T-136: Seventeen exact rational packings from SQUISH's third request

S3 V3 C3 Upper bound confirmed

2026-10-09 published · Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · 17 cases, $n=131$$n=131$ to 307

Seventeen complete rational source certificates by Chaoweeraprasit, using SQUISH, reported on jlevy/squares#481, prove finite upper bounds at the exact sides they state: $s(131)\le 546262522801617822590053861036247525257980632054448711125685429647597444905863823/45713647219586261662645849733884420493261937951141483500332220381589307940237228$, $s(153)\le 123797999524180008078530976954463155877532124256188212074430292658566758958161323/9617597300166052747813209874613658336103061290746859396882208758750786210512186$, $s(154)\le 784162558492446266593660385683137819091796768322490163735346888314210507575675725/60679277154496515024381071712664165344389803410694160363187109568787250164584599$, $s(207)\le 735531374822863515605402999283143325118010869312518967079087271050773613271161277/49412586952851805135714733794522095254473298746909591292001747563872946931057236$, $s(209)\le 352086775394428731924221642717430604621536094874634535695725650032320439933977965/23556905311577298619235189039173243928057287546251527278924911938419541052151072$, $s(232)\le 1114406289130974410117782886660181901248480109604626847722783024235385594726280057/70677745564963034331738560467960182867819656746924792855394670571264398270484380$, $s(236)\le 50320092412194411351092950431375593124874735166028004482989966459031138777005556/3171976347778525974082487945502584792871670535146640434897907865344648863834385$, $s(237)\le 985175957045048952983043545219609058238043896611796404750765221827457100311536267/61949105502245043847087036993442184322886619069992659024865961513616685288782651$, $s(259)\le 341518036803803444006790647841670042984586128471540050420967157601697370040453146/20584066845374091244555983652288807088661645599927706951717180608041569765032899$, $s(263)\le 432047778146841439029580692611188349728625751403534114235890133210701007973167612458309659642793497978693924904370413739637576535113305805473258454023835719181271711048486015594317660810620737576607410036454490449350514789375965798922408137/25819846521744160772224038706040075666707832377280732314567257517966255239622299571357684220252697011172510448203687628392777898509200720976473115105878476541217151532309155922966611211236819896892204567384190375033297252622850484642835750$, $s(269)\le 564564911646728951775552019108871435401015878808547246372660532154424222542885997/33403216161756613117962569049021857533205635354720854135770446125559372407730671$, $s(270)\le 161730688683764189720413661089911172609666557649301664556207086991163746546642589/9553029482827216570142998318928918813620116649281833748281870132253732451342836$, $s(292)\le 3761323845345218139515171916971344444825061926100378742572376994966931648579540403209467016041816040337818900346425264425539321771370681685330920067362079436395820610782571906200300814877227551992417206915107675733654665670276860831914801/213816326056741634814067355276646187722839206218411265487734233051189296444092664982158379275294035512776959997090425431151515816435932969086410258658043702048010698581133434199424606616799121971830257866615621151317801149797139930258434$, $s(302)\le 820999445009309740922029456921019877028889439831661589361637087272075109744743203/45937671990376526418357407123080370268809104789707150587775569201223938448317194$, $s(303)\le 162263195772134120155792974649693427899481189466645405582211912387896660025441601/9058371171012706945350582640902165821875350160207702968863672420698760940328800$, $s(305)\le 20660996865663684228769265396330188251261172861603842154544163047966189801381587/1150953769306329632110500736581956206891780101292689658665490428813909216741750$ and $s(307)\le 60196283485579249701801076299725636673865869456384222766836544824201358530035066/3347797278057252711799171413359076616946062908168377655562490378685987266056665$.

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, and the issue's 15-digit displays are not roundings of it. The author reports that SQUISH's exact Fraction checker accepts all seventeen and Ellsworth's check_packing.py each 50-digit export, and that each certificate was drawn from a 60-digit contact solution with clearance 1e-32, about 1e-14 at 263 and 292.

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 51 retained jobs, the seventeen positives with their duplicate-square and outside-container controls, were decided again here on 10 October 2026 by both maintained exact routes, from exact facts derived from the pinned certificates, and equal their retained rows. A third exact route, half-extent containment and exact intersection area, decided the seventeen 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 10 October review found no blocking defect.

Each side was below the ceiling its case held, and below every other report then pending at its count except at 209, 237, 270 and 305, where Couzo's later certificates of issue 488 (T-140) are smaller, at 153, 236, 263, 269, 292, 302 and 303, where Daniel's later regularized certificates (T-146) are smaller, and at 153 and 207, where PingYou Ltd's later certificates of issues 504 and 503 (T-148, T-147) are smaller still; at 270 and 305 the author's own later packings of issue 500 (T-151) are smaller than T-140's, at 305 Couzo's issue 506 refinement of that packing (T-152) is smaller again, at 232 Ostashev's later composition of issue 528 (T-158) is smaller, and at 154 Daniel's later packing of issue 552 (T-160) is smaller.

On 11 October 2026 the case records adopted this entry as the house at 131, 154, 232, 259 and 307, T-140 at 209, 237, 270 and 305, T-146 at 236, 263, 269, 292, 302 and 303, which regularizes these packings there and credits them to this entry, T-147 at 207 and T-148 at 153; on 12 October T-151 replaced T-140 at 270, T-152 replaced it at 305, T-158 replaced this entry at 232, and T-160 replaced it at 154.

Credit Chaoweeraprasit (https://github.com/itsnaka/squish-certs), using SQUISH and its Mondrian composition method, after Couzo, Xu, Göbel, Schadt, Ellsworth and Hajba, whose packings supplied pieces, as the issue's credit line names them. Per count, the issue lists the pieces: the Kingbird catalogue's packing of 5 and former records at 11, 50, 152, 202 and 241, Couzo's packings of 182 and 180, Xu's of 123, 129 and 175, and SQUISH's own earlier packings. The source states that the solver, its verification and the issue were written with Claude Opus 5.5 under the author's direction.