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, Goebel, Schadt, Ellsworth, Hajba, Levy · 17 cases, $n=131$$n=131$ to 307

Seventeen complete rational source certificates by Nate 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 David 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 is below the ceiling its case holds, and below every other report pending at its count except at 209, 237, 270 and 305, where Francisco Couzo's later certificates of issue 488 (T-140) are smaller, and at 153, 236, 263, 269, 292, 302 and 303, where Evan Daniel's later regularized certificates (T-146) are smaller; the case records are unchanged until a house is adopted, so each is pending adoption.

Credit Nate Chaoweeraprasit (https://github.com/itsnaka/squish-certs), using SQUISH and its Mondrian composition method, after Francisco Couzo, Ryan Xu, Frits Göbel, Thomas Schadt, David Ellsworth and Károly 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, Francisco Couzo's packings of 182 and 180, Ryan 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.