T-147: A packing of 207 squares from two placed patches of Xu's packing of 70

S3 V3 C3 Upper bound confirmed superseded by T-173

2026-10-11 published · PingYou Ltd after Xu, Daniel, Levy · $n=207$$n=207$

One complete rational source certificate by PingYou Ltd, reported on jlevy/squares#503, proves a finite upper bound at the exact side it states: $s(207)\le 14.8809603730436756319360591381170501178639$$s(207)\le 14.8809603730436756319360591381170501178639$, the fraction 148809603730436756319360591381170501178639/10^40 the issue prints.

The certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in SQUISH's JSON rows; its side and centres already carry the guard dilation 10000000001/10000000000 it states, and every square keeps side 1.

The author reports that three exact Fraction checkers of its own, by vertex projections, by centre and support-radius separating axes and by rational polygon clipping, accept all 21,321 pairs and every wall; that its reconstruction script rebuilds every pose and the side from the n70 parent; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept its corners Witness/v2. The checkers, the reconstruction and the conversion were prepared in one workflow. Those statements are the author's; no source program ran here.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 11 October 2026 by both maintained exact routes, from an exact fact derived from the pinned certificate, and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided it with six controls and found the fact to be the packing the pinned certificate and the pinned witness state. The source ran the two maintained routes on its own witness, but neither is its generator's code, and it did not run the third route, so the replay is independently re-implemented. The 11 October review found no blocking defect.

The side was 7.03e-03 below the ceiling the case held, Chaoweeraprasit's packing as Gupta refined it (T-127), and is 4.55e-03 below SQUISH's T-136. On 11 October 2026 the case record adopted this entry as the house at 207, its reported and verified upper bounds moving together, T-127's certificate kept as history and T-136 verified and not selected; private-worker custody (RV-6) was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) for the n207 assembly, as the source asks: two differently placed patches of Xu's n70 construction (issue 432), keeping 60 and 69 whole squares, with upright grids of 42 and 36 squares. The patches come from Daniel's exact n70 certificate (https://github.com/evand/square-packing, the regularized n-070 certificate T-146 retains), credited to him for its exact retention. Measured pose by pose against that certificate, 60 of its squares reappear exactly in place and 69 under its reflection about the n70's box, transposed and moved by (6, 6), as the source says. The source names SQUISH's n207 of issue 481 only as its comparison, and claims neither the n70 packing nor patch composition in general.

The source as pinned makes no statement about AI assistance. As first published on 11 October 2026, its README and the issue said that the verification, the comparison review and the package preparation were assisted by OpenAI Codex; minutes later a revision titled "Correct attribution in submission materials", the one the issue names, removed that sentence, and the issue was edited to match, with no reason given. Asked which account stands, the author answered on jlevy/squares#503 on 12 October 2026, for this certificate and the n153 of issue 504 (T-148): "Both solutions were AI assisted by 'PY', an unreleased agentic AI offering by PingYou Ltd."