T-160: A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly

S3 V3 C3 Upper bound confirmed superseded by T-222

2026-10-12 published · Daniel after Couzo, Levy · $n=154$$n=154$

One complete rational source certificate by Daniel, reported on jlevy/squares#552, proves a finite upper bound at the exact side it states: $s(154)\le 12922566401934059503377267965457/10$^30 = 12.922566401934059503377267965457.

The certificate gives a rational side and, for each of the 154 unit squares, a rational centre and the tangent of its half-angle, in the author's text format: the 80-digit KKT point of the packing with every centre scaled by 1 + 1e-20 about the origin corner and the side rounded up at 30 places.

The author reports that his two exact rational checkers, verify_cert.py by separating axes and verify_cert2.py by polygon intersection, accept it with least certified clearances of 1e-20 between squares and 5e-21 to the walls; he states that the KKT point has positive multipliers on its load-bearing contacts, the least 1.5e-4, a reduced Hessian positive semidefinite modulo 45 exact flat motions, a first-order MILP jammed in all 98 corner-to-corner branches and 8 rattlers, numerical evidence by his own statement and not proofs. Those statements are the author's; no source program ran here.

All three retained jobs, the positive with its 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 certificate with six controls and held the 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.

The side is 5.04e-04 below the ceiling the case held, SQUISH's third request (T-136), adopted on 11 October 2026 and on main since 12 October; 4.00e-03 below T-127, issue 470's pose and T-115; 5.73e-03 below T-113; and 9.10e-03 below T-098 and T-056: the smallest exact side the record holds or has pending at 154.

On 12 October 2026 the case record adopted this entry as the house at 154, its reported and verified upper bounds moving together and T-136 kept as history, with T-127, T-115 and T-113 verified and not selected; private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit Daniel (https://github.com/evand/square-packing) with the packing and its certificate, its start credited to Couzo's 5 October packing of 155 squares as the author asks. Measured here, the packing is another arrangement than T-136's: no symmetry of the box pairs the two square for square, and under the best placement found 32 of T-136's squares land within 1e-9 of one of these, one to the digit.

The author states that the packing came from a time-travel test of his size-chaining explorer, the quality-diversity basin explorer of issues 465 and 489, restarted from this register as it stood on 5 October, and that the chain never saw T-136's packing; in his comment of 12 October he states that it descends from Couzo's packing of 155 squares with one square removed, after 17 productive rounds at 154. The measurements here are consistent with that statement and do not establish it: 43 squares of Couzo's 155, as T-098 solved it, reappear to the digit, but so do 45 of Couzo's 154, and at a tenth of a unit the 155 is the nearer of the two, 114 squares against 87; the round-by-round trace is not published.

The search code, the run and the request were done with Claude (Claude Code), which Daniel directed and reviewed.