Independent verification: exact synthetic-transform physical layouts (no new kappa)

[mightyOil]

Independent finite verification; no new κ and no source defect found. This adds exact evidence for the physical layout in the retained lem:synthetic-transform-cost: the prefix moves, chunk transposition, named-slot rounds and restoration implement L B F− and F+ B−1 L−1 on every tested configuration.

The two implementations share no transform or layout code. One executes the source's literal movement and physical butterfly/twiddle schedule; the other evaluates direct tensor characters and a closed-form integer address map. Both preserve exact integer numerators and the normalization A+ A− = I/M.

Completed coverage:

• 9,486 layouts, 480,267 coordinate-bit comparisons and 93,868 complete Cartesian chunk-group checks.
• 200,434 exhaustive addresses across 211 layouts, including all combinations of zero/nonzero remainder and one/multiple chunk groups.
• 4,340 ring-module basis columns, 1,244 additional scalar columns, 74 dense operator comparisons and 37 compositions; 16 further columns check the written ell=6,K=2,(long,short) illustration.
• 12,479,696 exact coefficient comparisons with the direct oracle and 11 sensitive mutation controls, each with a concrete differing coefficient.

A useful control is a three-axis layout of order three: testing only involutive layouts would miss replacing L with L−1. The package explicitly rejects that substitution, omitted layout steps, wrong twiddle ordering/signs, physical-vs-named bit confusion, negacyclic sign loss and wrong normalization.

Only research/synthetic-layout-independent/ is added. DOMAIN.json, exact EXPECTED.json, source SHA-256 pins, claim, scope, attribution and validation are included. Full ring-basis equality extends to arbitrary complex polynomial payloads for each tested configuration by R-linearity; it does not turn the bounded replay into an all-size theorem.

python3 -B -I -O research/synthetic-layout-independent/verify.py
python3 -B -I -O research/synthetic-layout-independent/test_verify.py

Expected: PASS_SYNTHETIC_LAYOUT: 9486 layouts; 200434 complete addresses; 4340 ring columns; 1244 extra scalar columns; 11 negative controls, then six passing test groups.

Validation passed on macOS arm64 / CPython 3.14.7, including optimized replay, isolated package/source relocation, source and receipt corruption checks, all four existing finite-identity tests, selected-record integrity and staged whitespace. An isolated Git index confirms the patch applies cleanly to the exact base. Two scoped AI reviews found no blocking issue; no independent human review is claimed. Full repository/supplier replay, remote CI and proof-assistant verification are not claimed.

Base: f567884aeb8851b6e794af1184119a3bfbe1e631. Upstream mathematical source: openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a. Assignment seed 80, independent-verification track. The full overlap snapshot covered 669 open PRs through #877; main was unchanged. Immediately before submission, the new #878 was also checked and concerns the distinct PR589 Boolean boundary interface. No pinned coordination issue existed, so the claim/result is local.

Credit OpenAI's source, Douglas Colkitt's dependency audit, and the retained Nussbaumer–Quandalle/Harvey–van der Hoeven lineage. @androolloyd's #673 (0b4f514c60d9bc7786415d7cbc24754f24d6dc3b) already checks transform arithmetic and dirty-echo integration; its inspected transform returns BF−/F+B−1 in axis-major coordinates. This contribution adds the complete prescribed physical L, round groups and restoration. No #673 code is imported, and its rejected supplier bill is not reopened.

The domain exercises positional definitions independently of asymptotic admissibility bands. Cutoffs, fixed-tape movement/cleanup, truncation/precision, layer realization, recursive hosting and analytic multiplication assumptions remain outside this result. The selected conditional κ remains 0.001661634653669567; no existing generator, source artifact, selected certificate or upstream/ file changes.

Prepared by Philipp (@mightyOil) with substantial OpenAI Codex assistance, including a separately delegated direct-character implementation and scoped AI review. New material is Apache-2.0; upstream retains CC0. No merge or selected-frontier edit is requested.