From 1d55e9b86984082e5cdc5b8d05d5de6cb944a43a Mon Sep 17 00:00:00 2001 From: Gareth Davidson Date: Tue, 13 Oct 2026 10:18:58 +0100 Subject: [PATCH 1/2] research: prove quartic precision for balanced dirty commutators --- research/dirty-prefix-precision/CLAIM.json | 82 + research/dirty-prefix-precision/EXPECTED.json | 29866 ++++++++++++++++ research/dirty-prefix-precision/PROOF.md | 173 + research/dirty-prefix-precision/README.md | 139 + research/dirty-prefix-precision/SOURCES.json | 96 + research/dirty-prefix-precision/VALIDATION.md | 110 + research/dirty-prefix-precision/core.py | 106 + research/dirty-prefix-precision/matrices.py | 124 + research/dirty-prefix-precision/numeric.py | 92 + .../dirty-prefix-precision/test_precision.py | 71 + .../vendor/pr628/DIRTY-PROOF.md | 128 + .../vendor/pr628/FACTORS.json | 402 + .../vendor/pr628/NOTICE.md | 24 + .../vendor/pr628/PROOF.md | 269 + .../vendor/pr628/SOURCE.json | 18 + .../vendor/pr628/check.py | 178 + .../vendor/pr628/check_budgets.py | 59 + .../vendor/pr628/dirty.py | 133 + .../vendor/pr628/dirty_budgets.py | 38 + .../vendor/pr628/interval_kernel.py | 61 + .../vendor/pr628/scan.py | 123 + .../vendor/pr628/weighted.py | 118 + research/dirty-prefix-precision/verify.py | 170 + 23 files changed, 32580 insertions(+) create mode 100644 research/dirty-prefix-precision/CLAIM.json create mode 100644 research/dirty-prefix-precision/EXPECTED.json create mode 100644 research/dirty-prefix-precision/PROOF.md create mode 100644 research/dirty-prefix-precision/README.md create mode 100644 research/dirty-prefix-precision/SOURCES.json create mode 100644 research/dirty-prefix-precision/VALIDATION.md create mode 100644 research/dirty-prefix-precision/core.py create mode 100644 research/dirty-prefix-precision/matrices.py create mode 100644 research/dirty-prefix-precision/numeric.py create mode 100644 research/dirty-prefix-precision/test_precision.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/DIRTY-PROOF.md create mode 100644 research/dirty-prefix-precision/vendor/pr628/FACTORS.json create mode 100644 research/dirty-prefix-precision/vendor/pr628/NOTICE.md create mode 100644 research/dirty-prefix-precision/vendor/pr628/PROOF.md create mode 100644 research/dirty-prefix-precision/vendor/pr628/SOURCE.json create mode 100644 research/dirty-prefix-precision/vendor/pr628/check.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/check_budgets.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/dirty.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/dirty_budgets.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/interval_kernel.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/scan.py create mode 100644 research/dirty-prefix-precision/vendor/pr628/weighted.py create mode 100644 research/dirty-prefix-precision/verify.py diff --git a/research/dirty-prefix-precision/CLAIM.json b/research/dirty-prefix-precision/CLAIM.json new file mode 100644 index 0000000000..380f9f11eb --- /dev/null +++ b/research/dirty-prefix-precision/CLAIM.json @@ -0,0 +1,82 @@ +{ + "pass": 6, + "assignment_seed": 63, + "drawn_at_utc": "2026-10-13T08:36:24.867393+00:00", + "track": "adversarial mathematical audit", + "base": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "status": "validated; ready for submission", + "branch": "research/pass06-audit-seed63", + "claimed_at_utc": "2026-10-13T08:46:31.924453+00:00", + "candidate_selection": "63 modulo 3 = 0", + "candidates": [ + { + "object": "Uniform precision growth in PR628 balanced dirty commutators", + "method": "Bound ideal prefixes and their inverses through recursive restoration, then sum perturbed leaf errors", + "chosen": true + }, + { + "object": "Minimum all-dirty echo call count at fixed scratch", + "method": "Exact tree recurrence audit", + "chosen": false, + "reason": "PR730 and PR744 already supply complementary exact optimization and independent coverage" + }, + { + "object": "Restored-source precision at arbitrary cyclic stride", + "method": "Audit PR846 FIFO recursion under a borrowed-array contract", + "chosen": false, + "reason": "PR846 explicitly leaves dirty-array realization open; resolving it needs a new construction beyond this bounded audit" + } + ], + "statement": "Audit whether the N-leaf trajectory growth used in PR628 DIRTY-PROOF.md is necessary for balanced binary commutators. Establish a uniform ideal-prefix and inverse-prefix bound and a complete rounding-error budget, retaining arbitrary initial dirty arrays and the unchanged leaves.", + "expected_deliverable": "Written proof, exact prefix and error-propagation evidence, source-bound integer budget comparison and actual rounded-leaf/kernel replay; no new kappa.", + "domain": "PR628 balanced binary product-shear tree, every n>=1 and arbitrary composable bounded linear factor maps. Numerical evidence uses pinned cyclic leaves and nonaliased immutable phases. Higher fanout, different allocation, clean storage and physical supplier integration outside scope.", + "stopping_rule": "Complete this one prefix-bound/error recurrence and its declared deterministic validation; no optimizer or supplier construction", + "expected_finish": "Approximately 90 minutes; reserve final quarter for validation and handoff", + "coordination": "Only open non-PR issue #650 requests Discussions. No coordination issue exists; this claim is local.", + "related_prs": { + "628": "Original rounded leaf, dirty composition and conservative N-step precision bound", + "663": "Exact balanced binary leaf count and independent nonlinear restoration audit", + "673": "Multiway echo and phase-alias repair; same conservative per-leaf growth technique", + "730": "Ordered mixed-tree cost optimization, different scope", + "744": "Ternary fixed-space verification, different scope", + "862": "Cubic precision with owned widened words; explicitly excludes arbitrary dirty-bit restoration" + }, + "initial_open_pr_count": 660, + "latest_pr_at_claim": 867, + "final_overlap_check": { + "at_utc": "2026-10-13T09:16:19.773721+00:00", + "open_pr_count": 669, + "latest_number": 876, + "upstream_head": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "pr628_head": "0768f97d06c623b8211e651543c66135796b62bf", + "finding": "No overlapping balanced dirty-prefix precision audit found. Concurrent PR862 has an owned-widened-storage contract; new PR868-876 concerns other geometry, supplier or verification work." + }, + "result": { + "statement": "The unchanged balanced binary dirty commutator has a sufficient O(p+d+r^4) word-size bound, improving its inherited O(p+d+r^5) budget; no new kappa.", + "prefix_exponent": "E(1)=1; E(n)=n+max(floor(n/2)+E(ceil(n/2)),ceil(n/2)+E(floor(n/2))) <= 4n-3", + "radix64_example": { + "target_bits": 128, + "initial_magnitude_bits": 20, + "old_exact_count_bits": 516978511, + "new_prefix_bits": 73245212, + "leaf_calls": 4000, + "dirty_arrays": 6 + }, + "small_radix_limit": "The old bound can be tighter; no optimal guard claim." + }, + "validation": { + "python_versions": [ + "3.12.3", + "3.14.4" + ], + "uniform_budgets": 1740, + "dense_prefix_traces": 18, + "rounded_generic": 48, + "rounded_kernels": 40, + "mathematical_controls": 10, + "unit_tests": 4, + "package_mutation_controls": 6, + "minimal_isolated_optimized_replay": "passed", + "details": "VALIDATION.md" + } +} diff --git a/research/dirty-prefix-precision/EXPECTED.json b/research/dirty-prefix-precision/EXPECTED.json new file mode 100644 index 0000000000..938cea2dd1 --- /dev/null +++ b/research/dirty-prefix-precision/EXPECTED.json @@ -0,0 +1,29866 @@ +{ + "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "dense_prefixes": [ + { + "K": 2, + "all_columns_and_dirty_restoration": true, + "factors": 1, + "gaussian_inverse_checks": 1, + "inverse_prefix_maximum": 3, + "leaf_calls": 1, + "prefix_exponent": 1, + "prefix_maximum": 3, + "prefix_stream_sha256": "101b48dbe04f7d5df340d79639bf3fcaa098c228a8fe69493f097beec07766a9", + "sign": -1, + "subword_maximum": 3, + "subwords_checked": 1 + }, + { + "K": 2, + "all_columns_and_dirty_restoration": true, + "factors": 1, + "gaussian_inverse_checks": 1, + "inverse_prefix_maximum": 3, + "leaf_calls": 1, + "prefix_exponent": 1, + "prefix_maximum": 3, + "prefix_stream_sha256": "101b48dbe04f7d5df340d79639bf3fcaa098c228a8fe69493f097beec07766a9", + "sign": 1, + "subword_maximum": 3, + "subwords_checked": 1 + }, + { + "K": 3, + "all_columns_and_dirty_restoration": true, + "factors": 2, + "gaussian_inverse_checks": 4, + "inverse_prefix_maximum": 13, + "leaf_calls": 4, + "prefix_exponent": 4, + "prefix_maximum": 10, + "prefix_stream_sha256": "b8e01a371ff991f2c28e9608b54164bab2ee73da2f17af88001b5919a6b7082e", + "sign": -1, + "subword_maximum": 13, + "subwords_checked": 10 + }, + { + "K": 3, + "all_columns_and_dirty_restoration": true, + "factors": 2, + "gaussian_inverse_checks": 4, + "inverse_prefix_maximum": 13, + "leaf_calls": 4, + "prefix_exponent": 4, + "prefix_maximum": 13, + "prefix_stream_sha256": "a6bbc1fe5ccf24a803606e09c51d05045b70dcfdb93d4261010f554cf89119d2", + "sign": 1, + "subword_maximum": 13, + "subwords_checked": 10 + }, + { + "K": 4, + "all_columns_and_dirty_restoration": true, + "factors": 3, + "gaussian_inverse_checks": 4, + 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"name": "endpoint_only_prefix_bound", + "rejected": "prefix norm bound" + } + ], + "recurrence": { + "exact_predecessor_words": 128, + "predecessor_instructions": 191342, + "recurrence_cases": 10000, + "recurrence_sha256": "c4692468db03108a8175072c1da467ba52547e653bd90f3bc41b9d528602e66a" + }, + "result": "Balanced dirty commutator: sufficient O(p+d+r^4) word size, unchanged leaf count and dirty arrays; no new kappa", + "rounded_generic": [ + { + "boundary_checks": 1, + "coefficient_perturbation_checked": false, + "direction": -1, + "dirt": -7, + "error_squared_log2_upper": -104, + "exact_evidence_sha256": "4796db2d678928c910f30ac57f80d3984717b7c74c8c229afc901247b5359084", + "execution_bound_log2_upper": -33, + "factors": 1, + "inverse_cancels": true, + "leaf_calls": 1, + "length": 1, + "prefix_exponent": 1, + "radix": 3, + "scalar_steps": 3, + "source_and_dirty_restored": true, + "target_bits": 32, + "working_bits": 53 + }, + { + "boundary_checks": 1, + 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"shear_growth_bits": 66978856, + "target_bits": 1048576, + "working_bits": 308036808348 + } + ] +} diff --git a/research/dirty-prefix-precision/PROOF.md b/research/dirty-prefix-precision/PROOF.md new file mode 100644 index 0000000000..69c98a21d1 --- /dev/null +++ b/research/dirty-prefix-precision/PROOF.md @@ -0,0 +1,173 @@ +# Balanced dirty commutators: prefix bounds and quartic precision + +This audits the conservative per-leaf growth estimate in PR628's +`DIRTY-PROOF.md`, pinned and retained under `vendor/pr628/`. Its old bound is +valid. We give a smaller sufficient execution budget for its **unchanged +balanced binary dispatcher and unchanged rounded leaves**. No minimum +precision, physical implementation or new multiplication exponent is claimed. + +## 1. Ideal prefixes, including their inverses + +Let `A_1,...,A_n` be arbitrary composable linear maps on a common normed +space, with operator norms at most `K >= 0`. Use the maximum norm on the +joint source, target and borrowed arrays. Assume distinct ancestor and child +arrays, with one reserved connector per recursion depth, as in PR628. All +initial arrays may contain arbitrary data. No factor commutativity is used. + +Set `C=1+K`. A complete m-factor ideal child is a shear with multiplier +`A_(j+m-1)...A_j`, and restores its source and deeper scratch. Its norm and +its inverse's norm are at most `1+K^m <= C^m`. This uses the ordinary ideal +commutator identity, not an assumption about rounded linearity. + +Define + +```text +E(1) = 1, +a = floor(n/2), b = ceil(n/2), +E(n) = n + max(a + E(b), b + E(a)). +``` + +**Prefix lemma.