shea256 · Oct 10, 2026
diff --git a/.github/workflows/pr256-fourier-audit.yml b/.github/workflows/pr256-fourier-audit.yml
@@ -0,0 +1,37 @@
+name: PR256 complex proof audit
+
+on:
+  push:
+    paths:
+      - 'research/pr256-fourier-audit/**'
+      - '.github/workflows/pr256-fourier-audit.yml'
+  pull_request:
+    paths:
+      - 'research/pr256-fourier-audit/**'
+      - '.github/workflows/pr256-fourier-audit.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  finite-audit:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 10
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.13', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Replay finite program, source bindings, and retained evidence
+        run: python3 -B research/pr256-fourier-audit/verify.py
+      - name: Reject corrupted programs, statements, and evidence
+        run: python3 -B research/pr256-fourier-audit/test_audit.py
+      - name: Require unchanged tracked artifacts
+        run: git diff --exit-code
diff --git a/research/pr256-fourier-audit/MANIFEST.json b/research/pr256-fourier-audit/MANIFEST.json
@@ -0,0 +1,29 @@
+{
+  "description": "Immutable audit inputs and historical evidence; new audit code is reviewed through the PR diff.",
+  "files": {
+    "evidence/five_stage_lean_receipt.json": "195df11c9eb43f56a671bd5f90c96f810bac8ea954cde85dc44c100e04190604",
+    "evidence/five_stage_openai_origin.json": "ee6e89f2a099324c21da1e7385d2b5c82b64cdbfd225f91e076cd1c8320738c9",
+    "evidence/five_stage_regeneration_receipt.json": "a47ada8a3095ce1ffc58c43dac9d7dd564069caee184c6f0af4dd830af94a8e4",
+    "evidence/logs/axioms.log.gz": "c9811ae280c91401bf56660ca3d56fcda28ceee727fb7ab633925278aff0c25d",
+    "evidence/logs/build.log.gz": "d25a3cf89b770b7a0c7f0ad32adbf2a960d195eb557a3254103e07f39d6326c4",
+    "evidence/logs/compare_fourier.log.gz": "b379cde884336eb5139bbbd3ab46883e66bc9058390a9223b2c318d61a4c2598",
+    "evidence/logs/compare_wht.log.gz": "0fc07ebd9dfbaeb4ff419cb4e31a068a69a1075bf3af730d183e9a2e659dfbc1",
+    "evidence/logs/regenerate_certificate.log.gz": "fe780019ca79162cfcd68e284d73db0d79456715e9c55298e58449cf1954c7e2",
+    "evidence/logs/regenerate_fourier.log.gz": "6ed176c454f047733b22fed8c76c1f9966b630edd8f9e4c0291b3f0b36ad0a1c",
+    "evidence/logs/regenerate_program.log.gz": "3f2e7bae2862b423d3bdcf2b00f889763bde77a7aa79a1da14eef83f474adb8e",
+    "evidence/logs/toolchain.log.gz": "f0d1c3b28558d3555ba5e0322441cbcbcdc20b163d531a1ac14b5e0ace4bcf57",
+    "evidence/pr256_source_files.json": "3a8341035c7ef0ede3d8d43812aa4bb2ffc78287c8c3424d74b4b98cb8bb7103",
+    "evidence/proof_source_files.json": "6010ff79f3128533fddc076e223225d58b59d22761746b6119f7ccfd51071ac9",
+    "expected.json": "295a3f5c0e6843db7fec76b25609d93e0a7aeb60fb79c394e1851ebb6a8789f1",
+    "inputs/FOURIER_NOTICE": "9419dd03a2f52cf8bdfc4d18216ce29f263ab583b27f01ced8e420985bcb7c81",
+    "inputs/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "inputs/SOURCE.json": "00f5689fba8d1edc1741376c14a5ad6cdf5e0718dd24af9881949a6bc7cd2cf1",
+    "inputs/fourier-proof-source.tar.gz": "871f47a16d2571dbcebc2ef61914e85d96b7b44f6d59579da246e13294d5658b",
+    "inputs/gcert-program-233/NOTICE": "c1c95d22cfb0f455aca64b4bee4cab900829641b8f041280e7aa88034491c87d",
+    "inputs/gcert-program-233/SOURCE.json": "9ce463476af549c04de4705b1bf0c8f7f18d473dc0817883066ad5e2136d90b5",
+    "inputs/gcert-program-233/certificate/expected.json": "c0e3d245bf7e458ff86288eb47672491c9efee28181b570f1a3f17c8d915f766",
+    "inputs/gcert-program-233/certificates/gcert1-p11-pr233-flow.json.gz": "09ceba1c0ea17282da155b937c8463669b913443c20f45fe2a7e6e024cb2d33a",
+    "inputs/gcert-program-233/gx/gx.py": "6f3dc8bb639933227a8dae181ec4a542e85c7276be9c9e330f49518e7b630b95"
+  },
+  "schema": 1
+}
diff --git a/research/pr256-fourier-audit/NOTICE b/research/pr256-fourier-audit/NOTICE
@@ -0,0 +1,28 @@
+PR256 program / Fourier-proof reproduction audit
+Copyright 2026 Ryan Shea. New audit code and documentation: Apache-2.0.
