T-187: A smaller packing of 146 squares from Daniel's chain search, started from Xu's packing

S2 V3 C3 Upper bound confirmed

2026-10-14 published · Daniel after Xu · $n=146$$n=146$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(146)\le 12.583782190151860342292508597077$$s(146)\le 12.583782190151860342292508597077$.

All three retained jobs, the certificate with its duplicate-square and outside-container controls, were decided again here on 16 October 2026 by both maintained exact routes and equal their retained rows, and the frozen comparison rebuilt equal. No route here is the source's code, so the replay is independently re-implemented. The 16 October review found no blocking defect.

The author reports that his two exact rational checkers, verify_cert.py (separating axes) and verify_cert2.py (polygon intersection), accept the certificate and its regularized drawing. The side is the 80-digit KKT point S* = 12.58378219015186034216... scaled by 1 + 1e-20 about the origin corner; the source reports a local minimum strict modulo 59 exact flat motions, 69 corner-corner contacts with no corner-corner MILP descent, and 13 rattlers, and measures it against #505's packing as a different local minimum that agrees to about 1e-7 except a cluster of 10 squares tilted near 42 degrees. Those are numerical evidence, as the source says, not proofs. No source program ran here.

Credit Daniel (https://github.com/evand/square-packing), whose chain search found the packing in a time-travel test started from the public packings of 10 October, and credit Xu's packing of 146 (issue 432, T-125) as its start, as the author asks: two random large perturbations of it and a re-polish, then one exact descent step. The author names Barnett's refinement (T-150, issue 505) as the record it improves and calls the result a different local minimum from it.

Measured pose by pose by the 16 October review, the 146 pairs one to one with Xu's packing under the identity, nine squares moving beyond a tenth (at most 0.478) and 137 staying within one, and the same with Barnett's T-150 (at most 0.453): another arrangement of Xu's design and a refinement of neither, which agrees with the author's start and his different local minimum. Issue 606 states: "The search, the checks and this request were done with Claude (Claude Code). Daniel directed and reviewed the work."