Research audit: phase-weighted Fourier layout criterion (no new kappa)

[bbMisk]

Reusing a phase-weighted Fourier routine for both forward passes and conjugating it for the final pass preserves the retained convolution identity only when every phase is 1 and the read map is a self-adjoint group automorphism. Even the constant phase i produces i times the desired answer. With separate forward and inverse weights, affine layouts work under an exact character-phase condition.

This mathematical audit / scoped interface theorem extends Ross Richardson's PR940 from unweighted permutations to three monomial maps. It proves the full necessary-and-sufficient criterion, including agreement of the two forward address maps and the weighted adjoint for unequal cyclic factors. The original source's coordinate multipliers remain valid; no source defect or new kappa is claimed.

Only research/weighted-fourier-layout/ is added: written proof, independently implemented literal cyclotomic oracle, verifier, deterministic receipt, source/task/replay metadata and focused tests. No pending code is imported, and the selected frontier and frozen sources are unchanged.

Evidence and reproduction

• Complete declared census: 357,184 reduced candidates (744 admitted) and 22,688 shared-routine cases (nine admitted, all unweighted).
• Separate literal Fourier/convolution calculation: 10,662 basis pairs, five dense pairs, 1,136 mismatched-address witnesses and nine exact semantic counterexamples.
• Eleven focused test groups, including all 32,768 unreduced two-point cases, arbitrary reciprocal phase controls, correct affine and non-involutive source examples, false classifiers, source drift and output preservation.
• Registered Python 3.12.13 and isolated optimized Python 3.14.2 checks passed on macOS arm64. An offline source export in a path with spaces passes; corrupted source and expected certificate are rejected. Selected-record integrity and clean patch application to refreshed main pass. Hosted CI is separate.

python3.12 scripts/research_package.py --package research/weighted-fourier-layout

Expected: PASS_WEIGHTED_FOURIER_LAYOUT, eleven passing tests, and PASS_REGISTERED_RESEARCH_PACKAGE research/weighted-fourier-layout outcome=completed.

The universal result is the written proof; finite replay is separate evidence. No proof-assistant or independent human verification, rounded Gaussian implementation, physical routing cost, supplier adoption, cold supplier replay or multiplication theorem is claimed. All inherited implementation and analytic obligations remain. Open next questions are in the README/TASK metadata.

Pins, overlap and credit

Base: 410026dc5513b87db92ae98571d1d5984c69a983. Refreshed main e6f9716621f4e01fc190617934cdfaf072bd976f has identical audited inputs and selected kappa. Source: openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a, retained 07-resampling.tex, eq:source-permutation-cancel. Predecessor PR940: fccac36fa9bcc063c4fda733299a22349ce10e56; its proof was inspected and credited. Exact Git blobs and SHA256 pins are in SOURCES.json.

OS seed 71 assigned adversarial audit; after the overlap review, the seed sets phase and dense-input fixtures for this weighted extension. Claim #1463. The final full snapshot covered 829 open PRs and all paginated discussion records without retrieval warnings; the only related match beyond this package/claim was PR940. This adds weighted/affine scope rather than another unweighted permutation replay.

Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in mathematics, implementation, verification and prose. Credit Ross Richardson, OpenAI's original manuscript, and Douglas Colkitt / CrocSwap's retained construction and normalization audit. Apache-2.0; no literature-priority claim for elementary character algebra.

[Mister-Meeseeks]

Batch review at f24bb18c20f7bdb3cbc27a1b6a010babf427d6cc.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact character-coefficient proof classifies phase-weighted Fourier convolution layouts. Independent complete finite census and literal convolution controls pass; phases cannot repair the shared-routine contract, while separately paid affine phases remain possible.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.