+Prepared with substantial OpenAI Codex assistance.
+
+This work adapts the checks and retains the original reproduction evidence from
+shea256/fourier-transform-below-nlogn at a101b363b38e6bf7d309042c94ba5fa18a424fad.
+It contributes source/program/proof consistency checks and a separate reproduction,
+not a new exponent, network, formal framework, or proof of integer multiplication.
+
+OpenAI supplies the original exact arithmetic model and Fourier reduction. Doug
+Colkitt supplies early parameter/network work and the community repository; Rohan
+Arun and Swapnil Jain supply earlier community improvements and the research lineage.
+Jacob Sussman supplies the five-stage geometry, frame formalization, certificate
+format/checker, and formal Fourier framework. Chafik Boukhalfa supplies the PR200/233
+pairing, PR256 explicit program, and the stronger instantiation of that framework.
+The circuit retains icekylinx's source-assisted construction, eumemic's v4 modules,
+Avi Eisenberg's reconstruction/pairing, an664's sharing work, and the earlier
+contributors named in the preserved source notices. The incoming package credits
+SovereignSteak and hcg890 for five-stage composition and ledger work.
+
+inputs/gcert-program-233/NOTICE and inputs/FOURIER_NOTICE are retained byte for byte;
+inputs/LICENSE accompanies them. The complete proof archive retains its licenses,
+notices and source-specific OpenAI attribution. Do not reinterpret this audit as
+authorship of the imported constructions or proofs.
+
+Metadata clarification: PR209 was contributed by Jacob Sussman and PR237 by Rohan
+Arun. The PR256 notice groups those numbers with Rohan Arun; the original notice
+is preserved unchanged. This note supplies the checked attribution alongside it.
diff --git a/research/pr256-fourier-audit/README.md b/research/pr256-fourier-audit/README.md
@@ -0,0 +1,123 @@
+# PR256 complex program: reproduction and proof consistency audit
+
+This adds a separate reproduction of Chafik Boukhalfa's PR256 and its instantiation
+of Jacob Sussman's Lean framework. **No new multiplication kappa, network, or
+Fourier exponent is claimed.** The contribution is a reproducible audit linking
+the two source packages, exact arithmetic checks, and retained build evidence.
+
+## Pinned sources
+
+| Source | Revision | Role |
+| --- | --- | --- |
+| [CrocSwap PR256](https://github.com/CrocSwap/integer-mult-bounds/pull/256) | `db3f75cdc5f03f3131d48fb220f6bf1958404ff3` | Explicit complex program and source regeneration |
+| [Fourier proof PR2](https://github.com/jacobalansussman/wht-power-saving-lean/pull/2) | `fd19e46b728faa414cbfbf91b01a6a0b7903375d` | Sussman's framework with Boukhalfa's PR233 instantiation |
+| [OpenAI formal sources](https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean) | `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb` | Original model, Fourier reduction, and challenge |
+| [Downstream reproduction](https://github.com/shea256/fourier-transform-below-nlogn/tree/a101b363b38e6bf7d309042c94ba5fa18a424fad) | `a101b363b38e6bf7d309042c94ba5fa18a424fad` | Ryan Shea's checks, source audit, and build receipts |
+
+The proof extends Sussman revision `f010392c923279e3dd59ef3aa5fedad23affc5fd`.
+Lean is 4.34.1; Mathlib is `d13f23b723b8a846827a245b89c10fc7d3f11612`.
+The complete 456-file proof source archive is included, without compiled objects.
+Only the PR256 inputs needed for the finite audit are copied here; its complete
+producer is available at the pinned revision above. Original notices are retained.
