T-141: Exact rational certificates at 132 and 308 from Evan Daniel's third record hunt

S3 V3 C3 upper bound confirmed

2026-10-10 published · Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · $n=132,308$$n=132,308$

Two complete rational source certificates by Evan Daniel, reported on jlevy/squares#489, prove finite upper bounds at the exact sides they state: $s(132)\le 11.985680198845808811522231964102$$s(132)\le 11.985680198845808811522231964102$ and $s(308)\le 17.998269879526255875387992744507$$s(308)\le 17.998269879526255875387992744507$.

The same release states $s(343)\le 18.978232523635611279730512106023$$s(343)\le 18.978232523635611279730512106023$ and $s(344)\le 18.994450514277401521114135796864$$s(344)\le 18.994450514277401521114135796864$, beyond the case corpus; the source register tracks those as beyond-horizon rows, proved by the same evidence.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Evan Daniel's format, each side printed in full. The author reports that his two exact rational checkers, verify_cert.py and verify_cert2.py, with different geometry, pass all four, with least certified clearances of 1e-20 for pairs and 5e-21 for walls. Each side is an 80-digit KKT point with positive multipliers on the load-bearing contacts, a reduced Hessian positive semidefinite modulo exact flat motions and a jammed first-order MILP in every corner-corner branch; those are numerical evidence, as the source says, not proofs.

All 15 retained jobs of the release's five-count replay, the four positives with their duplicate-square and outside-container controls among them, were decided again here on 10 October 2026 by both maintained exact routes and equal their retained rows, and a third exact route, half-extent containment and exact intersection area, decided the four with six controls each. No route here is the source's code, so the replay is independently re-implemented. The 10 October review found no blocking defect.

Each side is below the ceiling its case holds, at 308 the grid's 18; the case records are unchanged until a house is adopted, so each is pending adoption.

Credit Evan Daniel (https://github.com/evand/square-packing), whose quality-diversity basin explorer with a two-parent crossover move found all four. The packings at 308, 343 and 344 build on Kevin Fang's packings from issue 484 in the Arslanov-Mustafin-Shangitbayev two-wedge family, and the author asks that the construction be credited to him: 344 is his packing polished and moved by a corner-corner descent, 308 his packing polished and moved seven explorer steps, and 343 his 344 less one square, in a different basin from his 343. The packing at 132 is the author's own lineage from issue 465. The search code and the request were written with Claude under the author's direction.