T-151: Nine exact rational packings from SQUISH's fourth request

S3 V3 C3 Upper bound confirmed superseded by T-174, T-186, T-219 and T-224

2026-10-11 published · Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · 9 cases, $n=176$$n=176$ to 305

Nine complete rational source certificates by Chaoweeraprasit, using SQUISH, reported on jlevy/squares#500, prove finite upper bounds at the exact sides they state: $s(176)\le 305005022674381157367132529416905207296020948646520876053762749858683599318120744/22081242074898135703849455266036121393870562834861102970956326327326447327968587$, $s(181)\le 249935253501271447722872166122379040683260708184410574609224560771762048914945913/17915453134884420632048333509167003378124582375369499899268894615896769573805255$, $s(238)\le 516829548917904437965878091074886828293320996185960477809932175908043454736320288/32455795222015655007007970530566514770773816146192500722754413705465356352461141$, $s(239)\le 957883664741088666359826291012097714608415772761965190010564315177693226164065987/60061263979374471914223634546066156202761470614164034734590750736057272204798672$, $s(270)\le 761203984658924994781689956746506132479479002945859948235062760911587068049389066/44966630820689093059794938239713705461989197412012889888820554168494459778549053$, $s(271)\le 226641509159806817533826968807522186618590497703207479506050881660303454046913630/13371877327550822127276902291903146156943895843133007855104572641371568124309293$, $s(272)\le 1114257766245565718930315998681473968716597475579279287292886911189991499936935381/65677527145799837468063782221943718992249135546143429464936478230690701220023471$, $s(304)\le 510485576549308947359639122176771480874060688471611521000826585202734096682803420/28471575075230807279259985772792709540176911602103069986239820873572704667056489$ and $s(305)\le 176574386584524522718242075735181894343653439251445905894414633471908824717084407/9836714596766635588546138051753839920746733217534271639970138607199828771763358$.

Each certificate gives a rational side and, for each unit square, a rational centre and the tangent of its half-angle, in SQUISH's JSON; each side is the exact fraction the source's summary.csv prints. The issue's 15-digit sides are those fractions cut, below each by less than 1e-15, so they are not bounds, and each certificate's s_decimal is not a rounding of it either; the record states the fractions. The author reports that SQUISH's exact Fraction checker accepts all nine and Ellsworth's check_packing.py each 50-digit export at epsilon 1e-50, that each certificate was drawn from a 60-digit contact solution with clearance 1e-32, and that an hour of nearby search and a deep squeeze moved none by more than 7e-07. Those statements are the author's; no source program ran here.

At every count squares touch the closed box, a least wall clearance of exactly 0, which the record's closed-box convention admits; no two squares meet, and the bound also holds if $s(n)$$s(n)$ is read as an open-box infimum.

All 27 retained jobs, the nine positives with their duplicate-square and outside-container controls, were decided on 11 October 2026 by both maintained exact routes, from exact facts derived from the pinned certificates, twice by the import lane and at 176, 270, 271 and 272 a third time by the review, each time equal to its retained row. A third exact route, half-extent containment and exact intersection area, decided all nine with six controls each and found each fact to be the packing its pinned upstream file states. No route here is the source's code, so the replay is independently re-implemented. The 11 October review found no blocking defect.

Each side is below the ceiling its case held, by 8.70e-04 at 239 to 1.00e-02 at 176, and below every other report pending at its count, the author's own issue 481 packings (T-136) and Couzo's issue 488 refinements of them (T-140) at 270 and 305 among them, except at 304 and 305, where Couzo's issue 506 certificates (T-152), built from these packings, are smaller.

On 12 October 2026 the case records adopted this entry as the house at 176, 181, 238, 239, 270, 271 and 272, each case's reported and verified upper bounds moving together and each earlier house kept as history; T-127 stays verified and not selected at 238 and 239, as do T-140, which had held 270 since 11 October, and T-136 at 270. Private-worker custody was closed in the same change (packing/devtools/house-custody-2026-10-11.json). At 304 and 305 T-152 holds the case and this entry stays verified and is not selected.

Credit Chaoweeraprasit (https://github.com/itsnaka/squish-certs), using SQUISH and its Mondrian composition method, with the packings the pieces came from as the source names them: Couzo's $s(127)$$s(127)$ (issue 451, T-128), interlocked with the Kingbird catalogue's $s(5)$$s(5)$, at 176; the catalogue's former records at 11, 19 and 182, at 304, 270 and 272; SQUISH's own $s(209)$$s(209)$ of issue 401 as refined by Gupta (T-127) at 239, and of issue 481 (T-136), whole, at 270 and 304; and SQUISH's earlier unfiled packings of 154, 240, 272 and 305 in the rest.

Measured where the record retains the piece, Couzo's 127 in the 176 and the issue 481 $s(209)$$s(209)$ in the 270 and the 304 reappear to the eye, most of their squares within a tenth of a unit under one placement, and not by pose: no tilted square keeps its pose, since each composed packing was solved again as a whole, so "whole" names the arrangement and not the coordinates.

The source states that SQUISH's solver, its verification and the issue were written with Claude Opus 5.5 for coding and research, that the Mondrian composition method was planned, ideated and formulated by the author and then implemented with Claude Opus 5.5, and that the project is directed, managed, reviewed and steered by Chaoweeraprasit.