Research audit: sharp T37 integrality penalties (no new kappa)

research/t37-integrality-audit/CLAIM.md

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+# Local sprint claim
+
+- Track: adversarial mathematical audit; OS assignment seed **65**.
+- Base: `407d52ff15b2217b6e22bce44728b4c3907b6515` (origin/main fetched
+  2026-10-13). Branch: `research/audit-seed65`.
+- Object: the integer-count/equality step in PR942's fixed-U8 all-replica
+  signed-dual relaxation, especially nonmultiples of 37.
+- Method: derive the credit-slack lattice; certify a 37-state residue
+  potential with exact rational interval edge bounds; exhibit relaxation
+  witnesses and controls that remove one premise at a time.
+- Deliverable: scoped proof, deterministic standard-library checker,
+  source pins, exact certificate and negative controls. No new kappa,
+  supplier, physical owner assignment or frontier edit.
+- Expected finish: 2026-10-13 09:25 America/Los_Angeles. Reserve the final
+  quarter for verification, independent review and packaging.
+
+The three candidates were (0) the prime-power cyclic inverse circle proof,
+(1) the disjoint-helper matching floor, and (2) integer consequences of the
+all-replica signed-dual support inequality. Seed `65 % 3 = 2` selects the
+third. The inverse family already has several open audits; the matching
+floor has a complete integer-dual review. The selected obligation has a
+qualitative equality statement, but no quantitative residue certificate.
+
+Curiosity Bot checked the issue listing and current PR metadata immediately
+before this local claim. No pinned sprint coordination issue was visible;
+the two open issues were #913 and #650. No issue was opened to reserve work.
+Queries for integrality and all-replica work found no competing quantitative
+residue audit. This is bounded coverage, not a claim to have read every PR.
+
+Overlap: Jonathan Capalbo's #942 proves the all-T support bound and
+constructs T37. Chafik Boukhalfa's #890 exhausts a finite T<=22 relaxation.
+Joel Pulikkan's open #962 independently constructs T37 and identifies its
+optimal normalized profile. This audit credits those results and adds the
+slack lattice, a uniform sharp numerator penalty in the arithmetic
+relaxation, and all 37 residue-class certificates. It does not repeat their
+construction or claim priority for T37 optimality.
+
+Delegated review is confined to this same integrality step and certificate.
+Stopping rule: prove/check the finite 37-state certificate and declared
+controls, then stop; no broader supplier or parameter search.
+
+## Result and handoff
+
+Completed 2026-10-13. The audit derives the 1200-unit slack lattice and
+sharp off-37 numerator penalty, with all 37 residue certificates. Both
+checkers passed normally and under `-I -O -B`: 1,163 bin patterns, 1,591
+edge inequalities, 37 witnesses, and 8 primary / 10 independent rejection
+controls. The fresh overlap check also distinguished PR974's interface
+budget study. No physical construction or new kappa is claimed.
+
+Submitted one focused research-audit PR:
+[CrocSwap/integer-mult-bounds #992](https://github.com/CrocSwap/integer-mult-bounds/pull/992).
+No coordination issue was available, so this is the single local claim
+update with the result and link. The selected frontier remains unchanged.