Research audit: exact scope of retained Fourier layout cancellation (no new kappa)

[RossRichardson]

This mathematical audit / scoped negative result establishes the exact extension criterion for the retained eq:source-permutation-cancel: reusing the forward read layout on the conjugate Fourier pass works exactly for self-adjoint group automorphisms, under the standard product-group character pairing. A general automorphism requires its adjoint on that pass. The source's stated coordinate multipliers satisfy the criterion. No source defect or new κ is claimed.

The package adds a written all-group proof, a complete finite census and explicit failure witnesses. For example, four-point bit reversal fixes zero and is involutive, but for u=δ₀, v=δ₁ the resulting coefficient at 1 is (2−2i)/64 instead of 4/64. A separate inverse-layout contract F+ P⁻¹ remains valid for arbitrary permutations; this audit concerns when it can be replaced by Pρ F+.

Completed evidence:

• 127,616 permutations across all cyclic groups of orders 2–8 and the presentations (2,2), (2,3), (2,4), (2,2,2). Literal character-matrix entries and reconstruction from group generators agree on every classification.
• Exact cyclotomic convolution replay on 2,960 accepted basis pairs, 205 dense adjoint pairs, and 186 further positive-control basis pairs. All 1,425 cyclic unit multipliers through modulus 68 pass the interface test.
• Eight exact convolution counterexamples, a separate nonunit admission rejection, four positive controls, and seven supplementary failure-sensitivity tests including five deliberately faulty implementations.

Reproduce using only Python's standard library:

python3 -B -I -O research/source-permutation-audit/verify.py
python3 -B -I -O research/source-permutation-audit/test_verify.py

The first prints PASS_SOURCE_PERMUTATION: 127616 permutations; 2960 census basis pairs; 205 dense adjoint pairs; 8 exact counterexamples; 1 admission rejection; 4 positive controls; the second runs seven passing tests.

Verified on macOS arm64 with CPython 3.12.12 and 3.14.6. The deterministic receipt agrees under optimized isolated execution. Relocation into a minimal source closure with spaces in its path succeeds; source and expected-receipt corruption are rejected. The four existing finite-identity tests, selected-record integrity at the base, staged whitespace and patch application to refreshed main pass. Full repository verification, supplier cold replay, remote CI and proof-assistant verification are not claimed.

Exact base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. Mathematical source: openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a, upstream/build/sections/07-resampling.tex. Required source pins and historical context pins are distinguished in SOURCE.json. Main advanced during the audit to 47481c89fd30a43f8f882eb4b391e89c29273fd0, selecting #890 at κ = 0.001661957774647029; the audited source and manifest are unchanged. Only research/source-permutation-audit/ is added. All inherited numerical, compiler, realization, hosting, tape and analytic obligations remain.

OS-random assignment seed 68 selected adversarial audit. The local claim records three candidate tasks, the fixed stopping rule, completed coverage and result link. No pinned coordination issue existed. The full overlap refresh covered 708 open PRs through #937; an immediate pre-submission check also inspected #938 and #939, which concern dirty fanout stars and E8 star-copy ratios. Credit OpenAI's source and Douglas Colkitt's prior normalization audit. Chafik Boukhalfa's #479 concerns streamed chirp phases; Philipp / mightyOil's #879 checks physical schedules with matching inverse layouts. Their inspected heads are pinned; no pending code is imported, and this contribution does not duplicate those checks.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing. New material is Apache-2.0; source licenses are unchanged. No independent human review or novelty priority for elementary Fourier duality is claimed. The finite replay is separate from the written universal argument and is not a formal proof of multiplication.

[Mister-Meeseeks]

100-PR batch review at fccac36fa9bcc063c4fda733299a22349ce10e56.

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

Reviewed scope: Fourier source-permutation cancellation iff the output uses the adjoint automorphism; same layout requires self-adjointness.

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.