** Every ideal primitive prefix `P_j` of either sign of the +balanced word satisfies + +```text +||P_j|| <= B, ||P_j^-1|| <= B, B = C^E(n). +``` + +The empty prefix is included. At a binary node write U for the left child's +source-to-connector shear and V for the right child's connector-to-target +shear. The positive chronological word is `U,V,U^-1,V^-1`; the negative word +is `V,U,V^-1,U^-1`. A prefix consists of completed children and a prefix +inside the next child. The largest completed-child degree before an active +b-child is `2a+b=n+a`, and before an active a-child is `a+2b=n+b`, allowing +both signs. Induction and submultiplicativity give the stated recurrence. +For the inverse of that prefix, reverse and invert the completed factors +and the active child prefix. The same product of norm bounds applies. We +do not infer an inverse-prefix bound merely from a forward-prefix norm. + +Inductively **`E(n) <= 4n-3`**: the two bounds become `2a+5b-3` and +`5a+2b-3`, both at most `4(a+b)-3`, since the balanced positive sizes satisfy +`a <= 2b` and `b <= 2a`. For powers of two the exact recurrence is +`E(n)=3n-2`. The recurrence is a sufficient majorant, not the exact maximum +norm or a lower bound on required precision. + +Every ideal propagator between two primitive boundaries consequently has +norm at most **`B^2`**, since it is `P_j P_i^-1`. This is where the inverse +prefix bound is needed. A small final product alone would not control the +intermediate dirty computation. + +## 2. Rounded execution with arbitrary initial dirt + +Suppose each executed leaf is a deterministic signed shear that restores its +source exactly, computes its increment independently of its target, and +approximates its ideal linear map with error at most + +```text +a u (||source||+1), u=2^-w, a>0. +``` + +Here `a` denotes the error constant, no longer the left subtree size. Let +`D` bound the norm of **all** initial arrays. Let N be the actual complete +leaf count, including inverse and restoration calls. PR663 gives, equivalently +to the elementary recursion `N(n)=2N(floor(n/2))+2N(ceil(n/2))`, + +```text +N(n) = 3 q n - 2 q^2, q=2^floor(log2 n). +``` + +Let M be the maximum actual joint norm over all leaf boundaries. Inserting +the actual leaf errors into the ideal linear recurrence is exact even though +each error depends nonlinearly on the current source. Variation of constants +and the prefix lemma give + +```text +M <= B D + delta (M+1), delta = N B^2 a u. +``` + +If `delta <= 1/2`, then + +```text +M <= 2 B D + 1, +execution error <= 2 N B^2 a u (B D+1). +``` + +The last error is for the whole joint state against the exact ideal word; +it therefore also bounds the target error. Exact source and dirty restoration +still follow from the predecessor's deterministic shear/inverse identity, +with immutable nonaliased phase/factor data. This proof neither assumes a +rounded child is linear nor claims its output is independent of dirt. + +For any desired execution accuracy `2^-t`, `t>=0`, it suffices to use + +```text +w >= t + ceil(log2(2 N B^2 a (B D+1))). +``` + +This also implies `delta<=1/2`. Retain the input grid and every separate +leaf-admission condition. All bounds are independent of cyclic length L. + +## 3. The inherited prime-power inverse and the complete bill + +For PR628's n=r-1 prepared factors, the inherited leaf lemma has +`K=r/beta`, `a=13(r+1)/beta^3`, and requires `u<=beta^2/2`. Its root-gap +budget supplies `beta >= 2^-g` with `g=O(r^3)`. If `D<=2^d`, set + +```text +A = 3g + ceil(log2(13(r+1))), +H = g + ceil(log2(r+1)), +w = max(A, 2g+1, p+3+ceil(log2 N)+A+d+3 E(n) H). +``` + +Indeed `a<=2^A`, `B<=2^(E H)` and `BD+1<=2^(EH+d+1)`, so the execution +error is at most `2^(-p-1)`. The smallness and leaf closure conditions hold. +If initial inputs have a finer dyadic grid, replace w by the maximum of w +and their fractional precision; an array cannot be silently coarsened. + +Keep PR628's **unchanged coefficient/phase preparation** at target +`p+1+d`. Its perturbation bound for the original source contributes at most +`2^(-p-1)`, hence the total error is at most `2^-p`. Initial dirty data +enters the execution budget D; it does not enter the ideal operator's +coefficient perturbation. No unimplemented preparation algorithm is supplied +by this audit. + +Since `E(n)=O(r)` and `N=O(r^2)`, this proves the alternative sufficient guard + +```text +w = O(p + d + r^4), +``` + +instead of the old `O(p+d+r^5)` growth bound. Integer magnitude bits at leaf +boundaries are `O(d+E log(1+K))=O(d+r^4)`. The inherited causal passes and +closure scalars add `O(log(beta^-1)+log r)` magnitude bits inside a leaf. +The bound is on complete word size after also retaining any finer input grid. + +All **N leaf calls remain paid**: each retains validation/access, causal, +output and undo passes. With inherited fixed-precision arithmetic and +preparation, the local neighboring-layout bill becomes + +```text +soft-O((L r^2 + r^3) (p + d + r^4)), +``` + +plus charged phase access and the predecessor's arbitrary-stride routing. +Borrowed storage remains `ceil(log2 n)` length-L arrays, arbitrary initially, +with `O(L log r)` words; scalar arithmetic, indexing and factor descriptors +retain the predecessor's charges. No clean length-L array, free copy, or +discarded restoration call is introduced. The rational reference code is +not an implementation or benchmark of the asymptotic bit-cost algorithm. + +This improves one local precision upper bound. It supplies no admitted +common-label/Clifford or reversible-bit word, physical owner layout, +complete-transform recurrence or multiplication reduction. The selected +supplier and κ are unchanged. PR862's cubic guard uses owned widened words +with reserved zero bits; its storage contract differs from this dirty-array +result. General multiway or optimized unequal trees are outside this proof. + +Credit Chafik Boukhalfa / chafreaky for PR628, 0xkilgore for PR663's exact +count and nonlinear restoration audit, androolloyd for PR673's multiway and +phase-alias work, and the original sources retained in the vendor notice. +The perturbation argument is a standard variation-of-constants estimate; +no literature-priority claim is made. Written for Gareth Davidson (@bitplane) +with substantial OpenAI Codex assistance; Apache-2.0. Finite evidence below +supports this handwritten argument, not formal verification of the full +multiplication theorem. diff --git a/research/dirty-prefix-precision/README.md b/research/dirty-prefix-precision/README.md new file mode 100644 index 0000000000..5ce83ef209 --- /dev/null +++ b/research/dirty-prefix-precision/README.md @@ -0,0 +1,139 @@ +# Quartic precision for balanced dirty commutators + +**Mathematical audit / improved sufficient local precision bound; no new κ.** +The balanced dirty-array inverse in +[#628](https://github.com/CrocSwap/integer-mult-bounds/pull/628) admits +**O(p+d+r⁴)** word size, improving its published **O(p+d+r⁵)** sufficient +bound. Here r is the radix, p the requested accuracy, and `2^d` bounds every +initial source, target and dirty array. Retain a finer initial input grid if +one is supplied. The dispatcher, leaf operations, number of calls and +borrowed arrays are unchanged. + +The old proof bounds growth across every leaf call. This audit instead bounds +ideal prefixes and their inverses using the restored structure of complete +recursive children, then propagates the actual nonlinear leaf errors through +those linear maps. The full proof and complete inherited bill are in +[PROOF.md](PROOF.md). + +For n factors, set `C=1+K`, where each ideal factor has norm at most K: + +```text +E(1)=1; E(n)=n+max(floor(n/2)+E(ceil(n/2)), ceil(n/2)+E(floor(n/2))), +E(n)<=4n-3; B=C^E(n). +``` + +Every ideal prefix and inverse prefix has norm at most B, so every interval +of the ideal word has norm at most `B²`. With N complete leaf calls and +per-leaf error `a u (||source||+1)`, the resulting execution error is at most +`2 N B² a u (B D+1)`, provided `N B² a u<=1/2`. This covers arbitrary initial +dirt and target-dependent trajectories without assuming rounded leaves are +linear. Exact dirty restoration still uses the predecessor's deterministic +inverse contract and nonaliased read-only phases. + +## Exact budget comparison + +At `p=128`, `d=20`, using PR628's inherited root-gap exponent: + +| Radix | Old guard after crediting #663's exact leaf count | New prefix guard | +|---|---:|---:| +| 4 | 680 | 1,341 | +| 9 | 31,106 | 32,514 | +| 16 | 461,599 | 266,372 | +| 64 | 516,978,511 | 73,245,212 | +| 256 | 544,452,103,903 | 19,177,910,612 | + +These are conservative **sufficient fractional-bit budgets**, not measured +minimum precision, speedups or useful hardware word sizes. The earlier bound +is better at small radices; either valid bound may be used. For radix 64 the +new budget improves even the exact-count baseline, beyond the earlier +reduction of PR628's `4n²` allowance. Its 63 factors still execute **4,000** +leaves and use six arbitrary dirty arrays. Their prefix exponent is 188. + +The complete neighboring-layout cost becomes +`soft-O((L r²+r³)(p+d+r⁴))` under the unchanged preparation and arithmetic +assumptions. All three leaf passes, validation/source access, inverse calls, +`ceil(log2 n)` borrowed arrays, clean scalar arithmetic, indexing, descriptors, +phase access and arbitrary-stride routing remain charged. + +## Reproduce + +Python 3.11+, standard library, no network or Git required, read-only by default: + +```sh +python3 -B -I research/dirty-prefix-precision/verify.py +python3 -B -I research/dirty-prefix-precision/test_precision.py +``` + +Expected main marker: + +```text +PASS_DIRTY_PREFIX_PRECISION budgets=1740 prefix_traces=18 rounded_generic=48 rounded_kernels=40 controls=10 kappa=unchanged +``` + +[EXPECTED.json](EXPECTED.json) contains the deterministic receipt. An explicit +`--write /new/path.json` reconstructs it and refuses an existing destination. +The [validation record](VALIDATION.md) describes executed commands and package +controls. + +- **10,000** factor counts check the resource majorants and #663's count formula. + The independently emitted positive/negative words match the immutable #628 + dispatcher for n=1..64: **128 words, 191,342 instructions**. +- **18** dense integer traces with noncommuting 2×2 factors check every prefix + and inverse prefix: **11,142 boundaries**. Separate rational Gauss-Jordan + inversion checks 66 selected prefixes. All **5,104** nonempty contiguous + subwords in the fixtures through eight factors are also built directly and + compared with their prefix/inverse-prefix products. +- **1,740** exact integer budgets cover all 116 prime-power radices from 3 + through 512, five target precisions and three initial magnitude bounds. + Both the original and exact-call-count baselines are recorded. +- **48** generic and **40** actual kernel executions use the unchanged #628 + rounded leaves. All ten pinned kernel certificates are checked, with dense + expanded-polynomial solves, nonzero initial dirt, both cycle orientations, + and cycle lengths one and three. Every executed leaf restores its source; + all 6,304 leaf boundaries satisfy the new magnitude bound. Every complete + inverse cancels exactly. All 88 executions have nonzero rounding error + bounded by the new execution budget. Kernel tests also charge coefficient + perturbation and prove total error at most `2^-128`. +- Ten controls reject altered budget obligations, omitted restore, wrong + inverse order and using the final operator norm as a prefix bound. Four + tests exercise exact logarithm boundaries, finer