Independent verification: signed packed Gaussian convolution (no kappa claim)

[Ambassators]

Result

Independent verification / scoped arithmetic proof; no new kappa. This audits packed_lower from Chafik Boukhalfa / chafreaky's PR #489 at 8922e3b3f0fd7772b0fb42028eb2532984c3ab42. An independently written integer-dot-product/divmod oracle agrees on the complete declared finite domain. No source bug was found in that domain.

Base: 66849e1. Contribution: b5ad616. Only research/gaussian-packed-convolution-audit/ is added. The selected #626/#543 record, its conditional kappa 51110230636731/31250000000000000, existing generators and upstream/ are unchanged.

Reproduce

Python 3.9+, standard library only, offline:

python3 -B research/gaussian-packed-convolution-audit/verify.py

Expected: PASS_INDEPENDENT_SIGNED_PACKED_CONVOLUTION.

• 22,620 input pairs, 45,240 packed calls, 183,212 output coordinates, 180,960 integer products.
• Exhaustive: every pair of length-1 through length-4 vectors over the documented ternary numerator alphabets at precisions 0, 1 and 3 (22,140 pairs), both triangular orientations.
• Stress: 480 prescribed pairs, lengths through 32, precisions through 64 fractional bits, zero inputs, cancellation, carry boundaries and seed-derived mixed values. Numerators reach 128 bits; packed operands reach 8,219 bits.
• Eight executed arithmetic mutants rejected with exact expected/mutant outputs, plus three input-admission controls. Exceptions do not count as arithmetic detection.

The oracle uses neither packing nor source rounded(), Gaussian factors, bits=None, or the original Fraction-matrix checker. Its upper-triangle oracle directly sums upper entries. The pinned primitive and its reversal wrapper are tested, not the full seven-step apply function. EXPECTED.json is recomputed and compared in full.

PROOF.md separately gives an elementary all-length radix argument, including the zero-input case and the necessity of rounding only completed coordinates. At one fractional bit, H=(1,1), X=(1,-2) has degree-one output zero; separately rounding positive/negative products incorrectly gives -1/2. This is a rejected implementation mutation, not a defect in #489. The written proof is not machine-formalized, and finite tests do not establish its universal quantifiers.

Validation

Passed on Darwin arm64 / CPython 3.9.6:

python3 -I -B research/gaussian-packed-convolution-audit/verify.py
python3 -B research/gaussian-packed-convolution-audit/test_package.py
make selected-record-check
git diff --cached --check

Fresh and original receipts are byte-identical (SHA256 8ca661b0127a3a469186f3c2adcde105781d02a958ee44bfd6cc8a0edde1a0c7). Package controls cover unrelated-directory replay with spaces, changed source, changed/missing evidence, existing-output preservation and refusal of -O because inherited admission uses assertions. Vendor files are byte-bound to git show at the source head and checked by SHA256/Git blob hashes before and after replay. Documentation links and the new-files-only boundary pass.

No full make verify, selected cold replay, Linux execution, remote CI success or Lean proof is claimed. Gaussian approximation/Schur transfer, weighted tensor evaluation, physical tape/routing and inherited all-size/analytic multiplication assumptions remain outside this audit. No new exponent or all-size multiplication theorem is asserted.

Coordination and attribution

OS seed 79, independent-verification track. The local claim records three candidates and one evidence-driven pivot: neighboring concurrent symbolic Gaussian claims already covered the initial q-polynomial obligation, so this contribution instead audits packing. Their code/results are not imported. Final overlap refresh through #688 found no duplicate; #684's Gaussian domain/budget audit explicitly excludes packing. No sprint coordination issue was found, so the claim is local.

Credit chafreaky's #489 implementation/checks, the RaD/hipotures phase-cell lineage, #481 boundary work, and the classical sources credited by #489. Prepared for Ryan Pinto (@Ambassators) with substantial OpenAI ChatGPT / GPT-6 Astra Pro assistance in research, reasoning, implementation, verification and writing. Apache-2.0 with original source license retained. This is independent implementation evidence, not independent human peer review.

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