T-165: A tighter certificate at 176 from Couzo, published with no issue

S2 V3 C3 Upper bound confirmed superseded by T-186

2026-10-12 published · Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · $n=176$$n=176$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(176)\le 13.807637226802279523050752140325$$s(176)\le 13.807637226802279523050752140325$.

All 42 retained jobs of the packet, its fourteen certificates with their duplicate-square and outside-container controls, were decided again here on 14 and 15 October 2026 by both maintained exact routes and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the certificate with six controls each and held the retained file to its upstream bytes, fetched at the pin. No route here is the source's code; the author's runs of this repository's sqpack verifier are his and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

Couzo published it in his repository on 12 October with no issue, 2.54e-10 below his issue-561 certificate (T-161); it is unchanged at the revision issue 586 pins, where the couzo-certificates-2026-10-13 packet retains it, and the certificates README there prints 13.807637226803 and calls it an exact optimum from Daniel's exact contact solver. No source says whether it is a new packing or a refinement of issue 561's; measured pose by pose by the 14 October review, it refines issue 561's packing (T-161): every square pairs with one of that packing's within 3.8e-03, none turned beyond 0.17 degrees, and the side is 2.54e-10 smaller.

Both maintained exact routes accept it here and refuse its two controls.

Credit Couzo (https://github.com/franciscouzo/square-packing); issue 561 credits Ostashev's arrangement as the start of its 176. The commit makes no statement about AI assistance; the author's issues of 12 and 13 October say OpenAI Codex assisted the search and verification and that he directed the work.

At 176 PR 574's certificate (T-174) is smaller, by 1.18e-02, and was held for the owner from 15 October until the house rule of 16 October 2026 lifted the hold. Meanwhile this entry held the case, the smallest certificate there that the hold left in. The adoption of 16 October 2026 (PR #657) made Polkovskyi's T-186, smaller than T-174, the house.