T-159: Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout

S2 V3 C3 Upper bound confirmed superseded by T-146

2026-10-11 published · Calle after Xu, Levy · $n=102$$n=102$

An exact rational certificate derived here from Calle's decimal layout of 102 unit squares, reported on jlevy/squares#529, proves a finite upper bound at the side the layout prints: $s(102)\le 10.605828696455852$$s(102)\le 10.605828696455852$.

The source's file is a layout, not a certificate: each centre and each rotation in radians is printed to 18 significant digits, and a rotation printed in radians has no rational cosine and sine. The certificate is this repository's: the printed side with no dilation, every centre as printed, and each half-angle tangent rounded down at 40 places, which turns a square by less than 2e-40 radians against margins of 2e-12.

The author reports that a 60-digit decimal separating-axis check of his own, run once here from scratch and reproducing its two gaps, finds least gaps of 1.9e-12 between squares and 2.0e-12 to the walls, and that his search engine's exact rational-interval certificate accepts the layout; the engine is not public and that certificate cannot be rerun.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 12 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 and derived it again from the layout fetched at the pin with its own reader and arithmetic, square for square. The certificate is the project's derivation and no route is the source's code, so the replay is independently re-implemented. The 12 October review found no blocking defect.

The side is above the certificate the case holds, Daniel's regularization of Xu's packing (T-146), by 2.43e-11, and below Xu's own certificate (T-125), the case's ceiling until 11 October 2026 and the one the issue compares with, by 5.48e-11, the issue's figure; so it moves no case, the house at 102 is unchanged, and the entry is superseded by T-146 from the day it was registered.

Credit Calle (https://github.com/riocalle/my-repo, University of New Hampshire) with this arrangement and its layout, found by his own search engine from scratch, and Xu (T-125) with finding the design and width first, as the issue asks: Xu published on 8 October 2026 and the author's search finished its first packing of the design about ten hours later. Measured pose by pose, every square pairs with one of Xu's under a half turn of the box, 40 within 1e-9 and none turned, and 62 moved, 19 of them beyond a tenth of a unit and the largest by 0.494, 11 turned by up to about 43 degrees: another arrangement of the design, not a refinement or a regularization of Xu's packing.

The issue and README state AI assistance: the author designed and directed the project and its experiments, and Claude wrote and ran the code, the GPU search, the exact finishing step, the refinement search and both checkers.