` bad fraction is charged +locally through the volume-weighted recurrence, rather than incorrectly +reusing its old global `o(V)` argument based on `2^-K`. + +Finite tests cover dirty address algebra, both slot orders, modular inverses, +bad-set invariance, exceptional repair, actual two-piece stream rotation, +complete-field preservation through padding and recursive row splits, +recurrence summation for all exponent orderings, and negative parameter cases. +Source integration checks preserve legacy appendix references and distinguish +the bit and complex arities. These checks support the written conditional +proof; they do not prove the upstream theorem or replace mathematical review. + +Completion checks passed: all 163 tests, all 17 patch-application checks, +the nine-page standalone note build, and the combined manuscript build. +Publication preparation also ran the full suite in a separate snapshot with +all proposed files and confirmed exact regeneration against its Git index. +These are reproducibility and consistency checks, not independent proof review. + +Reproduce with `make verify` and `make compact-note`. The combined manuscript +can be materialized with `make_compact_control_patch.patched_files()` into a +copy of `upstream/build`, then compiled with Tectonic after omitting the three +original pdfTeX-only metadata commands in that disposable copy, as described +in [the reproduction instructions](../reproducibility.md). The pinned source +itself must remain unchanged. + +## Next research priority + +First seek independent review of the new primitive, especially the temporary +layout invariant, local repair summation and uniform tape bounds. A substantive +flaw in one of those steps would revoke the new headline. + +If the proof survives, improving the complex finite network is the next route +to a stronger dyadic bound. The incidence/shared-sum/paired bit constructions +cannot simply be copied into it: the scalar coefficients, binary phase labels, +residual orthonormal bases and signed endpoint corrections require a new audit. + +The previous [joint-frame obstruction](joint-frame-audit.md) remains valid +within its fixed-boundary, grouped-gate scope. Splitting central gates or +changing boundaries between invocations are still possible finite-network +experiments, but the compact-control movement goal has succeeded conditionally +without needing that fallback. Older research pages retain their chronology; +use this page when interpreting their stale numerical targets or priorities. diff --git a/docs/research/dimension30-stream.md b/docs/research/dimension30-stream.md new file mode 100644 index 0000000000..a52ed0822f --- /dev/null +++ b/docs/research/dimension30-stream.md @@ -0,0 +1,90 @@ +# Dimension 30 with batched stream roles + +This separate experiment uses the explicit even-dimension matching from +[PR #12](https://github.com/CrocSwap/integer-mult-bounds/pull/12), pinned at +`35d31e30f28bc5da0ae6a88e7b03d75ebc855534`. Its unmodified matching source, +generic-moment example and attribution are retained under `references/pr12`. +The F3 five-set motif is inherited from Zhihao Chen's PR #7; batching and +controlled tensor coordinates are IceKylin's PR #10 contribution. + +The selected h30 circuit splits the ground set into 16 and 14 points. It uses +the previously selected local template patterns, then rebuilds four-point +stars and applies the bilateral saturated-frame register compiler at ancestor depth 4. +The checked result is **13,056,812 auxiliary roles**, with center loss 12,180 +and no additional frame loss. Its batched arithmetic supports +**κ = 168478/10^12**, or about 103.74 times the starting aligned-bit witness. +This is conditional on the general compiler, tensor, batching and inherited +multiplication arguments. Exact finite checks do not formally verify them. + +## Finite checks + +The rewritten circuit has 142,506 inputs, 142,941 designated outputs and +14,381,873 retained additions. The physical compiler removes 1,468,002 +delivery roles from the 14,524,814-role binary embedding. It checks the full +allocation, every producer-frame transition and every protected output. +The physical rank sum is 399,994,068 in each invocation orientation. The +bilateral extension adds 35,652 dual-frame continuations. An independent +exact-rational dirty-scratch control exercises a strict 24-to-27-dimensional +dual transition and checks its coefficients, frames and wrapper. + +The repaired source/dual cut delays 60 source nodes. All resulting frames are +nondegenerate, the source region is ancestor-closed, and all retained pair +totals remain in the source region. The