+
+## Offline finite audit
+
+From the repository root, using Python 3.11 or later and no third-party packages:
+
+```sh
+python3 -B research/pr256-fourier-audit/verify.py
+python3 -B research/pr256-fourier-audit/test_audit.py
+```
+
+The first command:
+
+1. Checks source pins and runs Sussman's unmodified `gx.check1` on the explicit
+   PR233 program: 12,052 registers, 18,248 frames, 52,300 gates, and 69,683
+   invocation blocks. Exact scalar replay covers all source columns; restoration
+   of arbitrary dirty scratch is supplied by the Lean invocation theorem.
+2. Reconstructs the five-stage histogram and checks `m=110`, `W=14692`, rank
+   1613040, deficit 3080, 358975 children and maximum child width 46.
+3. Independently bounds the normalized moment using exact rational logarithm and
+   exponential enclosures at `a=7547361/10^10`. It checks the Lean engine's
+   finite-fill correction at `s=40`, the positive Fourier gap
+   `a - delta = 1/10^10`, and failure at `7547365/10^10` for this histogram.
+4. Requires the PR256 and Lean programs to agree in every mathematical field
+   (only the descriptive `derived_from` string differs), then checks the Lean
+   histogram and exponent literals. It compares the Fourier challenge with the
+   original after accounting for comments, seven renamed declarations and the
+   stronger exponent.
+5. Checks all 89 imported OpenAI Lean modules, the original challenge and the
+   toolchain against the 91-file origin ledger. These were fetched separately
+   from the immutable OpenAI revision during the recorded source audit.
+6. Binds the historical reproduction receipts to their compressed logs and all
+   456 archived source hashes. Checking a recorded receipt does not rerun Lean.
+
+Optional full-checkout comparisons require no network:
+
+```sh
+python3 -B research/pr256-fourier-audit/verify.py \
+  --program-source /path/to/pr256-checkout \
+  --proof-source /path/to/pinned-fourier-checkout \
+  --openai-source /path/to/pinned-openai-checkout
+```
+
+They compare the 20 program-package files, 456 proof-source files and 91 original
+OpenAI files with the pinned ledgers, irrespective of the checkout's branch name. The default audit
+does not rerun the producer or a full Clifford/DFT computation.
+
+## Reproducing the Lean build
+
+Install `leanprover/lean4:v4.34.1`, then run:
+
+```sh
+python3 -B research/pr256-fourier-audit/reproduce_lean.py \
+  --work-dir /tmp/pr256-lean-rebuild --lake lake --threads 2 \
+  --receipt /tmp/pr256-lean-rebuild-receipt.json
+```
+
+This extracts the complete pinned source (or checks existing source bytes),
+downloads the pinned Mathlib cache, regenerates the 41 certificate and 14
+Fourier-chain files, builds the proof, audits axioms, and runs upstream
+`tools/Compare.lean` for WHT, DFT and convolution. The two large scalar reductions
+are staged sequentially to reduce peak memory. Allow several GB of storage and
+substantial memory; the recorded reproduction used a 32 GiB macOS arm64 machine.
+
+The original fresh project build passed in 2625.5 seconds, with Mathlib's compiled
+cache. The WHT and Fourier comparisons passed in 33.7 and 42.6 seconds. The three
+audited final theorems used only `propext`, `Classical.choice`, and `Quot.sound`.
+The original combined build temporarily paused one large scalar reduction until
+the other completed; it then resumed successfully, without changing source.
+The script now stages those targets sequentially. The receipt retains the actual
+original commands and is not relabeled as a new run.
+
+The full PR256 producer regeneration also passed (188.6 seconds), using Python
+3.11, NumPy 2.3.5 and SciPy 1.17.0. To rerun that separately:
+
+```sh
+git clone https://github.com/chafreaky/integer-mult-bounds /tmp/pr256-source
+git -C /tmp/pr256-source checkout --detach db3f75cdc5f03f3131d48fb220f6bf1958404ff3
+OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 \
+  uv run --python 3.11 --with numpy==2.3.5 --with scipy==1.17.0 \
+  python -B /tmp/pr256-source/research/gcert-program-233/verify.py
+```
+
+## What this establishes for multiplication
+
+| Evidence | Established scope | Multiplication obligations remaining |
+| --- | --- | --- |
+| Exact program and histogram checks | The pinned finite complex program and its five-stage accounting agree with the formal input | No bit-side construction or outer multiplication assembly is checked here |
+| Lean project and statement comparisons | The upstream WHT and all-length exact DFT/convolution theorems in OpenAI's family-130 RAM model | Finite precision/recovery, bit costs, fixed-tape implementation and the inherited multiplication-specific all-size interfaces are not proved |
+| OpenAI source comparison | Imported model/reduction files match the identified OpenAI revision | Human assessment of formal statements and assumptions remains separate |
+| Receipt and log hashes | The recorded successful runs are bound to these sources | Hashes alone do not independently attest that an execution occurred |
+
+The Fourier theorem has `delta=7547360/10^10`. It is an exact-complex-arithmetic
+result with supplied roots and unrestricted coefficients, not a multiplication
+theorem or practical FFT benchmark. The historical reproduction ran neither the
+official Linux comparator nor a second kernel and is not independent human
+mathematical review. Boukhalfa later reported official-comparator success in
+[`837dc313`](https://github.com/chafreaky/wht-power-saving-lean/commit/837dc3130adc79f7b7ab4ddd56c15576bbd766b0);
+that documentation-only update is distinct from this pinned local reproduction.
