T-175: New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly

S3 V3 C3 Upper bound confirmed superseded by T-182, T-186 and T-188

2026-10-13 published · Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · $n=211,241,304$$n=211,241,304$

Three complete rational source certificates report finite upper-bound improvements at the exact sides they state: $s(211)\le 149936941865832465706359181817612514235389167555807511399813149936941865832465706359181817612514235389167555807511399813/10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$, $s(241)\le 9991958905656775259135410873409821230409663792812755550161009991958905656775259135410873409821230409663792812755550161/625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ and $s(304)\le 358588555161503499313326522779782297248782400899192783432303585885551615034993133265227797822972487824008991927834323/20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$.

All 12 retained jobs, four certificates with their 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 three certificates this entry cites 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 14 October review found no blocking defect.

The author reports that each packing passes its own C++ checker (sqverify --precise --certify, MPFR then GMP rationals) at delta = 1e-60, that sqverify --check-cert certifies the shipped certificates in GMP rationals with no dilation and no tolerance, and that a separately written 80-digit mpmath cross-check finds least separations of 5e-21 at 211 and 241 and 4.995e-21 at 176 and 304; the issue states that all three checkers are the author's, written with AI assistance, the first two sharing the C++ parser and geometry.

It also states that each side is the exact first-order optimum of its contact topology, with nonnegative KKT multipliers at a 60-digit solution reached by two routes that agree, and the load-bearing squares scaled by 1 + 1e-20. Those statements are the author's; no source program ran here.

Credit Ostashev (https://github.com/dostashev/square-packing) with the packings and their certificates, which the issue says are new arrangements.

At 176, 211 and 241 a packing whose tilted squares form two bands was re-hardened on a GPU with every square soft and the box held (soft-to-rigid, after Nakajima's idea), and the two bands merged into one V: at 176 seeded from the author's earlier composition of issue 528's 38-square piece and issue 562's 44-square piece, at 211 from Daniel's exact optimum of de Winter's packing, and at 241 from Daniel's exact optimum of Couzo's packing, the case's house. At 304 the same re-hardening relaxed the author's composition of a 55-square (h + 8) x h piece and a 68-square (h + 7) x h piece without forming a V. Each was solved to the exact optimum of its contact topology.

The 211 is a different packing from the author's T-149, which it replaces in his repository. The author asks that the seeds' authors (Couzo, Daniel, de Winter) and the composition pieces' sources be acknowledged.

The issue states AI assistance: an AI-assisted research loop the author drives with his own ideas, merging the bands at the junction being his; AI agents wrote and ran the search and verification code, and an agent extended the move from his compositions to published two-band packings.

Measured pose by pose by that review, the 211 and 304 are new arrangements and the 241 rebuilds the seed's design, every square paired and 15 moved beyond a tenth. On 15 October 2026 the case record at 304 adopted this entry as its house, its reported and verified upper bounds moving together and Couzo's certificate (T-152) kept as history; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). At 211 and 241 Couzo's issue 586 (T-182) is smaller and holds the cases.