input grids, the feedback + threshold and eight additional nonlinear-error executions. + +The finite domains and seed **63** are fixed in [CLAIM.json](CLAIM.json). +The all-n and all-cycle-length conclusions rest on the written proof and +inherited leaf/preparation theorems, rather than an extrapolation from tests. + +## Scope, provenance and credit + +Base: `f567884aeb8851b6e794af1184119a3bfbe1e631`; selected #799 remains at +κ=`1661634653669567/10^18`. The subject is the separate local #628 primitive, +head `0768f97d06c623b8211e651543c66135796b62bf`. Its consumed files and notice +are vendored verbatim and verified by SHA256 and Git blob identity in +[SOURCES.json](SOURCES.json). Original documentation is retained as historical +source, including its superseded sufficient precision estimate. + +Credit **Chafik Boukhalfa (@chafreaky)** for #628's rounded leaves and dirty +composition, **0xkilgore** for #663's exact count and nonlinear restoration +audit, and **androolloyd** for #673's multiway echo and phase-alias repair. +The declared contract here requires phase arrays distinct from every mutable +array; this audit does not claim to fix the known alias defect in vendored +#628. The original notices credit Ali Romman's #621 kernel and its sources. + +Related #689/#730/#744 optimize or audit other echo trees and call counts; +#714 uses one clean root checkpoint; #846 studies arbitrary strides. +Chafik's concurrent **#862** obtains cubic precision with owned widened words +and reserved low zero bits. That differs from the arbitrary-dirty-array +contract preserved here. None of those pending packages is imported. + +General multiway or unbalanced trees, variable leaf precision, common-label +or Clifford compilation, reversible-bit realization, physical ownership, +full-transform accounting and the multiplication reduction are outside this +result. Factor preparation algorithms and fast fixed-precision arithmetic +remain inherited hypotheses, not newly implemented software. No selected +artifact or `upstream/` file changes. No full theorem formalization or +literature-priority claim is made. + +Prepared for **Gareth Davidson (@bitplane) with substantial OpenAI Codex +assistance**. Apache-2.0; predecessor notice retained. All new code and the +proof were written by one AI agent; separate arithmetic methods are not +independent human review. diff --git a/research/dirty-prefix-precision/SOURCES.json b/research/dirty-prefix-precision/SOURCES.json new file mode 100644 index 0000000000..c95a6b17e2 --- /dev/null +++ b/research/dirty-prefix-precision/SOURCES.json @@ -0,0 +1,96 @@ +{ + "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "source_pr": 628, + "source_head": "0768f97d06c623b8211e651543c66135796b62bf", + "source_author": "Chafik Boukhalfa (@chafreaky)", + "files": [ + { + "source_path": "research/prime-power-cyclic-scan/dirty.py", + "vendored_path": "vendor/pr628/dirty.py", + "sha256": "09e7f41141594e6540828330d27be27c083ee6569660ae42c0cd519b09fc8d21", + "git_blob": "2b9471083706a37519a1af7598e012f5fecf2636" + }, + { + "source_path": "research/prime-power-cyclic-scan/weighted.py", + "vendored_path": "vendor/pr628/weighted.py", + "sha256": "8b0f93d131a30e64eb0ccd893779daffea79c0c00f1d2a30a91d23eb2bf0bff5", + "git_blob": "da90ee6e83b15164cabdbe4472a2c5d53706160e" + }, + { + "source_path": "research/prime-power-cyclic-scan/check.py", + "vendored_path": "vendor/pr628/check.py", + "sha256": "a3e46b556894d71a1c83bbbac01c50c1a541285fc9422a3567d5eb1d1ee86e7d", + "git_blob": "5ef6dd67b6c6cf949fedf16c0cd7b34ca4ccafc2" + }, + { + "source_path": "research/prime-power-cyclic-scan/scan.py", + "vendored_path": "vendor/pr628/scan.py", + "sha256": "1a9c01ca1a29a9be1e756858161202621db29d5d332a4266b4017a9502a50d7e", + "git_blob": "d620e493596f12578c1ce47f216c44c79b65de08" + }, + { + "source_path": "research/prime-power-cyclic-scan/interval_kernel.py", + "vendored_path": "vendor/pr628/interval_kernel.py", + "sha256": "9b7bb24640a9a7d3f83c7951dfef483ec933db4c50de6c0130b85f450eeddfbe", + "git_blob": "1182175a55cb871481de0f56ea9ae56e474705fe" + }, + { + "source_path": "research/prime-power-cyclic-scan/check_budgets.py", + "vendored_path": "vendor/pr628/check_budgets.py", + "sha256": "581ba58a30e05cb09dadb35be6e25bd92232b2137a0cc7eb5b83e2ee20c0bf02", + "git_blob": "ff21b6e1904b5b492fb4389ae1c49ca69588c002" + }, + { + "source_path": "research/prime-power-cyclic-scan/dirty_budgets.py", + "vendored_path": "vendor/pr628/dirty_budgets.py", + "sha256": "107d7e80b9b752625e7e817f1226c364f872be1facc5fd24c2e74dfdcd4714ce", + "git_blob": "a3ce42755ae55d2be5b6fd4c40d88027d304bc8d" + }, + { + "source_path": "research/prime-power-cyclic-scan/FACTORS.json", + "vendored_path": "vendor/pr628/FACTORS.json", + "sha256": "643d60ac24d5e67f6c9305175bd1c99bf5e42edfef6d1995c051e0b98ba446a7", + "git_blob": "d3894bd2a1660229f1a7ed16c9d982689effc96f" + }, + { + "source_path": "research/prime-power-cyclic-scan/DIRTY-PROOF.md", + "vendored_path": "vendor/pr628/DIRTY-PROOF.md", + "sha256": "bb7c2fdc1c4673d56531fd56fb8790c81a16813ac152c10f1afabc9efdd70d61", + "git_blob": "caf4aafc784a630d61ca7874a0bd19cbcb34e585" + }, + { + "source_path": "research/prime-power-cyclic-scan/PROOF.md", + "vendored_path": "vendor/pr628/PROOF.md", + "sha256": "595893609764c64a90d9b21b9cbd6d9e61980703f3deb43361105cf8c3d1d093", + "git_blob": "fb8fe7cb3bd036c69047ae7b4f4009a51b11825b" + }, + { + "source_path": "research/prime-power-cyclic-scan/NOTICE.md", + "vendored_path": "vendor/pr628/NOTICE.md", + "sha256": "abeacf1ad1491d5797cfcf1f3832799bbf3b24b99597229d7bc884059a51ec68", + "git_blob": "22fbeb8ef0a28dd6b3feea718492091319ceb96e" + }, + { + "source_path": "research/prime-power-cyclic-scan/SOURCE.json", + "vendored_path": "vendor/pr628/SOURCE.json", + "sha256": "38a425e2d3a72a5e39b18b54c728e40407e121d481a1c6423edd5c18e9b68a04", + "git_blob": "59bddaea4d66725bf675a67b6d2cf85a061b1fae" + } + ], + "selected_at_base": { + "source_pr": 799, + "source_head": "80e6463157cfc80b85af09e7e2f14748fe2dac60", + "kappa": "1661634653669567/1000000000000000000" + }, + "related_heads": { + "862": "57e0659150d323b784580110b36aa64a2dca2a33", + "846": "4bdb3e755ea7118170a7a34764df955880dec282", + "744": "302383e0c1ec71c50860362d3f394f85664c2445", + "730": "d222e348179cc02a54d6d26dbd2838944f62c6e5", + "714": "efad8a40094ab453c0525efef0b0eff2a5d8b850", + "689": "0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a", + "673": "0b4f514c60d9bc7786415d7cbc24754f24d6dc3b", + "663": "e0e546fd01b396478c68b6b6c5a16d87515983ba" + }, + "comparison": "Vendored predecessor files are immutable exact Git bytes. No source file is modified; original notice retained." +} diff --git a/research/dirty-prefix-precision/VALIDATION.md b/research/dirty-prefix-precision/VALIDATION.md new file mode 100644 index 0000000000..4ce14c712d --- /dev/null +++ b/research/dirty-prefix-precision/VALIDATION.md @@ -0,0 +1,110 @@ +# Executed validation + +Linux x86-64, 2026-10-13, CPython **3.12.3** and **3.14.4**. Base +`f567884aeb8851b6e794af1184119a3bfbe1e631`; audited #628 head +`0768f97d06c623b8211e651543c66135796b62bf`. All twelve retained predecessor +files match their SHA256 and Git object pins. New arithmetic and the proof +were written by one AI agent; no independent human review is claimed. + +## Replays + +The primary verifier generated the receipt with Python 3.12.3: + +```sh +/usr/bin/python3.12 -B -I -S research/dirty-prefix-precision/verify.py --write /tmp/new-dirty-prefix-receipt.json +``` + +The resulting bytes became `EXPECTED.json`. Default receipt comparison then +passed with Python 3.14.4: + +```sh +/usr/bin/python3.14 -B -I -S -O research/dirty-prefix-precision/verify.py +``` + +The expected marker on both is: + +```text +PASS_DIRTY_PREFIX_PRECISION budgets=1740 prefix_traces=18 rounded_generic=48 rounded_kernels=40 controls=10 kappa=unchanged +``` + +The four focused tests passed on both versions, with optimized Python on 3.14: + +```sh +/usr/bin/python3.12 -B -I -S research/dirty-prefix-precision/test_precision.py +/usr/bin/python3.14 -B -I -S -O research/dirty-prefix-precision/test_precision.py +``` + +These test 603 exact logarithm boundary cases, preservation of a finer incoming +grid, the feedback threshold, and eight literal nonlinear-error executions +with exact dirty restoration and whole-word inversion. Checks use explicit +exceptions, so `-O` does not remove them. + +`make selected-record-check` passed at the unchanged selected value +`1661634653669567/1000000000000000000`. Full repository regeneration, selected +supplier cold replay and Lean were not rerun for this isolated research package. +Hosted CI is separate from these local results. + +## Independent arithmetic paths and coverage + +The exact receipt contains: + +| Check | Executed coverage | +|---|---:| +| Resource recurrence and closed-form leaf count | n=1..10,000 | +| Literal equality with the frozen dispatcher | 128 signed words, 191,342 instructions | +| Dense integer prefix/inverse-prefix replay | 18 traces, 11,142 primitive boundaries | +| Separate rational Gauss-Jordan inverse | 66 selected prefix matrices | +| Direct contiguous-subword reconstruction | 5,104 intervals | +| Inherited and new integer precision budgets | 1,740 cases over 116 prime-power radices | +| Actual rounded generic executions | 48 | +| Actual certified kernel executions | 40, covering all ten source certificates | +| Observed leaf-boundary magnitude/source restoration | 6,304 boundaries | +| Mathematical rejection controls | 10 | + +Every rounded execution has a nonzero exact error within the new bound, so +the comparison is not supported solely by exact-arithmetic special cases. +Kernel target comparisons use independently expanded polynomial matrices +and dense solves; the source's coefficient perturbation is separately charged +to the total `2^-128` guarantee. Both cyclic directions, lengths one and +three, nonzero targets, positive and negative dirt, and exact inverse-word +cancellation are included. Generic fixtures are interface cases, not additional +certified prime-power kernel instances. + +The first pilot exposed an ineffective negative fixture: flipping every leaf +sign at **three** factors gives the correct negative *ideal* product. The +inverse-order rejection control now uses two factors, where it provably +differs. This was a control-fixture correction, not a failure of the prefix +lemma or error bound. The final ten controls all reject as recorded. + +## Standalone and package controls + +A copy containing only this package, in a path with spaces and with no Git +metadata, passed the default verifier using Python 3.12.3 with `-B -I -S -O`. +All package file hashes were unchanged after that run: default replay is +read-only. No site packages, network or third-party Python modules were used. + +Six further controls were executed in that disposable copy: + +| Modification | Result | +|---|---| +| Append a comment to vendored `dirty.py` | Rejected: source drift | +| Understate a recorded working-bit budget | Rejected: certificate mismatch | +| Alter a recorded prefix-stream digest | Rejected: certificate mismatch | +| Substitute a boolean for an integer receipt field | Rejected: certificate mismatch | +| Reverse the sign in the new inverse-prefix update engine | Rejected by the separate Gauss-Jordan comparison | +| Request `--write` to an existing receipt | Rejected before replay; original bytes preserved | + +Every mutation was restored; the final complete file inventory matches the +initial one. Python syntax, local Markdown links and staged whitespace were +checked. The existing source files and selected artifacts are untouched. + +## Limits + +The universal result comes from [PROOF.md](PROOF.md) plus the inherited +rounded-leaf and coefficient/phase preparation assumptions. Finite tests do +not formalize that result or the full multiplication theorem. They do not +implement the worst-case factorizer, fast fixed-precision arithmetic, a +fixed-tape schedule or an admitted supplier. The new guard is sufficient, +not optimal; it can be worse than the old guard at small radices. Initial +grid capacity, all dirty data, internal leaf magnitude bits and every access, +restoration, preparation and routing obligation remain charged. diff --git a/research/dirty-prefix-precision/core.py b/research/dirty-prefix-precision/core.py new file mode 100644 index 0000000000..9ca79fdc2f --- /dev/null +++ b/research/dirty-prefix-precision/core.py @@ -0,0 +1,106 @@ +"""Balanced dirty-word prefixes and exact execution-error budgets. