T-158: New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly

S3 V3 C3 Upper bound confirmed superseded by T-186

2026-10-12 published · Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · $n=175,232,264$$n=175,232,264$

Three complete rational source certificates by Ostashev, reported on jlevy/squares#528, prove finite upper bounds at the exact sides they state: $s(175)\le 27528348634973193258855646440084685651524191041221518308579027528348634973193258855646440084685651524191041221518308579/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$, $s(232)\le 31528348634973193258895646440084685651524191041221518308579031528348634973193258895646440084685651524191041221518308579/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ and $s(264)\le 167747351323878328426102645873217709808239488409494842884223167747351323878328426102645873217709808239488409494842884223/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in the source's sqcert format: the 60-digit optimum of its packing with the load-bearing squares scaled by 1 + 1e-20, then every centre and the side dilated by 1 + 1e-60. Each side rounds up at 38 places to the side the issue prints and at 15 to the README's.

The author reports that each packing passes his own C++ checker in GMP rational arithmetic, on the shipped certificates with no dilation and no tolerance and on certificates rebuilt from the 60-digit decimal packings, and an 80-digit mpmath cross-check written separately, all three written with AI assistance; he also states that each side is the exact first-order optimum of its packing's contact topology, with nonnegative KKT multipliers at a 60-digit solution. Those statements are the author's; no source program ran here.

All nine retained jobs, the three positives with their duplicate-square and outside-container controls, were decided again here on 12 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 three with six controls each and held each retained file to its upstream bytes fetched at the pin. No route here is the source's code, and the source ran none of them, so the replay is independently re-implemented. The 12 October review found no blocking defect.

Each side was below the ceiling its case held, by 2.95e-03 at 175 (Couzo's refinement of Xu's packing, T-140), 3.25e-03 at 232 (SQUISH's third request, T-136) and 3.44e-03 at 264 (the catalogue's 9 + 11 sqrt(2)/2, Goebel's pattern, verified by Daniel's exact ceiling, T-101, and at its exact side by T-155's regularized certificate of the same packing); at 175 it is also 2.94e-03 below Couzo's certificate of 11 October, which the issue names as the previous best and which the record holds in the issue-506 packet and registers as T-185, and 4.72e-03 below Xu's T-125. At all three counts it is the smallest exact side the record holds.

On 12 October 2026 the case records adopted this entry as the house at all three counts, each case's reported and verified upper bounds moving together and each earlier house kept as history, with T-140, T-130, T-138 and T-125 verified and not selected at 175; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit Ostashev (https://github.com/dostashev/square-packing) with the three packings and their certificates. The issue says each is a composition: two library rectangles in opposite corners of the box with aligned squares around them, each using his new packing of 38 unit squares in an (h + 2) x h rectangle, h = 5.7641743..., found by a GPU soft-to-rigid search after Nakajima's idea; at 175 in place of the pieces of Ellsworth's $s(171)$$s(171)$ template, cut from Schadt's $s(50)$$s(50)$; at 232 beside a 76-square piece of Friedman's $s(85)$$s(85)$; at 264 beside a 93-square piece of Daniel's exact optimum of de Winter's $s(126)$$s(126)$; each then taken to the exact optimum of its contact topology. He asks that the pieces' sources be acknowledged as the record sees fit.

Measured pose by pose, none of the three is a placement of the packing its case holds or of any report pending at its count: no symmetry of the box pairs the 175 square for square with T-140, with Couzo's certificate of 11 October or with T-125, the 232 with T-136, or the 264 with T-101's certificate. Where the record retains a named piece exactly, it reappears: 66 of the 171 squares of Daniel's certificate of Ellsworth's $s(171)$$s(171)$ (T-101) land on squares of the 175 to the ninth decimal under a half turn, none turned, and 88 of the 126 of his exact optimum of de Winter's $s(126)$$s(126)$ (T-098) on squares of the 264. Friedman's $s(85)$$s(85)$ and the 38-square block are the author's statement, not measured.

The issue states AI assistance: the author set up an AI-assisted research loop and drove it with his own ideas, the rectangle library and composition being his; AI agents wrote and ran the search, composition and verification code.