+
+See [NOTICE](NOTICE) for attribution and [VALIDATION.md](VALIDATION.md) for the
+checks of this contribution. The root selected result and all existing witnesses
+remain unchanged.
diff --git a/research/pr256-fourier-audit/VALIDATION.md b/research/pr256-fourier-audit/VALIDATION.md
@@ -0,0 +1,42 @@
+# Validation of this audit contribution
+
+Prepared against CrocSwap main `3b6b66891c0ac888521cf591fe306c6286601d4f`.
+This is a separate AI-assisted reproduction, not independent human mathematical
+review. The bound and proof remain the credited upstream authors' results.
+
+## Audit package
+
+- Offline finite verifier: passed on Python 3.11, 3.13 and 3.14.
+- Eleven audit controls (two positive and nine negative): passed; altered stock/child counts, an unrecorded
+  exponent, a weakened quantifier, an extra axiom, a missing build, a wrong source
+  revision and a changed build log are rejected. Optimized Python is refused.
+- Full-checkout comparison: all 20 pinned PR256 package files and all 456 proof
+  source files matched the separate checked-out revisions.
+- OpenAI origin: all 91 ledger files were downloaded again from the immutable
+  OpenAI revision and matched, including all 89 imported OAI modules.
+- Ported Lean reproduction wrapper: passed against the existing pinned build,
+  with `--cache-ready`. Both source generators reproduced their output; staged
+  scalar targets, combined build, axiom check, WHT comparison and Fourier
+  comparison all exited 0. This wrapper test reused the project build and is not
+  a second fresh build. The retained historical receipt records the earlier
+  successful fresh project compilation.
+- Source syntax and whitespace checks passed.
+
+## Existing repository checks
+
+`make entrance-bank-verify` passed: selected-record validation, the complete
+offline complex/bit construction, all 47 outer constraints, seven margins,
+21 assembly controls, and the bank scheduling supplement.
+
+The full `make verify` regression run was invoked. At submission, its community,
+producer, partial-gauge, three-stage-cover and paired-cube groups have passed;
+the remaining groups are still pending. This is not a claim that the full suite
+has passed. No existing source, certificate, selected-result pointer or root
+documentation is changed by the contribution.
+
+## CI scope
+
+The added workflow repeats the offline finite audit and audit controls on
+Ubuntu with Python 3.11, 3.13 and 3.14. It does not claim to run Lean or a second
+kernel; full proof reproduction is the separate documented command. Configuring
+the workflow does not itself constitute a successful CI run.
