T-167: A tighter rational realization of PingYou Ltd's packing of 207 squares

S2 V3 C3 Upper bound confirmed superseded by T-173

2026-10-12 published · PingYou Ltd after Xu, Daniel, Levy · $n=207$$n=207$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(207)\le 14.880960371558556192074311116$$s(207)\le 14.880960371558556192074311116$.

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 three exact checkers, its own rational polygon-projection checker and Daniel's unchanged verify_cert.py and verify_cert2.py, accept all 207 squares, 828 wall conditions and 21,321 pairs; that its package replays the recorded rational construction and owner lineage and checks the lossless conversion to its corners Witness/v2; 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 the n207 assembly and this refinement of its certificate, as the source asks: two differently placed patches of Xu's n70 construction (ry-xu, issue 432), taken from Daniel's exact n70 certificate, with upright grids of 42 and 36 squares. The source names SQUISH's n207 (issue 481) only as its earlier comparison. The pinned source makes no statement about AI assistance; the author's statement on issue 503 of 12 October, 'Both solutions were AI assisted by "PY", an unreleased agentic AI offering by PingYou Ltd.', covers the n207 assembly.

Measured pose by pose by that review, it is PingYou Ltd's T-147 packing under a reflection with eleven squares moved, by at most 0.076. At 207 Polkovskyi's issue 573 (T-173) is smaller and has held the case since 15 October 2026.