central loss is +`binom(30,2)*(30-2) = 12180`. These properties preserve the scalar interfaces +needed by the tensor-stage and controlled-basis arguments. + +PR #7's particular Euler-tour construction does not cover every even +dimension. PR #12 instead gives a Hamilton cycle in the line graph of a +complete graph. Consecutive edges share exactly one vertex; substitution +into the two-full-pair case gives an intersection-two permutation of all +five-sets. The certificate checks all 142,506 images and their intersections. + +The retained source-span checker stores 28 coordinates. Reproduction creates +isolated copies with 32 coordinates and records the original and generated +source hashes. This is a storage extension: a five-set row has norm√5, +so every square minor through dimension 32 is bounded by 5^16, below 2^61−1. +Modular rank therefore still agrees with rational rank. The star extractor +and independent DAG audit receive corresponding dimension-guard extensions; +their algorithms and the retained source files are unchanged. + +Candidate generation uses fingerprint interning for speed. After star +rewriting, an independent canonical-ZDD evaluation checks every addition is +disjoint, every source core is exact, and every output is the required full +intersection-two or pair-total family. The certificate accepts only that +final exact check; no collision assumption enters the claimed result. + +## Batched arithmetic + +The five recursive widths are 1, `m-4h`, `m-2h²`, +`m-2h²-2h+2`, and `h²`, with `h=30` and `m=h³`. +The controlled stage-two corner remains one block of width `h²`. +Rank masses are recomputed from the final role count. Positive rational +logarithm enclosures, rounded upwards to multiples of 10^-9, certify +`a_b = 336958/10^12`. Increasing this by 10^-12 fails the same conservative +moment comparison. + +The complex network remains the h28 PR #7 construction. Its PR #10 batching +saving is 7/10^7 and its whole-residual guard has `C1=11999/10000`. +Using `epsilon=499999915710/10^12`, `c=1`, `beta=1/1000`, +`delta=1/10^10`, `lambda=1-a_b+10^-16` and +`lambda'=1-a_b+2*10^-16` makes all 29 assembly constraints strict. +The minimum final margin is +`84239485748905098429/(5*10^26)`, above the selected κ by +`485748905098429/(5*10^26)`. + +## Reproduction + +From the repository root: + +```sh +python3 scripts/dimension30_stream_network.py --workdir /tmp/dimension30-check +python3 -m unittest discover -s tests -p 'test_dimension30_stream_network.py' -v +python3 -m unittest discover -s tests -p 'test_ternary_dimension_sweep.py' -v +``` + +The first command needs Python 3, a C++17 compiler and several GB of memory. +It reconstructs the plan, checks the final DAG, frames and physical allocation, +and writes separate producer and assembly certificates. It needs no saved +discovery files. `--quick` checks the arithmetic without rebuilding the +producer. The h26 and h32 screens were weaker. Their selected circuits were +not promoted to full exact producer certificates. diff --git a/docs/research/fast-gaussian.md b/docs/research/fast-gaussian.md new file mode 100644 index 0000000000..6e93faace3 --- /dev/null +++ b/docs/research/fast-gaussian.md @@ -0,0 +1,96 @@ +# Faster Gaussian resampling: conditional kappa = 1479/10^12 > 2^-30 + +October 8, 2026. Contributed by eumemic, prepared with assistance from Claude +(Anthropic). Conditional on the pinned upstream interfaces, the compact-control +construction, the compressed complex network and the written arguments below. +Not independently reviewed or formally verified. + +- [Proof note (PDF)](../../artifacts/fast-gaussian-note.pdf) and + [source](../../notes/fast-gaussian-note.tex) +- [New resampling lemmas](../../notes/fast-gaussian-resampling.tex) +- [Independent upstream patch](../../patches/fast-gaussian-30.patch) +- [Exact certificate](../../certificates/fast-gaussian.json) + +## What was binding + +After the compressed complex network, the completed-layer margin is +`g3 = epsilon (1 - lambda') < epsilon a_b`, and the retained Gaussian +resampling row has power `3/4 + delta + 5 epsilon/4`, which must stay below 1. +So `epsilon < 1/5` and `kappa < a_b/5 < 2^-30`. The guard +(`epsilon C1 < 1` with `C1 = 5 - 4 beta + zeta`) also bound near `1/5`, but +only because `beta` was kept tiny; the compressed complex network makes larger +`beta` available. + +The line cost `O(t p^(3/2+delta) alpha)` has two sources. The map `S'` sums a +window of `sqrt(p) alpha` terms per output. The Neumann series for +`N^{-1}` needs `p/(alpha^2 theta)` terms with `theta ~ 1/(4d)`, which the +original analysis balances against the window by taking +`alpha ~ (dp)^(1/4)`. + +## Two changes + +**Chirped correlations.