diff --git a/research/pr256-fourier-audit/evidence/five_stage_lean_receipt.json b/research/pr256-fourier-audit/evidence/five_stage_lean_receipt.json
@@ -0,0 +1,135 @@
+{
+  "axioms": {
+    "OAI.PowerSaving.RAM.hillsY_program": [
+      "Classical.choice",
+      "Quot.sound",
+      "propext"
+    ],
+    "OAI.PowerSaving.convolution_mainY": [
+      "Classical.choice",
+      "Quot.sound",
+      "propext"
+    ],
+    "OAI.PowerSaving.transform_mainY": [
+      "Classical.choice",
+      "Quot.sound",
+      "propext"
+    ]
+  },
+  "commands": [
+    {
+      "command": [
+        "/tmp/fourier-lean-toolchain/lean-4.34.1-darwin_aarch64/bin/lake",
+        "--version"
+      ],
+      "exit_code": 0,
+      "label": "toolchain",
+      "log_sha256": "36526a338068c343bedff5daa38fec328b4f722df1e8f0b5d026d4f0f908f3a3",
+      "seconds": 0.246
+    },
+    {
+      "command": [
+        "/opt/homebrew/opt/python@3.14/bin/python3.14",
+        "-B",
+        "tools/gx/regen_check.py",
+        "tools/certificate/gcert1-p11-pr233-flow.json.gz",
+        "P233",
+        "B2Gp233"
+      ],
+      "exit_code": 0,
+      "label": "regenerate_certificate",
+      "log_sha256": "76952939fe619d8f344c0f88579f25929a183bd57b5e58fe974fec990f644d77",
+      "seconds": 28.829
+    },
+    {
+      "command": [
+        "/opt/homebrew/opt/python@3.14/bin/python3.14",
+        "-B",
+        "tools/fourier/regen_check233.py"
+      ],
+      "exit_code": 0,
+      "label": "regenerate_fourier",
+      "log_sha256": "eee7ec63261d1ab58a527ada9ee3d47adfabb06385b49925fb831fbb74c0cb4e",
+      "seconds": 0.107
+    },
+    {
+      "command": [
+        "/tmp/fourier-lean-toolchain/lean-4.34.1-darwin_aarch64/bin/lake",
+        "build",
+        "Work.GCert.Data.ChallengeB2Gp233x",
+        "Work.GCert.Data.SolutionB2Gp233x",
+        "Work.Fourier233.UniformFourierChallenge",
+        "Work.Fourier233.Main",
+        "Work.Fourier233.Axioms"
+      ],
+      "exit_code": 0,
+      "label": "build",
+      "log_sha256": "3e3f4b0f0f9a8c95ede2da69d08615bbc5ff763c5a06cc460b9f3f292153d783",
+      "seconds": 2625.459
+    },
+    {
+      "command": [
+        "/tmp/fourier-lean-toolchain/lean-4.34.1-darwin_aarch64/bin/lake",
+        "env",
+        "lean",
+        "Work/Fourier233/Axioms.lean"
+      ],
+      "exit_code": 0,
+      "label": "axioms",
+      "log_sha256": "fd6ef1a75394038e0035b143c1135f4a4d0a6129982ac741bc8c80ebd3c145bd",
+      "seconds": 6.3
+    },
+    {
+      "command": [
+        "/tmp/fourier-lean-toolchain/lean-4.34.1-darwin_aarch64/bin/lake",
+        "env",
+        "lean",
+        "--run",
+        "tools/Compare.lean",
+        "Work.GCert.Data.ChallengeB2Gp233x",
+        "Work.GCert.Data.SolutionB2Gp233x",
+        "OAI.PowerSaving.WHT.wht_main_block_B2Gp233x"
+      ],
+      "exit_code": 0,
+      "label": "compare_wht",
+      "log_sha256": "a1300346aee4d7a36791f6d5772969e772aa86d7d36ba9777dc521ec94ecc4f8",
+      "seconds": 33.7
+    },
+    {
+      "command": [
+        "/tmp/fourier-lean-toolchain/lean-4.34.1-darwin_aarch64/bin/lake",
+        "env",
+        "lean",
+        "--run",
+        "tools/Compare.lean",
+        "Work.Fourier233.UniformFourierChallenge",
+        "Work.Fourier233.Main",
+        "OAI.PowerSaving.transform_mainY",
+        "OAI.PowerSaving.convolution_mainY"
+      ],
+      "exit_code": 0,
+      "label": "compare_fourier",
+      "log_sha256": "48ff3e708c9c38926a39ffd94f761d33245a6abe9aa5749e7233e8f14657fa97",
+      "seconds": 42.588
+    }
+  ],
+  "finished_utc": "2026-10-10T03:12:13.637233+00:00",
+  "fresh_project_build": true,
+  "independent_human_review": false,
+  "independent_kernel_run": false,
+  "machine": "arm64",
+  "mathlib_commit": "d13f23b723b8a846827a245b89c10fc7d3f11612",
+  "mathlib_from_cache": true,
+  "numerical_dft_executed": false,
+  "official_comparator_run": false,
+  "platform": "macOS-26.6.2-arm64-arm-64bit-Mach-O",
+  "schema": 1,
+  "source_archive_sha256": "871f47a16d2571dbcebc2ef61914e85d96b7b44f6d59579da246e13294d5658b",
+  "source_commit": "fd19e46b728faa414cbfbf91b01a6a0b7903375d",
+  "source_files_checked": 456,
+  "started_utc": "2026-10-10T02:26:36.346772+00:00",
+  "statement_comparison": "upstream tools/Compare.lean; WHT, all-length DFT and convolution passed",
+  "success": true,
+  "threads": 2,
+  "toolchain": "Lake version 5.0.0-src+5045d00 (Lean version 4.34.1)"
+}