Independent verification: complete signed Gauss normal-form census (no new kappa)

[yssite]

The general binary-residual Gauss normal form currently has four embedded-basis examples in tests/test_endpoint_gauge.py. This adds an independent, exact census of every symmetric binary polarization matrix and every normalized Z4 quadratic refinement through dimension four, including signed nonalternating forms and the singular boundary. Independent finite verification; no new kappa and no source defect found.

Only research/signed-gauss-census/ is added (five files). No predecessor code is imported or modified.

Research question and result

For polarization matrix B, does the stated complete-square kernel use the correct inverse, endpoint signs, bilinear term, and global fourth-root normalization? What fails if B is singular?

• 16,933 refinements, including 7,411 nonsingular forms and all 1,849,609 kernel entries.
• 9,522 singular forms satisfy the exact radical-annihilator support and squared-magnitude formula. They are not admitted as full-rank flat Walsh children.
• A Gaussian-integer Walsh butterfly agrees with a separate direct four-phase counting sum on every case: 4,227,601 direct sum terms, with every polarization pair also checked.
• Separate truth-table enumeration through dimension three and all-matrix enumeration/XOR elimination through dimension four confirm completeness and ranks.
• All 34 signed realizations of the original four fixtures agree with the coefficient representation.
• OS seed d207afb6a1a4565a4812f3e58a38ad45 supplies 24 additional cases in dimensions 5, 6, and 9; these are explicitly samples, not exhaustive coverage.

This supplies reviewable evidence for phase conventions that can survive magnitude-only checks while producing a wrong kernel. All arithmetic uses integer pairs, without floating point or external dependencies.

Reproduction and controls

Python 3.10+, standard library only, from the repository root:

python3 -I -B research/signed-gauss-census/verify.py
python3 -I -O -B research/signed-gauss-census/verify.py
python3 -I -B research/signed-gauss-census/test_verify.py -v
python3 -I -O -B research/signed-gauss-census/test_verify.py -v

Expected checker output:

PASS_SIGNED_GAUSS_CENSUS refinements=16933 nonsingular_refinements=7411 singular_refinements=9522 kernel_entries=1849609 sampled=24 controls=11 pins=2

The test module has six tests. Eleven mathematical controls reject missing/wrong global factors, B substituted for its inverse, wrong endpoint signs, an omitted bilinear phase, discarded signed refinement, missing Walsh stages, singular full-rank admission, a nonzero constant, wrong polarization, and a nonsymmetric matrix. Additional tests cover malformed presentations, source drift, type-sensitive receipts, and preserving output after failure.

Actual validation

On Linux x86_64, CPython 3.12.14:

• Both new commands and all six tests pass normally and under -O.
• Both original endpoint tests pass in both modes. At the original research base, all five selected-result tests also passed in both modes; that historical run is not a validation of the newer selected construction.
• The final committed package and original endpoint tests pass in an isolated sparse Git checkout. Each package test suite also runs a fresh optimized minimal replay in a path containing spaces.
• Normal/optimized receipt bytes match; all five source/context Git-object pins match at the original base. make selected-record-check passed on the original contribution checkout; git diff --check passes.
• On the latest upstream checkpoint below, an explicit-base three-way merge is conflict-free. The package and original endpoint tests exported from the merged tree pass all six normal/optimized replay commands; the two live mathematical source pins remain unchanged.

Full make verify, selected cold replay, Lean, and hosted CI were not run for this isolated mathematical-test package. There is no performance improvement claim.

Baseline, overlap, and remaining obligations

Exact research base: 2921874911956718fcaaaf9396fe8267b3cfa9f0. Rechecked on 2026-10-14 against main ef5f39403610d231a0c27e8a6b4f08c10209bcd6: both mathematical subjects are unchanged and the contribution merges cleanly. The selected conditional result has advanced to PR1094 on PR1087, 836330098538459/500000000000000000 = 0.001672660197076918, reviewed in docs/research/community-batch1099-review.md, with cold replay make frontier1094-verify. That cold replay was not run. The package retains PR942 as explicitly historical context at the original research base; these historical context pins are not moving-main checks.

Credit icekylinx's #24 normal form, Douglas Colkitt's maintained tests, and the original OpenAI manuscript. #831 (Maximooch, head 66c3bc5c2eb5a2c066b1119310b29d26350e1dde) already checks the 452 alternating refinements in dimensions 2 and 4; that subset is deliberately corroborated. The new coverage adds 6,958 nonalternating refinements, the zero case, and 9,522 singular boundaries, with separate completeness checks. #911 (ToikY) concerns embedded orthonormal residuals and signed Hamming phases. No priority is claimed for predecessor identities or overlapping cases; no pending code is imported. The 2026-10-14 overlap refresh screened 385 PRs created since the preceding check, all-state targeted searches, and 72 Discussion records with their retrieved comments. PR831 remains at the pinned head. Nearby #1294 concerns universal dirty-interval block identities and #1477 concerns phase-weighted Fourier/convolution layouts; their bodies and README patches were inspected and have different scope. No equivalent complete signed/singular census was identified; this bounded search does not guarantee novelty.

The arbitrary-dimension normal-form proof remains the credited source theorem. This census is not a formal proof, supplier admission, physical cost audit, or multiplication theorem. All-size/compiler, realization, hosting, precision/tape, and analytic obligations remain separate.

Open next questions

A scoped follow-up is proof-assistant verification of the arbitrary-dimension radical support criterion and Gauss normal form. Start with translation along radical vectors and the exact support condition, then prove the squared-magnitude formula and nonsingular phase normalization. This addresses the finite census's dimension limit. Source-to-statement correspondence and formal algebra library support remain blockers; no formalization or new theorem is claimed here. The package README records this question and its limits.

Prepared for yssite with substantial OpenAI Codex assistance in research, implementation, verification, and writing. No independent human review is claimed. New material is Apache-2.0 under the repository license; predecessor attribution and historical sources are preserved.

[Mister-Meeseeks]

Batch review at 4a6d01a0981c4d272dc5ff9754f68cc554ca1cb5.

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

Reviewed scope: Exact Gaussian-integer census checks every normalized signed quadratic refinement through dimension four, including singular radical boundaries. Larger dimensions are expressly sampled; this verifies the stated normalization on its finite domain, not the full supplier.

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.