T-152: Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares

S3 V3 C3 Upper bound confirmed superseded by T-174 and T-186

2026-10-11 published · Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · $n=304,305$$n=304,305$

Two complete rational source certificates by Couzo, reported on jlevy/squares#506, prove finite upper bounds at the exact sides they state: $s(304)\le 17.929565946848500974381965343574$$s(304)\le 17.929565946848500974381965343574$ and $s(305)\le 17.950466482137712273488570226115$$s(305)\le 17.950466482137712273488570226115$.

The same release states $s(338)\le 18.906436500551825926629088515923$$s(338)\le 18.906436500551825926629088515923$, $s(339)\le 18.929433883807662155272760949615$$s(339)\le 18.929433883807662155272760949615$, $s(344)\le 18.994353244943442354485256580362$$s(344)\le 18.994353244943442354485256580362$, $s(375)\le 38166979981362750351336743477/1917328312988500000000000000$, $s(376)\le 19.910276586884544180280281394636$$s(376)\le 19.910276586884544180280281394636$, $s(378)\le 19.942541777354810533331219672476$$s(378)\le 19.942541777354810533331219672476$, $s(379)\le 19.953572709538574550151361449798$$s(379)\le 19.953572709538574550151361449798$, $s(380)\le 19.987440766555387551986240574787$$s(380)\le 19.987440766555387551986240574787$ and $s(381)\le 19.99266328846356482088565989631$$s(381)\le 19.99266328846356482088565989631$, beyond the case corpus; the source register tracks those as beyond-horizon rows, proved by the same evidence, which replace the rows it held at 338, 339, 344, 375, 376, 378 and 379.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in Daniel's format; each side rounds up to the 12-place print of the issue. Ten are exact optima from Daniel's exact contact solver; the one at 375 is a rational witness, the author's decimal packing with centres dilated by 1 + 1e-11, as the issue says, about the point (1/2, 1/2) as measured here. The author reports that all eleven pass Daniel's verify_cert.py and a copy of this repository's sqpack verifier in exact rational arithmetic; those statements are the author's, and no source program ran here.

No square touches a wall: every exact optimum clears its walls by at least 1/(2*10^20) and its pairs by about 1e-20, and the witness at 375 by 1e-15 and 7.05e-12.

All 45 retained jobs of the fifteen certificates the two source commits add or change, the eleven requested ones with their duplicate-square and outside-container controls among them, were decided twice on 11 October 2026 by both maintained exact routes, and the twelve at 304, 305, 344 and 381 again on 12 October, each time equal to its retained row. A third exact route, half-extent containment and exact intersection area, decided all eleven with six controls each. One maintained route is the verifier the source ran; the independent rational corner checker and the third route are independently re-implemented. The 12 October review found no blocking defect.

Each side is below the ceiling its case held, Daniel's exact optimum of Couzo's packing (T-098), by 5.08e-03 at 304 and 2.49e-03 at 305, and below every other report pending there: issue 500's exact sides (T-151) by 9.11e-05 and 7.84e-05, and at 305 T-140 and issue 489's tie of it by 6.73e-04 and T-136 by 7.30e-04. On 12 October 2026 the case records adopted this entry as the house at both counts, each case's reported and verified upper bounds moving together and each earlier house kept as history (T-098 at 304; T-140, which had held 305 since 11 October), with T-151 verified and not selected at both counts and T-136 at 305. Private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json).

Credit Couzo (https://github.com/franciscouzo/square-packing), with the packings each started from, as the author asks: Chaoweeraprasit's SQUISH Mondrian packings of issue 500 (T-151) at 304 and 305, improved by Daniel's fq at 304 and by the author's search cycle with Ellsworth's refine_packing, basin hopping and LP at 305; Daniel's exact contact solver wrote both exact optima.

Measured pose by pose, both refine issue 500's arrangements rather than being new ones: every square matches one of issue 500's packing one to one under a reflection of the box, none moved beyond a tenth of a unit, with 243 and 257 squares moved beyond 1e-9 and 72 and 92 turned.

Beyond the case corpus, 344 starts from Fang's 344 (issue 484), improved by the author's basin hopping, and is measured as Fang's arrangement moved, as Daniel's Hunt 3 344 is; 338, 339, 375, 376, 378, 379, 380 and 381 start from the author's smaller packings with an L-shaped border row added and squares removed: 338 and 375 from his 303, itself from SQUISH's (issue 401), 339 and 376 from his earlier 304, 378 from 306, 379 from 342, and 380 and 381 from his 343 and 344, both from Fang's 344.

The issue states that the search code and the request were written with Claude (Claude Code), directed and reviewed by the author.