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

[woklerx]

Change and mathematical scope

The fixed-U8 all-replica proof in #942 identifies equality at multiples of 37 but does not quantify the integer penalty elsewhere. This research audit derives a sharp positive numerator penalty and certifies all 37 replica residue classes. No new kappa, physical construction, or selected-frontier change is claimed.

For integer residual counts with the retained rank and signed-credit constraints, the credit slack is a multiple of 1200. At the pinned endpoint z = 0.001664727258740678855773, every 37 ∤ T has excess at least

delta(2) = c(2) - (26/37)c(1) > 0.291373150447.

The relaxed profile T=29, {1:4, 2:1, 48:8689} attains that exact symbolic constant. A 37-state potential additionally gives certified residue-specific lower bounds, with explicit relaxed witnesses within 2e-24 in the committed receipt. The normalized moment penalty is divided by 3139584*T, so this is not a uniform root/kappa gap as T grows.

Validation

python -I -O -B research/t37-integrality-audit/verify.py
python -I -O -B research/t37-integrality-audit/independent_check.py

Both pass locally on Windows/CPython 3.14.5, also without -O. Coverage: all 1,163 legal bin patterns, all 1,591 graph inequalities, all 37 witnesses, eight primary and ten separately implemented rejection controls. Controls include fractional equality at T1, omitted credit, incorrect mass, erased strictness, missing edges and corrupt bounds.

The second checker imports no primary checker: it uses DP bins, Bellman-Ford, direct moment differences, and the other retained rational engine. Both existing arithmetic engines are reused and source-pinned; no new independent transcendental proof or full formal multiplication verification is claimed. Existing supplier/global regenerating suites were not rerun because this additive package changes none of their inputs.

Scope and provenance

• Exact base: 407d52ff15b2217b6e22bce44728b4c3907b6515.
• Selected source at base: Conditional κ = 0.001661960547713947: T37 whole-owner packing and all-T bound #942, 1960cfa919a0945d0ca4cb14ea96237000037903; conditional kappa 0.001661960547713947 unchanged.
• Dependency hashes, full proof, deterministic certificate, validation and seed-65 local sprint claim are included.
• Bin realizability, owner timing, physical hosting, compiler, precision/tape, bit-interface and analytic-transfer obligations remain inherited/outside scope. The witnesses are arithmetic relaxation witnesses, not owner constructions.
• Credits: Chafik Boukhalfa's Independent exact certificate of PR741 owner-profile optimality #795/Conditional κ = 0.001661957774647029: T21 whole-owner packing and fixed-family bound #890 prices and equations, Jonathan Capalbo's Conditional κ = 0.001661960547713947: T37 whole-owner packing and all-T bound #942 support bound and construction, and Joel Pulikkan's concurrent Conditional κ = 0.001661960547713947: T37 owner map on PR890's U8 word (independent cross-check of #942) #962 independent T37 construction. The new contribution is the slack lattice and quantitative residue certificates; it claims no priority for T37 optimality. The U8 and arithmetic predecessor notices are retained.

Prepared for woklerx with substantial OpenAI Codex assistance, including a separately implemented agent check. Apache-2.0; not independent human peer review. No changes to upstream/, supplier code, selected records or the review ledger.