T-161: Exact rational certificates at 156, 176, 237 and 271 from Couzo

S3 V3 C3 Upper bound confirmed superseded by T-186, T-190 and T-219

2026-10-12 published · Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · $n=156,176,237,271$$n=156,176,237,271$

Four complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(156)\le 12.979165853834276453965882038847$$s(156)\le 12.979165853834276453965882038847$, $s(176)\le 42253500388043689581217825957/3060154296728500000000000000$, $s(237)\le 15.902947773658185869705573326239$$s(237)\le 15.902947773658185869705573326239$ and $s(271)\le 16.948992356451205603226618436627$$s(271)\le 16.948992356451205603226618436627$.

The same release states $s(338)\le 18.906332809852093379991386331018$$s(338)\le 18.906332809852093379991386331018$, $s(344)\le 83284325592732658265440964873/4384829811311500000000000000$, $s(375)\le 581837142778534347520704447657618356033117642159/29235669412085804585614997608579500000000000000$, $s(376)\le 19.904949736498672043413871274032$$s(376)\le 19.904949736498672043413871274032$ and $s(378)\le 28547984384116369335179568707/1431512228978500000000000000$, beyond the case corpus; the source register tracks those as beyond-horizon rows.

All 27 retained jobs, the release's nine certificates with their duplicate-square and outside-container controls, were decided again here on 14 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 four certificates this entry cites with six controls each and held each 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.

The author reports that all nine requested certificates, fetched at the pin and checked against the digests and row counts the issue lists, pass this repository's sqpack verify.py and Daniel's verify_cert.py, at the revisions the issue names, in exact rational arithmetic, in 4.52 and 5.56 seconds; that 156, 237, 271, 338 and 376 are exact optima from Daniel's exact contact solver; and that 176, 344, 375 and 378 are rational witnesses of their decimal packings with centres dilated by 1 + 1e-11, 1 + 1e-12, 1 + 1e-10 and 1 + 1e-12. Those statements, and the author's replay script and receipt, are the author's; no source program ran here.

Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Ostashev's arrangements at 156 and 176, continued with the author's basin hopping and refinement and, at 176, through his web app; Chaoweeraprasit's SQUISH arrangements at 237 and 271, continued with the author's basin hopping and Daniel's fq refinement (237 improves the author's T-140 certificate).

Beyond the case corpus, 344 continues Fang's 344 (issue 484) from the packing reported in issue 506, and 338, 375, 376 and 378 the author's L-border constructions of issue 506 (338 and 375 from his 303, itself from SQUISH's issue 401; 376 from his earlier 304; 378 from his 306), refined with Daniel's fq at 338 and 376 and basin hopping at 378; the latest step at 375 came through the author's browser web app with Ellsworth's search. Daniel's optimizer, exact contact solver and verifier, Ellsworth's refinement and search ideas, and this project's sqpack verifier were used.

The author states that OpenAI Codex assisted the continuing search, the application work, the source audit, the verification orchestration and the request, that earlier search code and submissions were prepared with Claude Code, and that he directed the work.

Measured pose by pose by that review, the 156 continues Ostashev's packing (T-149), the 237 continues the author's T-140 design and the 271 refines Chaoweeraprasit's SQUISH packing (T-151), each paired square for square; the 176 moves Ostashev's issue-562 arrangement and pairs with no placement of it.

At 237 Polkovskyi's issue 593 (T-183) is smaller and held the case from 15 October 2026, and Couzo's T-190 has held it since 16 October 2026. At 156, 176 and 271 PR 574's certificates (T-174) are smaller, by 1.30e-19, 1.18e-02 and 1.69e-19, and were held for the owner from 15 October until the house rule of 16 October 2026 lifted the hold. Meanwhile this entry held the cases at 156 and 271, the smallest certificates there that the hold left in, and T-165, the author's re-solve of the 176, held 176. The adoption of 16 October 2026 (PR #657) made Polkovskyi's T-186, smaller than T-174, the house at all three.