** For `sigma = t/s = 1 + theta` and integers `a, b`, + + (sigma a - b)^2 = sigma theta a^2 + sigma (a - b)^2 - theta b^2. + +Every Gaussian weight at the points `sigma a - b` therefore factors into a +decaying input chirp, a kernel depending only on `a - b`, and a growing output +chirp. A block of `m` outputs is one correlation of `O(m)` fixed-point +numbers, computed exactly by packing integers and calling the established +`O(N log N)` multiplier. Blocks are chosen so the growing chirp stays below +`2^(O(p))`; the working precision is then `O(p)` and the cost is +`O(p^(1+delta))` per output, independent of `alpha`. The same algorithm +computes `S'` and the correction `E = N - I`, with the same truncation windows +and error interfaces as the cited lemmas. + +**Powers of the correction.** Along a path of steps `h_i` in the expansion +of `E^n`, put `Phi(j) = sigma (beta_j^2 + 1/4) / theta`. Each step satisfies + + c(j,h) - Phi(j+h) + Phi(j) = sigma h^2 + (sigma/theta) w (2y - w) >= sigma h^2, + +where `w` is the nearest integer to `y = beta_j + h theta`. Summing over the +path gives + + ||E^n|| <= exp(pi alpha^2 (1/(4 theta) + 1/2)) (2.01 exp(-pi alpha^2 sigma))^n. + +The original bound `||E||^n < 2^(-alpha^2 theta n)` gains `alpha^2 theta` +bits per term; this one gains about `4.5 alpha^2` bits per term after a +burn-in of order `alpha^2/theta` bits. The Neumann series needs at most +`ceil((p+1)/(4 alpha^2) + 1/theta)` terms. The certificate checks the step +identity exactly on several `(s,t)`, and the tests compare explicit `E^n` +against both bounds. + +## New parameters + +`alpha = floor(sqrt(b/(8d)))` keeps `gamma = 2 d alpha^2 <= b/4`, the bound +used by the retained rounding argument, and gives `alpha^2 theta_i >= 1` and +Neumann counts at most `30 d`. The Gaussian row becomes `O(d^2 p^delta)` per +bit, with power `2 epsilon + delta`. + + epsilon = 49999/100000, c = 9999/10000, beta = 19/25, zeta = 1/10000, + delta = 1/10^6, C1 = 19601/10000, + lambda = 1 - 29595/10^13, lambda' = 1 - 2959/10^12, + kappa = 1479/10^12. + +The minimum margin is `g3 = 147947041/10^17`. The guard needs `beta > 3/4` +at this epsilon and the leaf needs `(1 - beta) a_c > 1 - lambda'`; with the +original complex saving `418/10^12` no `beta` satisfies both. + +## Ceiling and next targets + +Now `g2, g3, g4 <= a_b min(epsilon, 1 - epsilon)`, so `kappa < a_b/2 < 2^-29` +for these networks. The witness is above 99.9% of that bound. Further progress +needs a stronger bit network. Two bit-network observations from this session, +not yet proved or certified: + +- The center wires can gather through total nodes of the side circuit and + scatter at the zero label, losing `h - 1` instead of `h` each. That raises + the bit deficit numerator from `v - 6h^2` to `v - 6h(h-1)`, about 6.5% at + `h = 50`. +- A globally consistent block structure for the paired exclusion circuits + merges about 1.3% more side additions at `h = 50`. + +A rank argument (the Hadamard product of the Gram and integer central counts) +bounds any center decomposition's loss below by about `h^2/6`, against the +present `h(h-1)`, but small exact searches found no family beating +`h(h-1)`. diff --git a/docs/research/fused-block-audit.md b/docs/research/fused-block-audit.md new file mode 100644 index 0000000000..a6020c2728 --- /dev/null +++ b/docs/research/fused-block-audit.md @@ -0,0 +1,204 @@ +# A complete fused parity-butterfly block and its recursion budget + +**Follow-up:** the [short-guard reduction](short-guard-audit.md) isolates a +gathering theorem that would permit cheaper sparse movement. Its compact +arithmetic is costed, but a sufficiently fast gatherer remains unproved. + +**The published result remains conditional kappa=2^-59.** This bounded round +constructs and checks an exact basis-change/butterfly/basis-change block with +no explicit outer layout permutation. Its cost is three coordinate-layer +calls, not a new sublinear primitive. A cut-rank argument shows why arbitrary +fusion of coordinate children and pointwise gates alone cannot supply the +required recursive contraction. + +This follows [the sparse/fusion audit](sparse-fusion-audit.md). The implementation +is [audit_fused_block.py](../../scripts/audit_fused_block.py); its exact checks +are recorded in [fused-block-audit.json](../../certificates/fused-block-audit.json). +No published witness or source patch is changed. + +## 1. Specify the complete operator and physical layout + +Take two address slots x and y, each of width fK bits, in a complete +lexicographic array. Their selected positions are rho+jK for 0<=j