T-149: New packings of 155, 156 and 211 squares, each with an exact rational certificate

S3 V3 C3 Upper bound confirmed superseded by T-186, T-188 and T-197

2026-10-11 published · Ostashev after Nakajima, Levy, Xu · $n=155,156,211$$n=155,156,211$

Three complete rational source certificates by Ostashev, reported on jlevy/squares#501, prove finite upper bounds at the exact sides they state: $s(155)\le 64762127034792280678268012662770039849775757530809316735783064762127034792280678268012662770039849775757530809316735783/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$, $s(156)\le 8111982883527766297069290937797918339306619994848173142743008111982883527766297069290937797918339306619994848173142743/625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$ and $s(211)\le 74981683606681578473292015533873289965462274509366086287257074981683606681578473292015533873289965462274509366086287257/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$.

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. These are the certificates of the repository as it stood when read, which replaced, the day the issue was filed, the three whose sides the issue prints; each issue side is above the retained one, so its bound follows.

The author reports that each packing passes its 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; the README also states that each side is the exact optimum of its packing's contact topology, with nonnegative KKT multipliers. 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 11 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 and its side to the README's print. No route here is the source's code, and the source ran none of them, so the replay is independently re-implemented. The 11 October review found no blocking defect.

Each side was below the ceiling its case held, by 7.78e-05 at 155 (SQUISH's update, T-115), 2.91e-03 at 156 and 1.62e-03 at 211 (Daniel's exact optima, T-098), and at 155 is below Couzo's T-128 by 7.35e-05. On 11 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-128 verified and not selected at 155; 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 name he agreed to on the issue on 11 October 2026. The README says each was constructed from scratch, a V-shaped tilted band at a chosen angle with aligned corner blocks chosen by an integer program, no existing record geometry used, then re-hardened on a GPU with Nakajima's soft-to-rigid method, polished by the fixed-angle linear program it credits to this project, and solved to the exact optimum of its contact topology; its certificates follow Xu's construction.

Measured pose by pose, none is a refinement of a credited packing: at 155 the squares pair one to one with SQUISH's (T-115) and Couzo's (T-128) only through the aligned corner blocks every packing of that family shares, and the band between them is at another angle, 50 squares turned by up to 27 degrees; at 156 and 211 no symmetry pairs the squares with the case's packing one to one.

The source and the issue make no statement about AI assistance, and the issue omits the template's section on it. On 12 October 2026 the author stated on jlevy/squares#528 that AI agents wrote and ran his search, composition and verification code; the verification code it names is the one retained here, the same byte for byte in both releases, and whether the statement covers these three packings' search is asked (think-e59i).