T-168: A further exact refinement of Xu's packing of 146 squares, after Barnett's

S2 V3 C3 Upper bound confirmed superseded by T-187

2026-10-12 published · PingYou Ltd after Xu, Barnett, Daniel, Levy · $n=146$$n=146$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(146)\le 6291891101975417145956733968525575683386443086533905641519426925380587272252740885402451051/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided again here on 14 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 each and held the derived fact to the pinned file it is derived from, fetched at the pin. No route here is the source's code; the source's runs of this repository's two maintained routes on its corners witnesses are its own and decide nothing here, so the replay is independently re-implemented. The 14 October review found no blocking defect.

The author reports that its exact separating-axis checker and a separate rational polygon-intersection checker accept all 146 squares, every wall and all 10,585 pairs, with duplicate and outside controls refused and the conversion to its corners Witness/v2 lossless; and that this repository's sqpack exact_verify and check_rational_witness_independent, run unchanged, accept that witness. Those statements are the author's; no source program ran here.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with this further refinement, as the source asks, of Barnett's refinement (T-150, issue 505) of Xu's packing of 146 (ry-xu, issue 432); the source credits Daniel with the contact-solver source and the earlier n146 regularization. The pinned source makes no statement about AI assistance.

Measured pose by pose by that review, it is Barnett's packing (T-150) re-solved, no square moved beyond 2e-06. At 146 PR 574's certificate (T-174) is smaller, by 1.26e-79, and was held for the owner from 15 October until the house rule of 16 October 2026 lifted the hold. Meanwhile this entry held the case, the smallest certificate there that the hold left in. The adoption of 16 October 2026 (PR #657) made Daniel's T-187, smaller than T-174, the house.