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance; Apache-2.0. +See PROOF.md. No predecessor dispatcher is used in this module. +""" +from fractions import Fraction as Q +from functools import lru_cache + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +@lru_cache(None) +def resources(n): + need(type(n) is int and n >= 1, "positive integer factor count") + if n == 1: + return 0, 1, 1 + a, b = n//2, n-n//2 + ha, na, ea = resources(a) + hb, nb, eb = resources(b) + return 1+max(ha, hb), 2*(na+nb), n+max(a+eb, b+ea) + + +def word(n, sign=1, first=0, source=0, target=1, scratch=2): + """Independent chronological word: (source, target, factor, sign).""" + resources(n) + need(type(sign) is int and sign in (-1, 1), "word sign") + if n == 1: + yield source, target, first, sign + return + a, b = n//2, n-n//2 + commands = [(a, first, source, scratch, 1), + (b, first+a, scratch, target, 1), + (a, first, source, scratch, -1), + (b, first+a, scratch, target, -1)] + if sign == -1: + commands = [(m, j, s, t, -q) for m, j, s, t, q in reversed(commands)] + for m, j, s, t, q in commands: + yield from word(m, q, j, s, t, scratch+1) + + +def ceil_log2(value): + value = Q(value) + need(value > 0, "positive logarithm argument") + power = value.numerator.bit_length()-value.denominator.bit_length() + return power + (value > Q(2)**power) + + +def execution_parameters(n, K, a, D, bits): + need(K >= 0 and a > 0 and D >= 0, "operator/error/magnitude bounds") + need(type(bits) is int and bits >= 0, "fractional precision") + _, calls, exponent = resources(n) + B = (1+Q(K))**exponent + delta = calls*B*B*Q(a)/2**bits + need(delta <= Q(1, 2), "feedback smallness") + magnitude = 2*B*D+1 + error = 2*delta*(B*D+1) + return dict(calls=calls, exponent=exponent, prefix_bound=B, + feedback=delta, magnitude_bound=magnitude, error_bound=error) + + +def working_bits(n, K, a, D, target_bits, input_bits=0): + need(type(target_bits) is int and target_bits >= 0, "target bits") + need(type(input_bits) is int and input_bits >= 0, "input grid") + need(K >= 0 and a > 0 and D >= 0, "operator/error/magnitude bounds") + _, calls, exponent = resources(n) + B = (1+Q(K))**exponent + bits = max(input_bits, 0, target_bits+ceil_log2(2*calls*B*B*a*(B*D+1))) + result = execution_parameters(n, K, a, D, bits) + need(result['error_bound'] <= Q(1, 2**target_bits), "execution target") + return bits + + +def kernel_budget(radix, target_bits, magnitude_bits, inherited): + """Integer exponents only; no exponentially large root-gap numbers.""" + need(type(magnitude_bits) is int and magnitude_bits >= 0, "magnitude bits") + need(type(target_bits) is int and target_bits >= 1, "target bits") + source = inherited.budgets(radix, target_bits+1+magnitude_bits) + inherited.check(source) + n, g = radix-1, source['g'] + scratch, calls, exponent = resources(n) + A = 3*g+ceil_log2(13*(radix+1)) + H = g+ceil_log2(radix+1) + w = max(A, 2*g+1, target_bits+3+ceil_log2(calls)+A+magnitude_bits+3*exponent*H) + oldH = g+ceil_log2(radix+2) + exact_count_old = max(A, 2*g+1, target_bits+2+ceil_log2(calls)+A+magnitude_bits+(calls-1)*oldH) + return dict(radix=radix, target_bits=target_bits, initial_magnitude_bits=magnitude_bits, + n=n, dirty_arrays=scratch, leaf_calls=calls, prefix_exponent=exponent, + gap_bits=g, local_error_bits=A, shear_growth_bits=H, + working_bits=w, old_exact_count_working_bits=exact_count_old, + coefficient_target_bits=target_bits+1+magnitude_bits) + + +def check_kernel_budget(row, inherited): + need(all(type(v) is int for v in row.values()), "integer budget fields") + correct = kernel_budget(row['radix'], row['target_bits'], row['initial_magnitude_bits'], inherited) + need(row == correct, "budget certificate mismatch") + # Check the exponent inequality separately from certificate equality. + n, E, N = row['n'], row['prefix_exponent'], row['leaf_calls'] + need(E <= 4*n-3, "linear prefix exponent") + gap = row['working_bits']-row['target_bits']-3-row['local_error_bits']-row['initial_magnitude_bits']-3*E*row['shear_growth_bits'] + need(gap >= 0 and N <= 2**gap, "rounded execution inequality") + need(row['working_bits'] >= 2*row['gap_bits']+1, "leaf closure precision") + return True diff --git a/research/dirty-prefix-precision/matrices.py b/research/dirty-prefix-precision/matrices.py new file mode 100644 index 0000000000..25ece6ffbb --- /dev/null +++ b/research/dirty-prefix-precision/matrices.py @@ -0,0 +1,124 @@ +"""Independent dense integer prefixes, inverse prefixes and subwords. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance; Apache-2.0. +""" +import hashlib +import json +from fractions import Fraction as Q + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def identity(n): + return [[int(i == j) for j in range(n)] for i in range(n)] + + +def multiply(a, b): + columns = list(zip(*b)) + return [[sum(x*y for x, y in zip(row, col)) for col in columns] for row in a] + + +def norm(a): + return max(sum(map(abs, row)) for row in a) + + +def prefix_step(matrix, factor, op): + source, target, _, sign = op + need(source != target, "non-shear instruction") + width = len(matrix) + for i in range(2): + matrix[2*target+i] = [matrix[2*target+i][j] + sign*sum( + factor[i][k]*matrix[2*source+k][j] for k in range(2)) for j in range(width)] + + +def inverse_step(matrix, factor, op): + # If P_new = S P, then P_new^-1 = P^-1 S^-1: update columns. + source, target, _, sign = op + for row in matrix: + for j in range(2): + row[2*source+j] -= sign*sum(row[2*target+k]*factor[k][j] for k in range(2)) + + +def gauss_inverse(matrix): + n = len(matrix) + a = [[Q(v) for v in row]+[Q(i == j) for j in range(n)] for i, row in enumerate(matrix)] + for col in range(n): + pivot = next(i for i in range(col, n) if a[i][col]) + a[col], a[pivot] = a[pivot], a[col] + value = a[col][col] + a[col] = [x/value for x in a[col]] + for i in range(n): + if i != col: + value = a[i][col] + a[i] = [x-value*y for x, y in zip(a[i], a[col])] + return [row[n:] for row in a] + + +def trace(factors, instructions, sign, arrays, bound): + width = 2*arrays + P, inverse = identity(width), identity(width) + maximum, inverse_maximum = 1, 1 + stream = hashlib.sha256() + positions = {0, len(instructions)//4, len(instructions)//2, 3*len(instructions)//4, len(instructions)} + gauss_checks, matrices = 0, [identity(width)] if len(factors) <= 8 else [] + inverses = [identity(width)] if matrices else [] + for j, op in enumerate(instructions, 1): + factor = factors[op[2]] + prefix_step(P, factor, op) + inverse_step(inverse, factor, op) + np, ni = norm(P), norm(inverse) + need(np <= bound and ni <= bound, "prefix norm bound") + maximum, inverse_maximum = max(maximum, np), max(inverse_maximum, ni) + stream.update(json.dumps([j, np, ni], separators=(',', ':')).encode()+b'\n') + if j in positions: + need(gauss_inverse(P) == inverse, "independent inverse-prefix mismatch") + gauss_checks += 1 + if matrices: + matrices.append([row[:] for row in P]) + inverses.append([row[:] for row in inverse]) + product = identity(2) + for factor in factors: + product = multiply(factor, product) + endpoint = identity(width) + for i in range(2): + for j in range(2): + endpoint[2+i][j] = sign*product[i][j] + need(P == endpoint, "ideal product or dirty restoration mismatch") + need(multiply(P, inverse) == identity(width), "final inverse mismatch") + subwords, subword_maximum = 0, 1 + if matrices: + for first in range(len(instructions)): + direct = identity(width) + for last in range(first+1, len(instructions)+1): + op = instructions[last-1] + prefix_step(direct, factors[op[2]], op) + need(direct == multiply(matrices[last], inverses[first]), "propagator identity") + value = norm(direct) + need(value <= bound*bound, "subword norm bound") + subword_maximum = max(subword_maximum, value) + subwords += 1 + return dict(leaf_calls=len(instructions), prefix_maximum=maximum, + inverse_prefix_maximum=inverse_maximum, prefix_stream_sha256=stream.hexdigest(), + gaussian_inverse_checks=gauss_checks, subwords_checked=subwords, + subword_maximum=subword_maximum, all_columns_and_dirty_restoration=True) + + +def vendor_word(vendor, n, sign): + """Observe the immutable predecessor dispatcher with its leaf replaced by a recorder.""" + arrays = [[vendor.c.ZERO] for _ in range(2+(n-1).bit_length())] + ids = {id(a): i for i, a in enumerate(arrays)} + result = [] + + def record(x, y, factor, phases, beta, bits, direction, q, stats): + result.append((ids[id(x)], ids[id(y)], factor, q)) + + saved = vendor.leaf + try: + vendor.leaf = record + vendor.compose(*arrays[:2], list(range(n)), arrays[2:], [vendor.c.ONE], Q(1, 2), 8, 1, sign) + finally: + vendor.leaf = saved + return result diff --git a/research/dirty-prefix-precision/numeric.py b/research/dirty-prefix-precision/numeric.py new file mode 100644 index 0000000000..ee64f2cc9b --- /dev/null +++ b/research/dirty-prefix-precision/numeric.py @@ -0,0 +1,92 @@ +"""Actual pinned rounded leaves, exact dense oracles and restoration checks. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance; Apache-2.0. +""" +import hashlib +import json +from fractions import Fraction as Q + + +def run_case(core, vendor, factors, beta, radix, length, direction, dirt, target, prepared=None): + n = len(factors) + scratch, calls, exponent = core.resources(n) + arrays = [[(Q(dirt+2*i+j, 4), Q(dirt-3*i-j, 8)) for i in range(length)] + for j in range(scratch+2)] + original = [a[:] for a in arrays] + D = max(abs(x)+abs(y) for arr in arrays for x, y in arr) + K, a = Q(radix)/beta, 13*(radix+1)/beta**3 + bits = max(core.working_bits(n, K, a, D, target+1, 3), core.ceil_log2(2/beta**2)) + bound = core.execution_parameters(n, K, a, D, bits) + core.need(Q(1, 2**bits) <= beta**2/2, "rounded leaf closure") + phases = [vendor.c.ONE, (Q(0), Q(1)), (Q(3, 5), Q(4, 5))][:length] + core.need(not any(phases is arr for arr in arrays), "phase alias outside contract") + # The predecessor's oracle expands the prepared polynomial and uses dense + # Gauss-Jordan elimination, rather than the lifting/commutator recurrence. + if prepared: + exact = vendor.w.solve(vendor.w.matrix(prepared['q'], phases, direction), original[0]) + perturbation = Q(prepared['certificate']['factorization_inverse_error_bound'])*D + core.need(perturbation <= Q(1, 2**(target+1)), "unchanged factor certificate insufficient for source scale") + else: + exact = vendor.exact_operator(original[0], factors, phases, direction) + perturbation = Q(0) + observed = {'leaves': 0, 'maximum_squared': max(vendor.c.norm2(z) for arr in arrays for z in arr)} + leaf = vendor.leaf + + def monitored(x, y, factor, ps, b, w, d, sign, stats): + before = x[:] + leaf(x, y, factor, ps, b, w, d, sign, stats) + core.need(x == before, "leaf failed exact source restoration") + observed['leaves'] += 1 + observed['maximum_squared'] = max(observed['maximum_squared'], max(map(vendor.c.norm2, y))) + + try: + vendor.leaf = monitored + stats = dict(leaves=0, scalar_steps=0) + vendor.compose(*arrays[:2], factors, arrays[2:], phases, beta, bits, direction, stats=stats) + finally: + vendor.leaf = leaf + core.need(stats['leaves'] == calls == observed['leaves'], "full leaf count") + core.need(stats['scalar_steps'] == 3*length*calls, "paid three-pass accounting") + core.need(arrays[0] == original[0] and arrays[2:] == original[2:], "exact dirty restoration") + core.need(observed['maximum_squared'] <= bound['magnitude_bound']**2, "actual boundary magnitude") + net = [vendor.c.sub(x, y) for x, y in zip(arrays[1], original[1])] + error2 = max(vendor.c.norm2(vendor.c.sub(x, y)) for x, y in zip(net, exact)) + core.need(error2 <= bound['error_bound']**2, "new execution error bound") + core.need(bound['error_bound']+perturbation <= Q(1, 2**target), "combined preparation/execution budget") + # Same grid and same deterministic leaf; negative whole word cancels exactly. + vendor.compose(*arrays[:2], factors, arrays[2:], phases, beta, bits, direction, -1) + core.need(arrays == original, "whole-word exact inverse") + evidence = json.dumps([net, exact, error2, bound['error_bound'], perturbation], default=str, separators=(',', ':')) + return dict(factors=n, radix=radix, length=length, direction=direction, dirt=dirt, + target_bits=target, working_bits=bits, leaf_calls=calls, prefix_exponent=exponent, + boundary_checks=observed['leaves'], scalar_steps=stats['scalar_steps'], + error_squared_log2_upper=core.ceil_log2(error2) if error2 else None, + execution_bound_log2_upper=core.ceil_log2(bound['error_bound']), + exact_evidence_sha256=hashlib.sha256(evidence.encode()).hexdigest(), + source_and_dirty_restored=True, inverse_cancels=True, + coefficient_perturbation_checked=prepared is not None) + + +def generic(core, vendor): + result = [] + for n in (1, 2, 3, 5, 8, 17): + factors = [(i%2 == 0, (Q(1, 4), Q((-1)**i, 8)), Q(3 if i == 0 else 1)) for i in range(n)] + for length in (1, 3): + for direction in (-1, 1): + for dirt in (-7, 11): + result.append(run_case(core, vendor, factors, Q(1, 2), 3, length, direction, dirt, 32)) + return result + + +def kernels(core, vendor, data): + result = [] + for row in data: + state = vendor.w.k.certify(row) + factors = [(inside, b, Q(state['radix'] if i == 0 else 1)) for i, (inside, b) in enumerate(state['factors'])] + for length in (1, 3): + for direction in (-1, 1): + # Both signs of initial dirt across both cycle orientations. + result.append({**run_case(core, vendor, factors, state['beta'], state['radix'], + length, direction, -7 if direction == -1 else 11, 128, state), + 'embedding': state['embedding']}) + return result diff --git a/research/dirty-prefix-precision/test_precision.py b/research/dirty-prefix-precision/test_precision.py new file mode 100644 index 0000000000..17e2158ff0 --- /dev/null +++ b/research/dirty-prefix-precision/test_precision.py @@ -0,0 +1,71 @@ +"""Boundary budgets and nonlinear perturbations; Apache-2.0. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance. +""" +import importlib.util +import unittest +from fractions import Fraction as Q +from pathlib import Path + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location('core', HERE/'core.py') +c = importlib.util.module_from_spec(spec) +spec.loader.exec_module(c) + + +class PrecisionTests(unittest.TestCase): + def test_exact_logarithm_boundaries(self): + for exponent in range(-100, 101): + power = Q(2)**exponent + self.assertEqual(c.ceil_log2(power), exponent) + self.assertEqual(c.ceil_log2(power*(1-Q(1, 2**20))), exponent) + self.assertEqual(c.ceil_log2(power*(1+Q(1, 2**20))), exponent+1) + + def test_finer_input_grid_is_preserved(self): + coarse = c.working_bits(2, Q(1), Q(1), Q(2), 8) + self.assertLess(coarse, 1000) + self.assertEqual(c.working_bits(2, Q(1), Q(1), Q(2), 8, 1000), 1000) + + def test_feedback_boundary(self): + self.assertEqual(c.execution_parameters(1, 1, 1, 0, 3)['feedback'], Q(1, 2)) + with self.assertRaisesRegex(ValueError, 'feedback smallness'): + c.execution_parameters(1, 1, 1, 0, 2) + + def test_nonlinear_leaf_errors_and_exact_dirty_inverse(self): + cases = 0 + for n in (2, 3, 5, 8): + h, _, _ = c.resources(n) + factors = [Q(2) if i%2 == 0 else Q(-1, 2) for i in range(n)] + for dirt in (-7, 11): + original = [Q(1, 2), Q(3, 8)]+[Q(dirt+j, 4) for j in range(h)] + D = max(map(abs, original)) + bits = c.working_bits(n, 2, 1, D, 8, 3) + u = Q(1, 2**bits) + bound = c.execution_parameters(n, 2, 1, D, bits) + values = original[:] + + def run(sign): + peak = max(map(abs, values)) + for source, target, index, q in c.word(n, sign): + z = values[source] + error = (-1)**index*u*(abs(z)+1) + values[target] += q*(factors[index]*z+error) + peak = max(peak, max(map(abs, values))) + return peak + + peak = run(1) + expected = original[0] + for factor in factors: + expected *= factor + actual_error = abs(values[1]-original[1]-expected) + self.assertLessEqual(actual_error, bound['error_bound']) + self.assertLessEqual(peak, bound['magnitude_bound']) + self.assertEqual(values[:1]+values[2:], original[:1]+original[2:]) + run(-1) + self.assertEqual(values, original) + cases += 1 + self.assertEqual(cases, 8) + + +if __name__ == '__main__': + unittest.main() diff --git a/research/dirty-prefix-precision/vendor/pr628/DIRTY-PROOF.md b/research/dirty-prefix-precision/vendor/pr628/DIRTY-PROOF.md new file mode 100644 index 0000000000..caf4aafc78 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/DIRTY-PROOF.md @@ -0,0 +1,128 @@ +# Cyclic inverse shears with dirty array restoration + +This extends the classical rounded cyclic-inverse primitive of PR628. It removes +its need for a clean length-L intermediate array. It does not supply a common-label +Clifford realization, reversible bit arithmetic, transform layout, or the complete +integer-multiplication reduction. No new kappa follows from this lemma. + +## Contract + +Input x, target y and h borrowed arrays are complex dyadic grids of spacing u=2^-w. +Every array has length L>=1 and may initially contain arbitrary data. The arrays +are distinct. Their initial infinity norms are at most D. Read-only phases tau_i +have modulus one. Phase generation/storage and coefficient preparation are paid; +there is no free phase table. The primitive adds an approximation to Q(W)^-1 x +to y, restores x and every borrowed array exactly, and needs only a constant number +of clean scalar arithmetic registers besides loop counters and factor descriptors. +Here (Wx)_i=tau_i*x_(i-d), initially for d=+/-1 in a cyclic layout, and +Q(z)=r^-1 product_(j=1)^n(z-rho_j), n=r-1, with stable reciprocal/inner factors +of norm at most 1-beta. The final increment can depend on the initial dirt within +the stated approximation error. Exact independence of the increment from dirt is +not claimed for rounded arithmetic. + +## One factor + +First consider T x = y defined by y_i=x_i+b_i*y_(i-d), |b_i|<=1-beta. +Choose the cyclic order in direction d. Let z be the zero-boundary causal solution +and p_i the product of b from the first position through i. Let c be the complete +product. Then y_i=z_i+p_i*s, s=z_last/(1-c). The stable cyclic equation itself +implies ||y||<=B/beta when ||x||<=B, so |s|<=B/beta. + +On the dyadic grid, perform the causal pass directly in x, skipping the first +position: x_i += round(b_i*x_previous). These are integer lifts. Reversing their +order and subtracting the same rounded source products restores x exactly, +including arbitrary nonzero initial values. During the first pass also accumulate +the rounded scalar product c_hat. Compute s_hat=round(z_hat_last/(1-c_hat)). +In a second pass accumulate rounded prefix products p_hat_i and add +round(m_i*(z_hat_i+round(p_hat_i*s_hat))) to the appropriate target position. +Finally undo the causal lifts. Each target position is written exactly once; +its existing value never enters the computed increment. Scalar arithmetic and +coefficient scratch can be cleared classically; no reversible arithmetic gate +claim is made. Three passes suffice, excluding validation scans and paid +coefficient access. O(1) clean scalar words and O(log L) indexing bits suffice. + +Nearest complex rounding has norm error at most u. Assuming u<=beta^2/2, +|z_hat-z|<=u/beta, |p_hat-p|<=u/beta, |c_hat-c|<=u/beta, and +|1-c_hat|>=beta/2. Using s=z_last/(1-c), rather than separately bounding the +numerator and denominator, gives + + |s_hat-s| <= u + (2/beta)*(u/beta + (B/beta)*(u/beta)) + <= 5*(B+1)*u/beta^3. + +Also |p_hat|<=1+u/beta<=3/2. Hence the corrected output before m multiplication +has error at most 12*(B+1)*u/beta^3. If |m_i|<=r, the final rounded multiplication +has error at most a*(B+1)*u, a=13*(r+1)/beta^3. None of these constants depends +on L. + +For an outer root, b_i=rho^-1*tau_i, m_i=-rho^-1. For an inner root, scan in +direction -d with b_i=rho*conjugate(tau_(i+d)); the value at position i is +multiplied by conjugate(tau_i) and added at i-d. This is +(W-rho I)^-1=(I-rho W^-1)^-1 W^-1. The scalar r is included in one leaf's +multiplier. Each exact factor has operator norm at most K=r/beta. No permutation +of the mutable source array is required for the inner factor. + +## Product of factors + +For exact linear maps A,B and a dirty scratch array z, the sequence + + z += B*x; y += A*z; z -= B*x; y -= A*z + +adds AB*x to y and restores z. Split the factor list in halves recursively; +use one distinct borrowed array per recursion depth. A negative call executes +the exact inverse instruction word (reverse operation order and invert signs). +It is not obtained by negating signs in the original order. + +The rounded construction still restores every borrowed array and x exactly. +Inductively each call preserves its input and all deeper scratch, and its increment +is independent of its target. The paired calls computing B therefore see the same +input and deeper initial dirt. Never use an ancestor input or target as deeper +scratch: the ancestor target may have changed between the paired calls. This is +an explicit alias exclusion in the executable contract. + +There are h=ceil(log2 n) borrowed arrays and at most N=4*n^2 leaf calls. Pad to the +next power of two for this count if desired (identity leaves require no rounded +inverse). The implementation splits without padding and checks the actual count. + +For the ideal elementary shears, the joint max norm grows by at most C=1+K per +leaf. With a*u<=1 the computed trajectory satisfies + + M_j+1 <= G^j*(D+1), G=C+1. + +Comparing the complete rounded word with the ideal word on the same initial +arrays, and summing the propagated leaf errors, gives + + E_N <= N*a*(D+1)*u*G^(N-1). + +The ideal word cancels all borrowed initial data exactly. This proves the target +error bound for every length and every initial dirt bounded by D, while actual +restoration is exact without an error allowance. Choosing + + w >= p+1+ceil(log2(N*a*(D+1)))+(N-1)*ceil(log2 G) + +and enforcing a*u<=1 and u<=beta^2/2 gives execution error <=2^(-p-1). +PR628's coefficient/phase perturbation bound, budgeted to <=2^(-p-1) for the +original x, then gives total <=2^-p. That perturbation bound scales with ||x||; +for general input bound X use p+ceil(log2(max(1,X))) in its preparation budget. +The initial borrowed dirt does not enter the ideal operator perturbation bound. + +PR628 has log2(beta^-1)=O(r^3). Therefore these deliberately loose budgets need +w=O(p+log(D+1)+r^5) bits, and O(r^2) passes. Inside a leaf, the causal state and closure scalars add at most another +O(log(beta^-1)+log r) magnitude bits. Maximum transient magnitudes thus add +O(log(D+1)+r^5) integer bits, already covered by this word-size bound. Standard +fast fixed-precision arithmetic yields soft-O((L*r^2+r^3)*(p+log(D+1)+r^5)) +bit work plus charged phase-oracle and routing costs. Factor preparation and +coefficient enclosures