Research: sharp boundary of the T37 owner relaxation (no new kappa)

[cbethin]

The uniform T37 supporting line in #995 is proved through b=1/3 and fails at b=1. This note determines its exact first failure, including the equality case and an explicit relaxed competitor. It adds a sharp characterization of the inherited relaxation; it does not change a supplier, the selected frontier, or kappa.

For the fixed PR795/890/942 whole-helper inventory and its signed-dual relaxation, the normalized T37 remainder profile is optimal for every positive integer replica count and every real 0 < b <= b_star, where b_star is the unique positive root of

37*(3/2)^b - 48^b - 36 = 0,
16/25 < b_star < 13/20.

Rank 32 is the first competing rank. All other noncritical ranks remain strictly supported through b=2/3. Below the boundary, integer equality requires a multiple of 37 replicas. At the boundary, equality also permits rank 32, with 37*n_1 + 32*n_32 = 6*T.

The explicit relaxed profile T=16, {0:27461, 32:3, 48:4792} attains equality at the boundary and has strictly lower normalized moment for every larger exponent. This is a countermodel in the necessary-constraint relaxation, not a physically realizable packing or a contradiction of PR942.

The complete argument uses strict concavity after x=48^b, exact finite endpoint signs, and the fixed mass/credit identities. An independent integral-ratio monotonicity argument checks the continuum step. Cartesian bin enumeration with integer radical bounds and a separate fill recurrence with bounded logarithm/exponential series independently check the crucial finite premises. The two checkers import neither each other nor retained moment engines. The same Codex-assisted author prepared them; this is not external human review or a proof-assistant verification.

Validation on macOS / CPython 3.9.6:

python3 -B research/t37-support-interval/reproduce.py --out /absolute/new/output

Expected final line: PASS_T37_SUPPORT_INTERVAL_REPRODUCTION.

• Complete coverage: 1,163 legal bin vectors and all 45 attainable ranks; no exponent sampling substitutes for the proof.
• Both implementations, 13 semantic rejection controls, and four positive controls pass normally and under -O.
• Committed receipt comparison, minimal source-closure relocation, source-drift rejection, and existing-output preservation pass.
• make selected-record-check, Markdown lint, and patch whitespace checks pass. Full native replay is outside this additive note's affected checks.

Concurrent work and usefulness. Credit @woklerx's #995, inspected at 93a8127b990f9755d0c80e729b40c969d6c611e4, for publishing the uniform theorem first. #1000 adds an elementary rational proof near the selected endpoint; #992 addresses integer penalties there. This note resolves #995's explicitly unlocated first failure and supplies the boundary mechanism and full-range relaxed countermodel. We claim no additional practical owner-search exclusion near the selected endpoint. None of those packages' code is consumed. All physical, compiler, hosting, precision/tape, analytic-transfer and bit-interface obligations remain separate.

Predecessor credit: Jonathan Capalbo #942; Chafik Boukhalfa #795/#890; Hao Yan's T16 lineage; Jeremy Kothe, NewBArch and ColdWitness/icekylinx's retained local-word lineage; Joel Pulikkan's concurrent #962. Exact pins and arithmetic-method proofs are included. Base 8bddbee377e2f3ce1f1247196c4b290d55767d20; all five consumed sources are unchanged at refreshed main 2921874911956718fcaaaf9396fe8267b3cfa9f0. Seed 72, adversarial audit. No pinned coordination issue was available, so the claim is local.

Prepared by Charles Bethin with substantial OpenAI Codex assistance. Apache-2.0. Status: mathematical research note, pending maintainer review.