T-150: Exact rational refinements of Xu's packings of 146 and 261 squares

S2 V3 C3 Upper bound confirmed superseded by T-174 and T-187

2026-10-11 published · Barnett after Xu, Daniel, Levy · $n=146,261$$n=146,261$

Two complete rational source certificates by Barnett, reported on jlevy/squares#505, prove finite upper bounds at the exact sides they state: $s(146)\le 116279649124311595126849961042493265136092670741573762928668277775998122712863472766532671969348798419026610394581005669/9240437194455113887096299456817024243377978045957194209468060456305894240222303442118770000000000000000000000000000000$ and $s(261)\le 281511998443714605777671262552678001053511713995864784318453182156213791529093472577894719178723409384395264262601/16878452773388004800535351238155816857400040173477095354881922318666694517637251382500000000000000000000000000000$.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Daniel's format; each side rounds up to the 18-place print of the issue, 12.583782203951046204 and 16.678779875343207747. The author reports that both pass four exact rational checks with zero acceptance tolerance, a separating-axis and a polygon checker of the same assisted project and Daniel's unchanged verify_cert.py and verify_cert2.py, and that Ellsworth's check_packing.py accepts 140-digit exports at tolerance 1e-90 as supplementary numerical evidence. Those statements are the author's; no source program ran here.

All six retained jobs of the two, each certificate with its duplicate-square and outside-container controls, were decided again here on 11 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 both with six controls each and found each fact to be the packing its pinned certificate states. No route here is the source's code, and the source ran none of them, so the replay is independently re-implemented. The 11 October review found no blocking defect.

Each side was below the ceiling its case held, Xu's certificate (T-125), by 7.36e-08 at 146 and 7.35e-11 at 261, and at 146 is below Daniel's regularization (T-146) by 7.35e-08. On 11 October 2026 the case records adopted this entry as the house at both counts, each case's reported and verified upper bounds moving together, Xu's certificates kept as history and T-146 verified and not selected at 146; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit Xu (ry-xu, https://github.com/ry-xu/square_packing, issue 432) with both packings, as the source does, and Barnett (https://github.com/RobertBarnettProjects/square-packing-research) with the refinements, by minimum-movement sequential linear programming followed by an exact rational contraction, and with the certificates. At 146 the source compares with Daniel's regularization of Xu's packing (T-146) and credits it there.

Measured pose by pose, both are Xu's arrangements: each certificate matches his square for square at the identity. The 261 is a precision refinement, no square moved by more than 9.1e-06 or turned by more than a sine of 2.4e-06. The 146 is a local re-optimization: no square moved by more than 5.0e-02, but 51 moved beyond 1e-3 and 24 turned by up to about 1.2 degrees, and it lies nearer his certificate than Daniel's view, so the refinement is of Xu's certificate.

The source discloses OpenAI Codex assistance with the solver, the certificate and checker code, the source review and the preparation.