T-195: A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155

S1 V3 C3 Upper bound confirmed

2026-10-15 published · PingYou Ltd after Schadt, Ellsworth, Daniel, Levy · $n=29$$n=29$

One complete rational source certificate reports a finite upper-bound improvement at the exact side it states: $s(29)≤24534865107674036688168549876397025785476132874112370099771467927504056975875987776681455985875024244254213572886097261458084288767455559197653938689488649902066228662684853391396663327014033932373723477506069710425226218750526706135891585179797517343019488443339867249715360890013756666854628979176900944993305442261602772842180838440053667420917480395180220066666602334511722453879594972998191138612834034189666510059905050426568116428378640731543810807747891542284864038369907643508459843509205620695587608368557533498173488519528447981793482358600152308521631459565592767014183052501199151931076562879250106887616015278050875978119717926899218680011808702390993996536195830670260972350387443818823426662369027496193945575599138014543670320983369783863929251334778825936608589267984001537304179324158789246196108802236193481133294548709295335474346256527544613659984746868470460857727484675678537346741164019856884567986913934704956927321686209550808604138457927403334411832210594036476801938126308790969489032002021963023705206483679471872149687890566611443899435653214191799667499764111469459385033095127931344323459448933275284125181277683793104256050531452804725334403790564783668180540557925744070464684545649032408833632838438910661813484030778674222303675402958374138153085439248296679730701126691111630742632653637849625799540379878830607499958400806445626950768552900180049729020839822237918053913692758374445410695986201178837254745307861725778970345267181802138568964810374228407899464496680162178116797606104631259924089248154437042373925033307272942769766239315424518195846205980008361535060213644205099149816040655058140387176369175347126651923289738983179736034453072062298517733929444973959326041866213931560671058617171889088120835739271365353588567878531477697526577302776627625153488856099874787185474253160414731224686151501675402728352580934099397733963238715051627968686911414459275184683424784163644191544707377182384601048542578683951517983159198963572073487749781532348390081793066311053790977197895164959450212581707715858563399258411304003599006319077167663800875019623826094756719222309800908141246359094069406089611981416987407570927946930756422932236652442311952880289753229889653917790471125660627044950535988555435869596551538259114765832766790488411832484772360835487671284213135964427186102124863000233796141278872662965770217771690365276248657719966519254396208/4134741101514774825128795556716042227961798785923906182461358460239918424782567413226407211403203065887739199609118084844533471798679416900735032687473222139680760065590940944078234018015370334443128772607933060444000826617769429646882194583220599963273124225726071777932738764595777987721482365783510337019481236506878836287721256689347799807863951388092667648197563925993938390556531505228608954855292930046726399665713751905730833652017210190663827825661322365063724262883246743380492604927786523721825931831886303155063551654710154578169238652367383102078792000130468658641539801741146889224943442810013209072949461457217339392927864030591169892883726839884880594167460487749243589056331690253699842043479578088884954988951340936860924955322743194977380476502898903410854524818203652027795198144692092594657450033887027432586825094110442844464606853681664319033486389387445736214034648895232429305200383123921501470838824817050243022153810427067912956942772219391824845560443747547400843195049198930533146285574982530233459642803642285852596410633979360461440217163898024968451883257550199336829150031495720132797637319132645987063404501212699321705824766887053681261752487880911742084111713948715586118826460029866243412955813075255128613682606937176468037935097555448704569843358216916775243025949351743902174698612665846213152609786378184952030306093106808751354257494437791937397557320503668105995208209305535886191424772995183160168155795973488716123209693421756294653489941963987963470670675939442599780713172144359901210230692152439814696253554319377531586894873990381499235730768339867996867328907448233810242342899234366597370788054658934954410215490093061628299835361415650502735274254384722944822121189889481733221480599225541033150521791602097236911676058943766150551365114305649138572343660857535228979083917905543053139843750484912743933307704134563365828870127440166022411140550453711645377040805715862280799298958039516958431092020836360669755785671136319313346695819974050474113917337370449084604146353831484738751871298385136659525010746969348827942534339711929337559777205045248731333573416596885461197747132145825993130411956264953672744857178743458111168300805276706096833515134514702136577188472029912623847531283031692405241550695989903352952233750694266905747073315282264735651376143483435574343954773982361734961798348911999812513553974602740493739669761924137585813615$.

All three retained jobs, the positive with its duplicate-square and outside-container controls, were decided here on 16 October 2026 by both maintained exact routes, and decided again equal to their retained rows: the positive passes both, and both refuse both controls. The source reports running those two routes itself on its corners witness, so on them the replay reproduces the source's own runs.

Two checks that share no code with any check the source ran decided the certificate again: the third exact route, half-extent containment and exact intersection area, with six controls, two of them 1e-40 and 1e-30 past exact contact; and the 16 October review's clean-room exact checker, from the release asset, with four controls, two of them 1e-3000 past contact. On those two the confirmation is independently re-implemented. The review found no blocking defect. The side is 1.57e-36 below T-155's, the case's verified upper bound, and 7.32e-90 below PR 574's N29 certificate, which T-174 does not register.

The author reports that standard-library exact routines of its own, by separating axes and by rational polygon clipping, accept all 29 squares and 406 pairs of the certificate and of its three comparators and refuse duplicate-square and outside-container controls, that the conversion to its corners Witness/v2 is lossless, and that this repository's sqpack exact_verify and check_rational_witness_independent, unchanged at the pull request's base, accept that witness. Those statements are the author's; no source program ran here.

Credit PingYou Ltd (https://github.com/BuddenPing/square-packing-certificates) with this certificate refinement, as the submission asks: an 800-digit realization of the packing's contact system from Daniel's source tools and data, scaled about a common centre and cropped exactly. The packing is Schadt's, improved by Ellsworth, as the Kingbird catalogue credits it, and the submission says it is no new construction; the count is a regularization of that packing. Measured by the review against T-009's witness, it pairs with that packing one to one, no square moved beyond 1e-9. On 16 October 2026 the author stated on the pull request that all of its submissions are 'PingYou Ltd, assisted by the PY agentic offering'.