Research audit: signed block extraction and coefficient-alias controls

[thakrarsagar]

The retained signed-ring-product lemma extracts coefficients from fixed-width two's-complement blocks with a one-bit carry. The existing signed-packing test repeatedly divides a signed Python integer, so it does not exercise that interface. This research audit adds an independent block decoder, direct-convolution oracle, source pins and exact receipt. The stated strict-bound cases pass; no source defect or new kappa is claimed.

The useful negative control is a coefficient alias at the forbidden positive half-radix boundary. With B=32 and a=c=[4,0], decoding returns [-16,1,0,0] instead of [16,0,0,0]. Integer recomposition and terminal-carry checks both pass. This shows why coefficient comparison or strict separation is necessary. It violates the retained lemma's radix choice and is not a counterexample to that lemma.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Assignment seed: 60, adversarial mathematical audit. Source: openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a, lem:signed-ring-product; exact local hashes are in SOURCE.json. Only research/signed-extraction-audit/ is added.

Completed coverage:

• 57,160 exhaustive strict-centered coefficient words and every residual identity.
• 139,776 local transitions, including q+carry=B.
• 6,561 exhaustive p=2, r=2 products against direct convolution.
• 180 larger seed-60 products and six negative controls.

Reproduce with Python's standard library:

python3 -B research/signed-extraction-audit/verify.py --check research/signed-extraction-audit/RESULT.json

Expected: PASS_SIGNED_EXTRACTION: 57160 words; 139776 transitions; 6561 exhaustive products; 180 sampled products; 6 negative controls.

Verification passed on CPython 3.14.8, both normally and with -O. All four existing test_finite_identities.py tests and make selected-record-check pass. Two in-memory code mutations are rejected. The written induction and checker received an additional scoped AI review. This is finite verification, not a machine-formalized theorem or a fixed-tape runtime implementation. Full make verify, supplier cold replay and remote CI success are not claimed. The selected construction, inherited all-size/physical/analytic assumptions and upstream source are unchanged.

The final overlap refresh found Ryan Pinto / @Ambassators's concurrent #690 at 8dd49f5e322f682502d9ab3c03d393a50f046e24. It audits packed_lower from Chafik Boukhalfa's #489. This contribution independently audits the retained upstream block/carry interface and adds the exact transition census and positive-half-radix alias control. No #690 code or result is consumed. The local claim records the candidate choices and overlap checks; no pinned sprint issue was available.

Credit OpenAI for the original CC0 lemma, Douglas Colkitt for the prior dependency audit and finite test, and the selected checkpoint's contributors. Prepared by Sagar Thakrar / @thakrarsagar with substantial OpenAI Codex assistance in research, code, writing and scoped AI review. New files use Apache-2.0.

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Independent signed-block extraction and carry audit. Strict coefficient bounds, convolution and half-radix alias controls pass; forbidden-boundary example is not a counterexample to the retained lemma.

The scoped package check passed at 7d857bb58fb55ada6d05364a4b6f01bba8e2283f. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.