are inherited from PR628. Borrowed storage is +O(L log r) words; clean arithmetic storage is O(w+log L) bits up to the chosen +multiplication implementation's workspace, plus O(r) read-only factor descriptors. +A machine with only a fixed number of dirty data arrays is not covered by this +contract. Arbitrary stride still incurs PR628's explicitly charged routing. + +## Executable evidence + +`dirty.py` uses exact rational arithmetic and explicit nearest dyadic rounding. +It checks generic variable-coefficient leaves, dense independently assembled +factor inverses, both orientations, nonzero targets, several initial dirty arrays, +exact source/scratch restoration, exact whole-word inversion, and the leaf count. +Finite checks support the implementation; the inequalities above are the all-L +argument. The exact-rational prototype does not itself implement the claimed +fast fixed-precision arithmetic cost. diff --git a/research/dirty-prefix-precision/vendor/pr628/FACTORS.json b/research/dirty-prefix-precision/vendor/pr628/FACTORS.json new file mode 100644 index 0000000000..d3894bd2a1 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/FACTORS.json @@ -0,0 +1,402 @@ +[ + { + "radix": 3, + "embedding": 1, + "roots": [ + [ + 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Attribution and scope + +Ali Romman's PR621, exact head 520774462b19b29af41554b74def31f9d01e997e, +supplies the prime-power kernel circle bound. Its kernel attribution is +OpenAI's exact-DFT preprint at adc7f1241b42e322a6451854ab7e4b4c146bf78a +(`loc:newton`, `loc:fourier-factorization`) and classical q-Pochhammer/Newton +factorization, including Alexey Kuznetsov's cited q-calculus work. This +package does not claim to originate those constructions. + +Pan's approximate-factorization theorem is used in the form stated by +Mehlhorn, Sagraloff and Wang; Brent's work supplies fast elementary-function +evaluation. Causal/anticausal recursive filtering is standard, as illustrated +by the Briand-Davy source listed in SOURCE.json. Sources are cited for these +specific tools, not as proofs of this package's complete claim. + +The scan rounding analysis, explicit budget composition, interval and +factor certificates, routed reference execution, and packaging were +prepared with substantial OpenAI Codex assistance for chafreaky. The dense +matrix solver and algebraic coefficient crosschecks provide independent +computational paths, not human peer review or formal Lean verification. + +No new multiplication exponent, literature priority, maintainer acceptance, +physical tape execution, or production performance result is claimed. +The code and documents use the repository's Apache-2.0 license. diff --git a/research/dirty-prefix-precision/vendor/pr628/PROOF.md b/research/dirty-prefix-precision/vendor/pr628/PROOF.md new file mode 100644 index 0000000000..fb8fe7cb3b --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/PROOF.md @@ -0,0 +1,269 @@ +# Uniform preparation and rounded inversion of a neighboring cyclic kernel + +This note combines PR621's circle lower bound, standard approximate +factorization, and the local two-scan rounding lemma. It concerns a +neighboring cyclic shift with gauge 1 and clean input/output work arrays. +It is not a complete multiplication algorithm, reversible word, arbitrary +stride tape schedule, or new kappa result. The proof uses the cited +factorization theorem; the accompanying numerical nomination program does +not implement that theorem's worst-case algorithm. + +## Statement and model + +Let r>=3 be a prime power, k coprime to r, zeta=exp(2 pi i k/r), and + + P(z)=sum_(j=0)^(r-1) h_j z^j, + h_j=1/product_(s=1)^j(1-zeta^s). + +Let S be either neighboring cyclic shift on L>=1 entries. Given p>=1 +and an exactly supplied dyadic complex vector x with ||x||infinity<=1, +there is an algorithm producing y with + + ||y-P(S)^-1 x||infinity <= 2^-p. + +Coefficient preparation is independent of L. Both the required coefficient +bit lengths and the execution precision are O(p+r^4), including magnitude +bits. The array execution uses O(rL) rounded scalar operations and O(r) +sequential full-array passes. With ordinary near-linear integer arithmetic, +including a conservative exact coefficient-product verification in setup, +the bit-operation bound is + + soft-O((Lr+r^3)(p+r^4)). + +Here soft-O suppresses logarithms in the displayed parameters. Input +fractional precision is assumed at most the execution budget below; +longer exact inputs require their reading/rounding cost and a separately +charged input-error budget. Clean arrays and standard arithmetic scratch +are allowed. The scan layout is sequential for S=Z or Z^-1; this statement +does not assert that an arbitrary modular-stride permutation is free. +The setup bound uses the bit-complexity model of the cited factorization +result. A concrete fixed-tape implementation of its setup is not supplied. + +## Explicit budgets + +Put n=r-1, D=phi(r), ell=ceil(log2 r), and + + d=1+2 ell+r n D, + g=d+ell+2r+2, beta=2^-g, + a=ell+n g, Abar=2^a, + e=max(d+1, a+1, p+2a+3), epsilon=2^-e, + b=max(n ell, e+r+3), + w=max(2g+ceil(log2(14n)), + p+(n+2)g+ceil(log2(28n(r+1)))), u=2^-w. + +All these are computable integer budgets. Their sizes satisfy +d,g=O(r^3) and a,e,b,w=O(p+r^4). No array length occurs in them. +The executable budget checker verifies the integer inequalities used below +for a range of radices and precisions; the algebra proves them generally. + +## Approximate factorization and preparation + +Define f=rP. Its leading coefficient is 1 and its coefficients are + + f_j=product_(s=j+1)^(r-1)(1-zeta^s). + +Thus ||f||_1<=H=2^r-1, and Cauchy's bound places all roots inside +|z|<=1+H<2^(r+1). Its logarithmic root bound is O(r). + +Pan's approximate-factorization theorem, in Theorem 2 and Corollary 1 of +Mehlhorn, Sagraloff and Wang [1], provides dyadics rho_1,...,rho_n with +||f-product_j(z-rho_j)||_1<=2^-b ||f||_1. Its factorization cost is +soft-O(n b+n^2 Gamma), with coefficient-oracle precision O(b+n Gamma), +where Gamma bounds the logarithmic root magnitudes. Repeated roots do not +require a distinct-root count for this factorization step. + +Let Q=(1/r) product_j(z-rho_j). Our choice b>=e+r+3 gives + + ||P-Q||_1 < epsilon/8. + +Only this coefficient error is needed. We do not isolate the true roots, +match approximate roots to true roots, or require simple roots. + +The coefficient oracle is also charged. Compute zeta to the requested +precision, its first r powers, and the backward products defining f_j. +There are O(r) complex operations. Products have magnitude at most 2^r; +guarding by O(r+log r) bits bounds coefficient errors. Fast elementary +function evaluation [2] supplies the required pi, sine and cosine values +with near-linear precision cost up to logarithms. Hence the oracle fits +soft-O(r(p+r^4)). This is an algorithmic preparation argument, not a +complexity claim about the Fraction/Decimal reference scripts. + +An independent certificate can expand the dyadic-root product exactly and +compare it with rigorous coefficient enclosures of total componentwise +error at most epsilon/8. Replacing complex moduli by the sum of absolute +real and imaginary parts costs at most a factor of 2, so the resulting +upper bound is still below epsilon. Exact expansion +of n B-bit roots has O(nB)-bit coefficients; a simple iterative expansion +costs soft-O(n^3 B) bit operations, B=O(p+r^4). This conservative setup +cost is included in the theorem. Faster expansion is unnecessary here. + +## Stability without individual root certification + +PR621 gives |P(z)|>=delta on |z|=1, where + + delta=1/[2rn H^(nD-1)] >= 2^-d. + +The coefficient certificate and e>=d+1 imply |Q(z)|>=delta/2 on the +unit circle. Also ||Q||_1<=H/r+epsilon<=H/r+1<=2^r. + +For any root rho of Q at radial distance t=||rho|-1|<1/2, integrate Q' +along the radial segment to the unit circle. On that segment |z|<2, so + + delta/2 <= t n 2^r 2^(n-1). + +Roots farther than 1/2 already have a fixed gap. Therefore every root of +Q lies at radial distance at least + + eta=min(1/2, delta/[2n 2^r 2^(n-1)]). + +Our g ensures beta<=eta/2. If |rho|>1, b_rho=1/rho has modulus at most +1-beta. If |rho|<1, b_rho=rho has modulus at most 1-beta. These comparisons +can be made exactly on the dyadic rho. Inverting an outer rho produces +an exact rational complex coefficient with O(p+r^4) numerator and +denominator bits; it need not be rounded to a dyadic. + +For each factor, + + (S-rho I)^-1 = -rho^-1 (I-rho^-1 S)^-1 if |rho|>1, + (S-rho I)^-1 = (I-rho S^-1)^-1 S^-1 if |rho|<1. + +Each stable inverse has norm at most beta^-1. Outer scalar factors have +modulus less than 1, while the inner shift has norm 1. Apply the scalar r +to the input first. Exact stage-j values have norm at most r beta^-j. +In particular, ||Q(S)^-1||<=r beta^-n<=Abar, uniformly in L. + +Since epsilon Abar<=1/2, a Neumann perturbation bound gives + + ||P(S)^-1-Q(S)^-1|| + <= Abar^2 epsilon/(1-Abar epsilon) + <= 2 Abar^2 epsilon <= 2^(-p-2). + +This is an operator bound for arbitrary input vectors, not just samples +or a particular Fourier mode. + +## Rounded execution + +Use the two-scan periodic first-order procedure proved in +SCAN-PROOF.md, with grid u and exact rational coefficients. +An inner factor scans in the opposite direction and includes the unit +cyclic shift. An outer factor includes a final rounded multiplication by +-rho^-1. The per-factor error is bounded by + + 7(B+1)u/beta^3 + +for input norm B. The constant 7 includes the outer scalar multiplication; +the original scan constant is 6. Copying and shifting dyadic entries is +exact. Multiplying the original dyadic input by integer r is exact. + +If E_j is the error after j factors, then E_0=0 and + + E_j <= E_(j-1)/beta + +7(r beta^(-(j-1))+E_(j-1)+1)u/beta^3. + +Set T_j=beta^j E_j. Since beta<=1 and j>=1, + + T_j <= (1+7u/beta^2)T_(j-1)+7(r+1)u/beta^2. + +The budget u<=beta^2/(14n) implies +(1+7u/beta^2)^n<=exp(1/2)<2. Consequently + + E_n <= 14n(r+1)u beta^(-(n+2)) <= 2^(-p-1). + +Adding the factorization perturbation yields at most 3*2^(-p-2)<2^-p. +All scan intermediates have O(p+r^4) bits including integer magnitude. +In particular, b_rho^L is accumulated with rounding at every step, so its +representation does not grow with L. The closure denominator stays at +least beta/2, by the single-factor proof. + +Exact rational multiply/add followed by dyadic rounding and the one +closure division use integers of O(p+r^4) bits. Standard fast integer +multiplication and division give the stated scalar bit cost. A bounded +number of sequential passes suffices per factor, including rewinds and +the unit cyclic rotation. Two alternating clean data arrays suffice for +the streaming values. A conservative space allowance is +soft-O((L+r^2)(p+r^4)) bits, including setup and arithmetic scratch. + +## Paid unit phases + +The same factor certificate works for U=tau S when tau is an exactly +supplied rational complex number of modulus 1. In an outer factor, the +scan coefficient becomes tau/rho and the final scalar remains -1/rho. +In an inner factor, it becomes rho/tau; after the backward scan and unit +shift, multiply by tau^-1. Both new coefficients retain the same margin +beta. The inner scalar has modulus 1, so the same constant 7 covers its +rounded multiplication. If tau has T representation bits, scalar +arithmetic costs use O(p+r^4+T)-bit operands; T is not silently free. + +More generally suppose a unit tau has a charged approximation oracle. +Choose a rational unit sigma with |tau-sigma|<=nu, where + + nu <= epsilon/(n 2^r). + +Then ||P(tau S)-Q(sigma S)||<=epsilon+n||Q||_1 nu<=2 epsilon. +Because e>=p+2a+3, we still have 2 epsilon Abar<=1/2 and + + ||P(tau