Audit (no κ): exact owner-fill optima on the T74 handoff line — #2374's fill is optimal over all fills; #2435's fill leaves +5.8e-10 (pattern optimum certified)

[thakrarsagar]

Audit / independent check; no κ claim. This PR certifies the exact pattern-integer optimum of the whole-helper owner fill for the two latest T74 claims, and compares each published fill with it.

claim	published fill idle	exact pattern optimum (unique within 1e-12)	gap
#2374 on #2322's words, κ = 0.0016971949509	{1:21978, 3:8592, 5:2552, 32:1, 48:24560}	the same	0 (every fill has b ≤ κ + 1e-15)
#2435 on its copy-buffer-raised words, κ = 0.0016985113437	{1:20824, 3:6695, 5:1, 48:25302}	{1:20822, 3:6694, 5:2, 48:25302}, root 0.0016985119240	+5.80e-10 available

• Conditional κ = 0.001697194950901770: exact owner fill on #2322's handoff construction #2374: its claim of "an exact window-free-optimal owner fill" is confirmed and strengthened to optimality over all legal whole-helper fills. Owner-map work on Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322's words is exactly exhausted; Research note (no κ): T74 #2322 words — owner fills capped at b < 0.001697195759 (#2374 within 8.1e-10); exact prices of handoffs, aliases and rank-1 climb merges #2402's cap allowed up to 8.1e-10.
• Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435: its "exact owner fill for the new inventory" is not the pattern optimum. The optimum moves two rank-1 remainders and one rank-3 remainder into one more rank-5 remainder. It uses, for example, two (32: 9,9,9) bins and 1,156 (48: 4,8,9,9,9,9) bins. The full pattern counts are in the receipt, as a concrete owner-level target worth up to +5.8e-10. Whether it can be realized under Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435's windows and families is not checked here.

Method (owner_optimum.py, standard library only, about 45 s).

• Relaxation. Every legal fill places each helper role whole in one idle bin of rank 8, 16, 32 or 48. That is an integer pattern-count vector over the 1,163 width patterns with fixed bin and inventory row sums, so it relaxes every legal fill. This is the Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 / Certify PR2339's whole-helper T74 packing cap below 0.0016870 #2360 relaxation, after Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum #1129.
• Steps, per claim:
  1. Certify the published idle multiset's root within ±1e-15 of the claimed κ, using 800-bit enclosures.
  2. Enumerate exactly, by budget-pruned DFS with exact rational basis solves, every integer solution that could beat the claim. Degenerate zero-remainder swaps are handled with an integrality and nonnegativity argument.
  3. Certify the best solution's root.
  4. At (best root) + 1e-15, an exact rational dual, feasible against lower enclosures on all patterns, leaves zero integer survivors.
  5. Control at (best root) − 1e-12: the optimum survives, and every survivor has its idle multiset.
• Inputs.
  • The Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 profile and inventory were recounted in Research note (no κ): T74 #2322 words — owner fills capped at b < 0.001697195759 (#2374 within 8.1e-10); exact prices of handoffs, aliases and rank-1 climb merges #2402.
  • Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435's two words, at head 324d7d35, were recounted with the Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 frame-climb rule. They match its reported profile {1:2359, 2:632, 3:330, 4:120, 5:38, 8:8} and inventory {4:33, 5:9, 7:62, 8:90, 9:240}.
  • Idle multisets and κ are as published.
• Review. An adversarial AI reviewer could not refute the Conditional κ = 0.001697194950901770: exact owner fill on #2322's handoff construction #2374 part. It reproduced the root to 60 digits and ran a Gomory group-relaxation check, which found the only correction within budget is one empty 32-bin. It also ran more than 48,000 local exchange moves with no improvement; its best alternative is 5.0e-10 below. Its two wording issues are fixed. The Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435 extension uses the same code path.
• Checks. --check-only, the 3.10 grammar check and the research tests pass. Run on CPython 3.14.4; the registered 3.12 replay was not run.

python3 -I -B research/t74-owner-fill-optimality/owner_optimum.py
# PASS_T74_OWNER_OPTIMUM pr2374:claim_is_optimum pr2435:optimum_gain=+5.803e-10 unique_within_1e-12 no_new_kappa

Scope. The pattern relaxation drops windows, families and ports, so the upper bounds cover every legal fill. Realization of any fill is not re-run, neither the claimants' fills nor #2435's optimum. Base: main 0fdd7d82. Card: independent-moment, scope refined; the assignment is the adversarial-audit track, seed 71.

Credits.

• Conditional κ = 0.001697194950901770: exact owner fill on #2322's handoff construction #2374 / Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435 / Conditional κ = 0.001697065994108880: owner relocations on #2322's handoff construction #2349 (Jeremy Kothe) and Improve conditional κ to 0.001697147064845350 with whole-role augmenting paths #2363 (mason-hale): the audited fills.
• Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 (Vlado Vojdanovski): the words.
• Certify PR2339's whole-helper T74 packing cap below 0.0016870 #2360 (Jeremy Diviney) and Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum #1129 (Ali Romman): the pattern-dual relaxation.
• Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 / Research note (no κ): T74 #2322 words — owner fills capped at b < 0.001697195759 (#2374 within 8.1e-10); exact prices of handoffs, aliases and rank-1 climb merges #2402 (this author): the tools, vendored with pins.

Prepared with substantial AI assistance (Claude Code, Anthropic, Claude Opus 5.5). No independent human review is claimed. Apache-2.0.

🤖 Generated with Claude Code