T-139: A more precise certificate of Ryan Xu's 126-square arrangement, from Evan Daniel

S2 V3 C3 upper bound confirmed

2026-10-09 published · Daniel after Xu, Chaoweeraprasit · $n=126$$n=126$

One complete rational source certificate by Evan Daniel, published in his repository and reported by no issue, proves a finite upper bound at the exact side it states: $s(126)\le 11.742640687119285146522492579501$$s(126)\le 11.742640687119285146522492579501$.

The 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. It certifies Ryan Xu's arrangement as Daniel reconstructed it from a picture and polished it: 58 of its 126 squares sit where Ryan Xu's certificate (T-125) puts them and 68 are displaced by up to 0.06, so it is a more precise certificate of the same arrangement, not T-125's poses refined in place. The source reports that its two exact checkers accept it and that its exactsolve finds the packing a local minimum at 15/2 + 3 sqrt 2, strict modulo exact flat motions; the certificate's side is 1.17e-19 above that closed form, which nothing here certifies.

All three retained jobs, the positive with its duplicate-square and outside-container controls, 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 certificate with six controls. No route here is the source's code, so the replay is independently re-implemented. The 10 October review found no blocking defect.

The side is 9.25e-11 below the ceiling the case holds, Ryan Xu's certificate (T-125); the case record is unchanged, so it is pending adoption.

Credit Ryan Xu (ry-xu, https://github.com/ry-xu/square_packing, issue 432) with the arrangement, and Evan Daniel (https://github.com/evand/square-packing) with its reconstruction from a picture and the rational certificate. The same commit reconstructs two packings of Nate Chaoweeraprasit's SQUISH, phases 14 and 10, at larger sides. The source says the packings are other people's and makes no claim; the commit names Claude Opus 5.5 as co-author.