n = 179 · Case Records · The Squares Project

n = 179 open ★ corrects Nagamochi 2005=

$13.39379695 … \leq s (179) \leq \frac{27790682139953}{2000000000000}$

The best packing known for 179 squares, side

13.89534106 …

, David Ellsworth 2025

13.39413.895

131415

13.37914.379

\sqrt{n}

\sqrt{n} + 1

Proven

$13.393796$ $\leq s (179) \leq$ $13.895342$

new result
exact

Citation record n-179

lowerKarakuş 2026, arXiv corrects Nagamochi 2005 (confirmed T-083)

upperEllsworth & Stead, Squares in Squares (confirmed T-101)

Open

optimality

Bounds

Best known packing

13.89534106…

13.89534106997649

Found by: David Ellsworth 2025
Improved by: David Ellsworth, Tej Stead
Construction: annealing
Source: [Kingbird]
Evidence: E-kingbird-upper-register

Minimal polynomial, degree 158

$8589934592 s^{158} - 16776142258176 s^{157} + 16266969590267904 s^{156} - 10441345150486577152 s^{155} + 4990890969418522689536 s^{154} - 1894890891956091860549632 s^{153} + 595233553645077573253726208 s^{152} - 159113050638468252758651699200 s^{151} + 36946881101748166646688276021248 s^{150} - 7570528011499805372416905315352576 s^{149} + 1385891578176852485661489685302607872 s^{148} - 228946999504531062403011118751740329984 s^{147} + 34413620162507686902871641974066772967424 s^{146} - 4739414969648759439001963325862742247604224 s^{145} + 601558903076466519248711186174615705104678912 s^{144} - 70728521883260905125811562082651513068453888000 s^{143} + 7737324864939840471669049009557872918683080196096 s^{142} - 790585503455216150483325645451760629989718481698816 s^{141} + 75710472593031465121679359297860921994821806424326144 s^{140} - 6816107669194451839326245571456597509717280087217274880 s^{139} + 578464353973418020310634721471147245527247969916282732544 s^{138} - 46392196301967821201942383879550938456841855963672659099648 s^{137} + 3523778022346774202729023048723499268217406682839658472144896 s^{136} - 254007557632703931702284138402948704046754094717702549410938880 s^{135} + 17408514944756592799011061062793080406734360236390718787664478208 s^{134} - 1136289408555428180648670583070362285374923700458618425908962263040 s^{133} + 70746350689980519032909089788896748172161993438186829164293644091392 s^{132} - 4207556129456759344308858415465053577041368877241068956738220497829888 s^{131} + 239354989863239805378215733640966714759031392886443184547834139634565120 s^{130} - 13039937701889133602522940121437467740380489339869710649632146169215844352 s^{129} + 681121984780387584834632874983009358436546556465233223958353011934178050048 s^{128} - 34146981518084020265286598813403020689873371210367006316656970694310255132672 s^{127} + 1644706783086263538695268413115181263837528839464397631913378383728624169385984 s^{126} - 76179115388511389338974966078823698047392964128905863436611198453092009978363904 s^{125} + 3396029760196995135414003157751932891401086851350859611839522194749793909454798848 s^{124} - 145830548160976026110107173781348917957600371659040062528734011247947209949832019968 s^{123} + 6036688398449164245442413664709896878159839227452502008663071643229606265751761059840 s^{122} - 241065003744893507570294914217808771866408232864440989017504748957474297075123445301248 s^{121} + 9292837675912940924990206604213586743959387631276652413153774089339334930583357885251584 s^{120} - 346033381587716048731439927283691733988482429528821553481170585146014499111469646522351616 s^{119} + 12453863096408680041169819937015904178551258329418542708340249530830191448369003863172251648 s^{118} - 433463976329682012516438503148299128977806426837989890510437299390776995591721276576617267200 s^{117} + 14598141148558474441489940471124243659727028033703700282193057362485627371545100422654973476864 s^{116} - 475946159678203763044019330301479855280496665057709049769636708941048408733535567507232605077504 s^{115} + 15029413454895856206745304655333305920339334147655697289316972268447305314445472965366575870803968 s^{114} - 459882887026930756111779196790484422309888489721984733293681816698747303334084015933748357602639872 s^{113} + 13641435856761237243270601890540721935285466013051320427467684656529581843459442845803621647070674944 s^{112} - 392425562908358813383988314609841618542004813430581599186965429400036128949453093614593364571474100224 s^{111} + 10952324736134402585630692527440145315759297609971129663107943912535605948752831894502037722046495045632 s^{110} - 296664656496686935491008198711516247552096799600597442599154951217702380384594259087430148235624233351168 s^{109} + 7801643682055627820367893516151454270806235476506774922390506271847870619103118170338703063269725788306432 s^{108} - 199255192025393249257281162633492073381264548839168047766331085822035220024893240940136840499276717715906560 s^{107} + 4943918829702069200344250555862617065881474793148741789066656917163448416852500326308433876888147427374462976 s^{106} - 119206223118371351740016288849065932977040679786443508334868736700308645062707993971803649427270223400500803584 s^{105} + 2793916168315879842843667884775778340310502447380959352704425599408207395601504948292393832756530514977101818880 s^{104} - 63669152659910654355612465707448636402220983316196460574685970889322290053994415590936999913019302314039117438976 s^{103} + 1411088290290660361761432231391106518395261428737833503161709379547625385506769563676428741123316171450744394798848 s^{102} - 30422219394618178272015665467316141149389049578610405565799548203573752318275023776696574468770663825110615009838848 s^{101} + 638168847016760942456272614874935437121589490396926535876115429490878101547691125109415495463730997511782606519275392 s^{100} - 13028042748128154152155749699183768405228176972983373957636735466263762707237542240576781695707329961351942799373862912 