S)^-1-Q(sigma S)^-1|| + <=4 Abar^2 epsilon<=2^(-p-1). + +The same rounded execution budget contributes at most 2^(-p-1), giving +the original target 2^-p. The phase needs O(p+r^4) approximation bits, +in addition to reading whatever representation defines it. + +Here is a constructive way to obtain sigma with exact unit modulus. +Choose C in {1,-1,i,-i} so v=C^-1 tau has Re(v)>=1/2. A coarse enclosure +suffices to choose such a chart: one coordinate of a unit vector has +absolute value at least 1/sqrt(2). Put t=Im(v)/(1+Re(v)). Its denominator +is at least 3/2. Approximate t by a dyadic t0 with |t-t0|<=nu/2 and set + + sigma=C (1-t0^2+2i t0)/(1+t0^2). + +This is exactly rational and unit. The stereographic map's derivative +has modulus 2/(1+t^2)<=2, proving |tau-sigma|<=nu. Obtaining t0 costs +the oracle request plus scalar arithmetic at the stated precision. +For a phase exp(2 pi i h/m), [2] supplies that oracle after the cost of +reading/reducing the integer phase description; a very long h or m is +an extra input cost and is not covered by a bound that omits its length. + +This section treats a single constant unit phase. WEIGHTED-PROOF.md +extends the same bounds to a unit-weighted neighboring shift with an +independent phase at each position. The full transform's collection of +phases still needs its composition and preparation accounting. + +## Scope and finite evidence + +For r bounded by a fixed power of log p, the cost is quasi-linear in p +up to polylogarithmic factors and polynomial dependence on that radix. +There is no extra factor proportional to p in the number of full-array +passes. This removes the repeated-Neumann-pass obstruction for this +neighboring, clean-array inverse primitive. + +The mixed-factor reference certificate at CERTIFICATE-PROOF.md checks +ten kernel/embedding cases through radix 16, including inner roots, exact +rational unit phases, and a certified rational approximation of +exp(2 pi i/7). It uses exact coefficient enclosures, exact factor products, +and an explicit rounding budget. These finite certificates do not implement Pan's +worst-case factorizer or prove its complexity. The uniform argument above +depends on [1], [2], PR621, and the proved scan lemma. + +Remaining work for a multiplication application includes the complete +collection of phase gauges and their paid preparation, the actual strides/layout of every kernel, +composition across the complete transform and recursion, its total +precision guard, input/output normalization, and restoration of dirty +scratch where required. None is inferred from this standalone theorem. + +## Sources and attribution + +[1] K. Mehlhorn, M. Sagraloff, P. Wang, *From Approximate Factorization to +Root Isolation with Application to Cylindrical Algebraic Decomposition*, +Theorem 2 and Corollary 1 (crediting V. Pan), +https://arxiv.org/pdf/1301.4870, PDF page 8. + +[2] R. P. Brent, *Fast multiple-precision evaluation of elementary +functions*, J. ACM 23 (1976), 242-251, +https://maths-people.anu.edu.au/~brent/pub/pub034.html. + +PR621 by aliromman, head 520774462b19b29af41554b74def31f9d01e997e, +supplies the kernel and circle bound. Causal/anticausal filtering and +approximate factorization are standard methods. The explicit budget +composition and reference certificates here were prepared with OpenAI +Codex for chafreaky. No literature-priority claim is made. diff --git a/research/dirty-prefix-precision/vendor/pr628/SOURCE.json b/research/dirty-prefix-precision/vendor/pr628/SOURCE.json new file mode 100644 index 0000000000..59bddaea4d --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/SOURCE.json @@ -0,0 +1,18 @@ +{ + "upstream_kernel_pr": "https://github.com/CrocSwap/integer-mult-bounds/pull/621", + "upstream_kernel_head": "520774462b19b29af41554b74def31f9d01e997e", + "upstream_proof_url": "https://github.com/CrocSwap/integer-mult-bounds/blob/520774462b19b29af41554b74def31f9d01e997e/research/prime-power-cyclic-inverse/PROOF.md", + "upstream_proof_sha256": "2b8056c88e66f692f407450fff0414d41bcd09db8597be22e274a6c3dc20eb9e", + "factorization_primary": "https://arxiv.org/pdf/1301.4870", + "factorization_result": "Theorem 2 and Corollary 1, PDF page 8, crediting Pan", + "elementary_functions_primary": "https://maths-people.anu.edu.au/~brent/pub/pub034.html", + "recursive_filter_primary": "https://www.ipol.im/pub/art/2019/257/article_lr.pdf", + "local_provenance": { + "stable-cyclic-scan": "bc36a9d44893f9cd25225b64b08f1f3e938a248f39c16fa50d71e84eb0a1b0ef", + "stable-cyclic-mixed": "df063d9ec36d95a65119af4518137f78935ac6fac0e88de11e6a7212492afe4a", + "uniform-neighbor-cyclic-inverse": "1f8fc021a466d03d62ccf708ef6cc83d2b38b340b50893fbcceb39c704922a71" + }, + "base_commit": "ed49217ad6244f1d3bfe4588f122fa861849739c", + "reference_runtime": "Python standard library; no external packages or historical workspace files required.", + "unimplemented": "Pan worst-case factorizer, fast elementary-function implementation, literal tape simulator, dirty-scratch restoration, complete multiplication reduction." +} diff --git a/research/dirty-prefix-precision/vendor/pr628/check.py b/research/dirty-prefix-precision/vendor/pr628/check.py new file mode 100644 index 0000000000..5ef6dd67b6 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/check.py @@ -0,0 +1,178 @@ +"""Rational proof of mixed inside/outside cyclic factor inversion.""" +from pathlib import Path +from fractions import Fraction as F +import argparse,importlib.util,json,math,sys +sys.set_int_max_str_digits(100000) +P=Path(__file__).resolve().parent +def module(name): + s=importlib.util.spec_from_file_location(name,P/(name+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m +c=module('scan');ik=module('interval_kernel') + +def prime_power(r): + if type(r)!=int or r<3:return False + p=next((d for d in range(2,math.isqrt(r)+1) if r%d==0),r) + while r%p==0:r//=p + return r==1 + +def certify(data): + r,k=data['radix'],data['embedding'] + c.need(prime_power(r) and type(k)==int and 00 and type(target)==int and target>0,'precision budget') + c.need(type(fb)==int and 0=F(1,2),'gauge chart') + parameters=[im/(1+re) for im in v[1] for re in v[0]] + mid=(min(parameters)+max(parameters))/2 + t=c.rounded((mid,F(0)),1024)[0] + sigma=c.mul(center,((1-t*t)/(1+t*t),2*t/(1+t*t))) + c.need(c.norm2(sigma)==1,'exact rational unit phase') + error=sum(max(abs(endpoint-coordinate) for endpoint in iv) for iv,coordinate in zip(box,sigma)) + c.need(error0) + for name in ('unit_root','missing_root','wrong_embedding','low_precision','bad_margin'): + row=json.loads(json.dumps(mixed)) + if name=='unit_root':row['roots'][0]=['1','0'] + if name=='missing_root':row['roots'].pop() + if name=='wrong_embedding':row['embedding']=row['radix'] + if name=='low_precision':row['working_bits']=8 + if name=='bad_margin':row['beta']='1' + try:certify(row) + except ValueError as error:controls.append(dict(name=name,rejected=str(error))) + else:raise ValueError('bad control accepted '+name) + state=certify(mixed);x=[c.ONE,c.ZERO,(F(1,2),F(1,4))] + bad=apply(x,state,1,omit_inner_shift=True) + try:residual_check(x,bad,state,1) + except ValueError as error:controls.append(dict(name='omitted_inner_shift',rejected=str(error))) + else:raise ValueError('inner-shift mutation accepted') + try:apply(x,state,1,(F(1),F(1))) + except ValueError as error:controls.append(dict(name='nonunit_gauge',rejected=str(error))) + else:raise ValueError('nonunit gauge accepted') + args.out.write_text(json.dumps(dict(status='passed',kernel_crosschecks=crosschecks,irrational_gauge=gauge_record,certificates=certificates,cases=cases,controls=controls),indent=2)+'\n') + print(json.dumps(dict(status='passed',certificates=len(certificates),cases=len(cases),dense_cases=sum(row['dense_comparison'] for row in cases),controls=len(controls)))) + +if __name__=='__main__':main() diff --git a/research/dirty-prefix-precision/vendor/pr628/check_budgets.py b/research/dirty-prefix-precision/vendor/pr628/check_budgets.py new file mode 100644 index 0000000000..ff21b6e190 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/check_budgets.py @@ -0,0 +1,59 @@ +"""Exact integer checks of the explicit uniform precision budgets.""" +from pathlib import Path +import argparse,json,math + +def need(ok,msg): + if not ok:raise ValueError(msg) +def ceil_log2(n): + need(type(n)==int and n>0,'positive integer log argument') + return (n-1).bit_length() +def prime_power_data(r): + need(type(r)==int and r>=3,'radix') + prime=next((d for d in range(2,math.isqrt(r)+1) if r%d==0),r) + q=r + while q%prime==0:q//=prime + need(q==1,'prime power') + return r-r//prime +def budgets(r,p): + D=prime_power_data(r);need(type(p)==int and p>=1,'target bits') + n=r-1;ell=ceil_log2(r);d=1+2*ell+r*n*D;g=d+ell+2*r+2;a=ell+n*g + e=max(d+1,a+1,p+2*a+3);b=max(n*ell,e+r+3) + w=max(2*g+ceil_log2(14*n),p+(n+2)*g+ceil_log2(28*n*(r+1))) + return dict(radix=r,target_bits=p,n=n,D=D,ell=ell,d=d,g=g,a=a,e=e,b=b,w=w) +def check(x): + r,p,n,D,ell,d,g,a,e,b,w=(x[k] for k in ('radix','target_bits','n','D','ell','d','g','a','e','b','w')) + need(D==prime_power_data(r) and n==r-1 and ell==ceil_log2(r),'kernel parameters') + need(d>=1+2*ell+r*(n*D-1),'delta bound exponent') + need(g>=d+ell+2*r and g>=2,'root gap exponent') + need(a>=ell+n*g,'inverse prefix bound') + need(e>=d+1 and e>=a+1 and e>=p+2*a+3,'factor perturbation budget') + need(b>=n*ell and b>=e+r+3,'approximate factorization precision') + need(w-2*g>=ceil_log2(14*n),'rounded recurrence growth budget') + gap=w-p-(n+2)*g-1 + need(gap>=0 and 14*n*(r+1)<=1<0,'distinct equally sized arrays') + c.need(d in(-1,1)and sign in(-1,1),'orientation and sign') + c.need(0=(beta/2)**2,'closure denominator') + seed=c.rounded(c.div(x[previous],denominator),bits) + power=c.ONE + for i in ordered(n,d): + power=c.rounded(c.mul(coefficient(i),power),bits) + value=c.add(x[i],c.rounded(c.mul(power,seed),bits)) + value=c.rounded(c.mul(multiplier(i),value),bits) + j=destination(i);y[j]=c.add(y[j],scaled(value,sign)) + # The scalar seed and coefficient scratch may be cleared by ordinary classical + # arithmetic. The dirty data itself is restored by exact inverse integer lifts. + first=0 if d==1 else n-1 + for i in ordered(n,-d): + if i!=first:x[i]=c.sub(x[i],c.rounded(c.mul(coefficient(i),x[i-d]),bits)) + if stats is not None:stats['leaves']+=1;stats['scalar_steps']+=3*n + +def leaf(x,y,factor,phases,beta,bits,d,sign,stats): + inside,b,scale=factor;n=len(x) + if inside: + lift(x,y,lambda i:c.mul(b,conj(phases[(i+d)%n])),beta,bits,-d,sign, + lambda i:(i-d)%n,lambda i:scaled(conj(phases[i]),scale),stats) + else: + lift(x,y,lambda i:c.mul(b,phases[i]),beta,bits,d,sign, + multiplier=lambda i:scaled(c.neg(b),scale),stats=stats) + +def compose(x,y,factors,scratch,phases,beta,bits,d,sign=1,stats=None): + """Balanced commutator; negative call is the exact inverse instruction word.""" + c.need(sign in(-1,1)and d in(-1,1),'orientation and sign') + c.need(factors and len(scratch)>=(len(factors)-1).bit_length(),'dirty depth') + arrays=[x,y]+scratch;c.need(len({id(v)for v in arrays})==len(arrays),'all ancestor arrays distinct') + c.need(len(x)>0 and all(len(v)==len(x)for v in arrays)and len(phases)==len(x),'array lengths') + c.need(all(c.norm2(t)==1 for t in phases),'unit phases') + if len(factors)==1:return leaf(x,y,factors[0],phases,beta,bits,d,sign,stats) + mid=len(factors)//2;B,A=factors[:mid],factors[mid:];z=scratch[0];rest=scratch[1:] + operations=[(x,z,B,1),(z,y,A,1),(x,z,B,-1),(z,y,A,-1)] + if sign==-1:operations=[(src,dst,fs,-sgn)for src,dst,fs,sgn in reversed(operations)] + for src,dst,fs,sgn in operations:compose(src,dst,fs,rest,phases,beta,bits,d,sgn,stats) + +def exact_operator(x,factors,phases,d): + # Dense rational oracle constructed independently from the lifting recurrence. + n=len(x);out=x[:] + for inside,b,scale in factors: + a=[[c.ZERO for _ in range(n)]for _ in range(n)] + if inside: + for i in range(n):a[i][i]=c.neg(b);a[i][(i-d)%n]=c.add(a[i][(i-d)%n],phases[i]) + else: + rho=c.div(c.ONE,b) + for i in range(n):a[i][i]=c.neg(rho);a[i][(i-d)%n]=c.add(a[i][(i-d)%n],phases[i]) + out=[scaled(z,scale)for z in w.solve(a,out)] + return out + +def certificate(n,beta,r,D,bits): + N=4*n*n;a=13*(r+1)/beta**3;G=2+F(r)/beta;u=F(1,1<=0,'initial magnitude bits') + base=b.budgets(r,p+1+d);g=base['g'];n=r-1;N=4*n*n + A=3*g+b.ceil_log2(13*(r+1));G=g+b.ceil_log2(r+2) + w=max(A,2*g+1,p+1+b.ceil_log2(N)+A+d+1+(N-1)*G) + return dict(radix=r,target_bits=p,initial_magnitude_bits=d,n=n,leaf_bound=N,dirty_arrays=(n-1).bit_length(),gap_bits=g,local_error_bits=A,growth_bits=G,working_bits=w,coefficient_target_bits=p+1+d) + +def check(x): + r,p,d=(x[k]for k in('radix','target_bits','initial_magnitude_bits'));base=b.budgets(r,p+1+d);b.check(base);g=base['g'];n=r-1;N=4*n*n;A=x['local_error_bits'];G=x['growth_bits'];w=x['working_bits'] + b.need(x['n']==n and x['leaf_bound']==N and x['dirty_arrays']==(n-1).bit_length(),'commutator resources') + b.need(x['gap_bits']==g,'inherited root gap') + b.need(A>=3*g+b.ceil_log2(13*(r+1))and G>=g+b.ceil_log2(r+2),'amplification exponents') + b.need(w>=A and w>=2*g+1,'local precision') + b.need(w>=p+1+b.ceil_log2(N)+A+d+1+(N-1)*G,'complete word error') + b.need(x['coefficient_target_bits']>=p+1+d,'scaled original-input perturbation') + return True + +def main(): + records=[];controls=[] + for r in range(3,257): + try:b.prime_power_data(r) + except ValueError:continue + for p in(1,32,128,1024,1048576): + for d in(0,20,1024): + x=budget(r,p,d);check(x);records.append(x) + for name,key,value in [('underprecision','working_bits',128),('insufficient_dirty_arrays','dirty_arrays',0),('underestimated_calls','leaf_bound',1),('omitted_dirt_magnitude','initial_magnitude_bits',2048),('omitted_input_scaling','coefficient_target_bits',128)]: + x=budget(9,128,20);x[key]=value + try:check(x) + except ValueError as error:controls.append(dict(name=name,rejected=str(error))) + else:raise ValueError('accepted corrupt budget '+name) + ap=argparse.ArgumentParser();ap.add_argument('--out',type=Path,required=True);a=ap.parse_args();a.out.write_text(json.dumps(dict(status='passed',records=records,controls=controls),indent=2)+'\n');print('PASS',len(records),'budgets',len(controls),'controls') +if __name__=='__main__':main() diff --git a/research/dirty-prefix-precision/vendor/pr628/interval_kernel.py b/research/dirty-prefix-precision/vendor/pr628/interval_kernel.py new file mode 100644 index 0000000000..1182175a55 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/interval_kernel.py @@ -0,0 +1,61 @@ +"""Outward-rounded rational enclosures for prime-power cyclic kernels.""" +from fractions import Fraction as F +import math +from functools import lru_cache + +BITS=1536 +SCALE=1<4:raise ValueError('angle bound') + itheta=(point(0),theta) + total=term=ONE + # Taylor remainder for complex exp: <=exp(4)*4^(N+1)/(N+1)! <128*... + n=0 + while True: + n+=1;term=cmul(term,itheta);term=(scale(term[0],F(1,n)),scale(term[1],F(1,n)));total=cadd(total,term) + remainder=F(128*4**(n+1),math.factorial(n+1)) + if remainder 0, 'empty cycle') + need(direction in (-1, 1), 'invalid direction') + need(0 < beta <= 1, 'invalid margin') + need(norm2(b) <= (1-beta)**2, 'uncertified stable factor') + u = F(1, 1 << bits) + need(u <= beta*beta/2, 'precision too low for closure') + order = list(range(len(x))) if direction == 1 else list(reversed(range(len(x)))) + z, power = ZERO, ONE + for i in order: + z = rounded(add(x[i], mul(b, z)), bits) + power = rounded(mul(b, power), bits) + den = sub(ONE, power) + need(norm2(den) >= (beta/2)**2, 'closure lower bound failed') + seed = rounded(div(z, den), bits) + y, previous = [ZERO]*len(x), seed + for i in order: + previous = rounded(add(x[i], mul(b, previous)), bits) + y[i] = previous + return y + +def dense_solve(x, b, direction): + """Independent exact Gaussian elimination on the cyclic matrix.""" + n = len(x) + a = [[ZERO for _ in range(n)] + [x[i]] for i in range(n)] + for i in range(n): + a[i][i] = add(a[i][i], ONE) + j = (i-direction) % n + a[i][j] = sub(a[i][j], b) + for k in range(n): + pivot = next(i for i in range(k, n) if a[i][k] != ZERO) + a[k], a[pivot] = a[pivot], a[k] + p = a[k][k] + a[k] = [div(v, p) for v in a[k]] + for i in range(n): + if i == k: continue + p = a[i][k] + a[i] = [sub(v, mul(p, w)) for v, w in zip(a[i], a[k])] + return [row[-1] for row in a] + +def residual_bound(x, y, b, beta, direction, bound): + # Neumann inverse norm <= 1/beta converts residual to forward error. + for i in range(len(x)): + residual = sub(sub(y[i], mul(b, y[(i-direction) % len(x)])), x[i]) + need(norm2(residual) <= (beta*bound)**2, 'residual error bound failed') + +def main(): + parser = argparse.ArgumentParser() + parser.add_argument('--out', type=Path, required=True) + args = parser.parse_args() + cases = [] + for beta in (F(1, 2), F(1, 16), F(1, 4096)): + for unit in (ONE, (F(-1), F(0)), (F(3, 5), F(4, 5))): + b = mul((1-beta, F(0)), unit) + for n in (1, 2, 3, 8, 17, 64): + x = [(F((i*7)%13-6, 8), F((i*11)%17-8, 16)) for i in range(n)] + B = max(abs(v[0])+abs(v[1]) for v in x) + for direction in (-1, 1): + bits = 64 + bound = 6*(B+1)*F(1, 1 << bits)/beta**3 + y = scan(x, b, beta, bits, direction) + residual_bound(x, y, b, beta, direction, bound) + if n <= 8: + exact = dense_solve(x, b, direction) + need(all(norm2(sub(v,w)) <= bound**2 for v,w in zip(y,exact)), 'dense comparison failed') + cases.append(dict(beta=str(beta), b=[str(v) for v in b], length=n, + direction=direction, bits=bits, forward_error_bound=str(bound), + dense_crosscheck=n<=8, residual_check=True)) + controls = [] + for name, x, b, beta, bits, direction in ( + ('unstable', [ONE], ONE, F(1,2), 64, 1), + ('underprecision', [ONE], (F(15,16),F(0)), F(1,16), 3, 1), + ('empty', [], ZERO, F(1,2), 64, 1), + ('bad_direction', [ONE], ZERO, F(1,2), 64, 0), + ): + try: scan(x,b,beta,bits,direction) + except ValueError as error: controls.append(dict(name=name,rejected=str(error))) + else: raise ValueError('negative control accepted: '+name) + x = [ONE, ZERO, ONE] + b, beta = (F(1,2),F(0)), F(1,2) + y = scan(x,b,beta,64,1) + y[0] = add(y[0],ONE) + try: residual_bound(x,y,b,beta,1,F(1,1<<40)) + except ValueError as error: controls.append(dict(name='corrupt_output',rejected=str(error))) + else: raise ValueError('corrupt output accepted') + args.out.write_text(json.dumps(dict(status='passed',cases=cases,controls=controls),indent=2)+'\n') + print(json.dumps(dict(status='passed',cases=len(cases),dense_cases=sum(c['dense_crosscheck'] for c in cases),controls=len(controls)))) + +if __name__ == '__main__': main() diff --git a/research/dirty-prefix-precision/vendor/pr628/weighted.py b/research/dirty-prefix-precision/vendor/pr628/weighted.py new file mode 100644 index 0000000000..da90ee6e83 --- /dev/null +++ b/research/dirty-prefix-precision/vendor/pr628/weighted.py @@ -0,0 +1,118 @@ +"""Exact certificates for varying unit phases on a cyclic neighbor shift.""" +from pathlib import Path +from fractions import Fraction as F +import argparse, importlib.util, json +P=Path(__file__).resolve().parent +s=importlib.util.spec_from_file_location('kernel',P/'check.py');k=importlib.util.module_from_spec(s);s.loader.exec_module(k) +c=k.c + +def scan(x,bs,beta,bits,direction): + c.need(len(x)>0 and len(bs)==len(x),'coefficient length') + c.need(direction in (-1,1),'direction') + c.need(0=(beta/2)**2,'closure denominator') + previous=c.rounded(c.div(z,den),bits);y=[c.ZERO]*len(x) + for i in order: + previous=c.rounded(c.add(x[i],c.mul(bs[i],previous)),bits);y[i]=previous + return y + +def apply(x,state,phases,direction,wrong_phase=False): + n=len(x);c.need(n>0 and len(phases)==n,'phase length') + c.need(all(c.norm2(t)==1 for t in phases),'unit phases') + y=[(v[0]*state['radix'],v[1]*state['radix'])for v in x] + for inside,b in state['factors']: + if inside: + inv=[(phases[(i+direction)%n][0],-phases[(i+direction)%n][1])for i in range(n)] + if wrong_phase:inv=[(v[0],-v[1])for v in phases] + bs=[c.mul(b,t)for t in inv] + y=scan(y,bs,state['beta'],state['bits'],-direction) + y=[c.rounded(c.mul(inv[i],y[(i+direction)%n]),state['bits'])for i in range(n)] + else: + y=scan(y,[c.mul(b,t)for t in phases],state['beta'],state['bits'],direction) + y=[c.rounded(c.neg(c.mul(b,v)),state['bits'])for v in y] + return y + +def matrix(q,phases,direction): + # Independently expand W^j entry-by-entry, including wrap-around phases. + n=len(phases);a=[[c.ZERO for _ in range(n)]for _ in range(n)] + for i in range(n): + weight=c.ONE + for j,v in enumerate(q): + col=(i-direction*j)%n;a[i][col]=c.add(a[i][col],c.mul(v,weight)) + weight=c.mul(weight,phases[col]) + return a + +def residual(x,y,a,bound): + for i,row in enumerate(a): + v=c.ZERO + for z,t in zip(row,y):v=c.add(v,c.mul(z,t)) + c.need(c.norm2(c.sub(v,x[i]))<=bound**2,'expanded weighted residual') + +def solve(a,x): + a=[row[:]+[v]for row,v in zip(a,x)];n=len(a) + for j in range(n): + i=next(i for i in range(j,n)if a[i][j]!=c.ZERO);a[i],a[j]=a[j],a[i] + z=a[j][j];a[j]=[c.div(v,z)for v in a[j]] + for i in range(n): + if i!=j: + z=a[i][j];a[i]=[c.sub(v,c.mul(z,t))for v,t in zip(a[i],a[j])] + return [row[-1]for row in a] + +def main(): + ap=argparse.ArgumentParser();ap.add_argument('--out',type=Path,required=True);a=ap.parse_args() + units=[c.ONE,(F(0),F(1)),(F(3,5),F(4,5)),(F(-5,13),F(12,13))] + cases=[];certs=[];generic=[];controls=[] + for beta in (F(1,2),F(1,16),F(1,4096)): + for n in (1,2,3,8,17,64): + bs=[c.mul((1-beta,F(0)),units[(i*i+3*i+1)%4])for i in range(n)] + # Alternate magnitudes as well as phases. + bs=[(z[0]/(1+i%2),z[1]/(1+i%2))for i,z in enumerate(bs)] + x=[(F(i%5-2,4),F(i%7-3,8))for i in range(n)] + B=max(abs(z[0])+abs(z[1])for z in x);bound=6*(B+1)*F(1,1<<64)/beta**3 + for d in (-1,1): + y=scan(x,bs,beta,64,d) + m=[[c.ZERO for _ in range(n)]for _ in range(n)] + for i in range(n):m[i][i]=c.ONE;m[i][(i-d)%n]=c.sub(m[i][(i-d)%n],bs[i]) + residual(x,y,m,beta*bound) + if n<=8:c.need(all(c.norm2(c.sub(v,t))<=bound**2 for v,t in zip(y,solve(m,x))),'generic dense') + generic.append(dict(beta=str(beta),length=n,direction=d,dense=n<=8)) + sigma,nu,phase_cert=k.irrational_gauge() + for row in json.loads((P/'FACTORS.json').read_text()): + if row['radix'] not in (4,9,16):continue + st=k.certify(row);A=F(st['certificate']['inverse_norm_bound']) + eps=F(st['certificate']['coefficient_error_bound'])+sum(j*(abs(v[0])+abs(v[1]))for j,v in enumerate(st['q']))*nu + c.need(A*eps<1 and A*A*eps/(1-A*eps)+st['budgets'][-1]<=F(1,1< Date: Tue, 13 Oct 2026 10:20:24 +0100 Subject: [PATCH 2/2] docs: record dirty-prefix audit submission --- research/dirty-prefix-precision/CLAIM.json | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/research/dirty-prefix-precision/CLAIM.json b/research/dirty-prefix-precision/CLAIM.json index 380f9f11eb..4237e0ba4b 100644 --- a/research/dirty-prefix-precision/CLAIM.json +++ b/research/dirty-prefix-precision/CLAIM.json @@ -4,7 +4,7 @@ "drawn_at_utc": "2026-10-13T08:36:24.867393+00:00", "track": "adversarial mathematical audit", "base": "f567884aeb8851b6e794af1184119a3bfbe1e631", - "status": "validated; ready for submission", + "status": "submitted", "branch": "research/pass06-audit-seed63", "claimed_at_utc": "2026-10-13T08:46:31.924453+00:00", "candidate_selection": "63 modulo 3 = 0", @@ -78,5 +78,7 @@ "package_mutation_controls": 6, "minimal_isolated_optimized_replay": "passed", "details": "VALIDATION.md" - } + }, + "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/877", + "submitted_at_utc": "2026-10-13T09:20:21.110701+00:00" }