s^{99} + 258885154036933942104389818290348573419743137983733979188104999484694459390927539400396852964333594797360240836965714048 s^{98} - 5008410076259728530318102435875013929833377669045555053858177483366919404615981677268236834880846299261048710179562544768 s^{97} + 94348012061177049107563103911587540381920203938465825263786673258359983815634043250302695674806092227319985594633260139072 s^{96} - 1730917247479842371424720714737313481645564168800276072926831012342943828146405322912582371486785157061919246556963482411008 s^{95} + 30931172318968345000510391271634143021960225811419361444157407558765687432612455423022976197652777023487049412091056193054594 s^{94} - 538461008921247047828394096469918761105490138278621128660199756491980567891515598094143218731586738989389802898022630537945282 s^{93} + 9132859636016669776434123526820418683345820135872048991647075051302437776490106765500102774227913215416288532364098317544576377 s^{92} - 150940731034288104524482051243706919477841797200631340591331264060921614223744641783199564155639633612402903498528508886421692376 s^{91} + 2431092723225673407156571175693900356728276521082021803696057749846673716964615398053322487268141955153664079493766711190523367752 s^{90} - 38162439006514562511240781099631012794063423429713011149374407886305067715039446472926753544562508542073763984769313518943302662832 s^{89} + 583915896096747053996996323059960512964444147521784546544897874144632048416760777438475697042259444982417339227884159274650080566872 s^{88} - 8709236879750766882453681576852485364331892633869673848337497705630544792658670004136918434746607892296869871638170874996236176890272 s^{87} + 126636138296866694051122107014506879529305094181982590350023687952080994178451330962893760156800903477391893139844183119963405174731720 s^{86} - 1795193256764781321107143575352115822674906359815187230466757950120469384965265392123944015635359271580223475545363042688847493168420232 s^{85} + 24812193769633022975987552214679572383978205991624460286003447138677810543985474608435487975275222609758642157607327594001502315958459044 s^{84} - 334379885533624578703556863326346402939779650373080655995774830345924042811245868539921732172903825222436928921068371256649101399883530592 s^{83} + 4393928708722487075933679180578558733970880676526077339436660870457607763640605948580043549813023248207984231479413380427817129209389812336 s^{82} - 56301097918088298093454650045058048305325020437030743507220175402717433615153301169494298139135597640308031170513509287099895633675977621824 s^{81} + 703462752987507036458562735537680816881719938319905275526127357220208343202431074663388402630958620800512242482602361617306412697171465424496 s^{80} - 8570998212587852930175651144642807599951786012003763264443061638473783465625066449136977513186466774826722981798461971142730664365464079767776 s^{79} + 101833272553373741235032302509902465160175868842192714927716949983438287370371757517279729031084781631828701810210342629097929400737755064293124 s^{78} - 1179815793231266370256972220737728299964183673435163549122473591852114400578693651592966415854379188333769518428156838829477322470700035721028852 s^{77} + 13329055385894109812266802844685919670343403013442360747858954950844495765925245539508872533955803839758431941455121041463239188788384040794109722 s^{76} - 146837123904824194961234084974570501452074717596883096185219452417662074277013860164683394262203627903037851493658441532250519741481678709897246544 s^{75} + 1577286318461684922273318209313735614922436386787475112599564366916347781535514689507250142185177687000877842997498123282902359277154337054386341344 s^{74} - 16519858966798022599229567943966886561256779982094971639491094232964622132629929717576975546887539814573777740422627348566156477860275093467997371248 s^{73} + 168695030333672345490476203873790848962135254168618363264435313243401187568723479693115443208092732240292594851909463273732026761763340086761948475704 s^{72} - 1679474458507329989132517634608895062662146615211082406584834311032383549434953154993938330781059104297788687699004179007132153574177220111962080876352 s^{71} + 16300111868518456662841726646960774763587242690483536485009292949048056061811590179679854095935055534369473477448896348699504174866227241750841783371312 s^{70} - 154212840210987916676264223038826690901022059678053923252841299665320079807138162308528417005218056409018087677069249693344539425903311562254252837685632 s^{69} + 1422085276643028178778425151076781001937427561650672360326457358558871404320146285660611707536494367536547236126225485697484012110320900404072541279168848 s^{68} - 12780970467642192978187441726351951362520318627942008798985171811086894522838035240545206709707819980554105113258480364544355112997250795941703026370717984 s^{67} + 111940809741399705638022999961087604053018040609108662108096247241336209302607133844273683790386223916878034641986559164044622962415217366546184744164865568 s^{66} - 955318185935579482847268208738986472032834002959414491513714703039136505391355303639528303526581170698399400387573804183334547249810226118890719377712183184 s^{65} + 7943030782933702910309362522584013062116396883034105859514763434357789909349630658556763168546741067520875540601486521938431840511417519419122375560804319064 s^{64} - 64334138351583170363889596947395233708333474902969306963812792711731280614703024068111601694276554925690614502238860859976387485281899326574007044293786236384 s^{63} + 507512583111608460086839337442881927840753060560625125290572604800021418518043074394586176931570009271593070042196264568576721782301148922151946459147790396230 s^{62} - 3898787781847780646276537706916970267887504154704503277342337527269522862362538935230206512081319538034556915315628136747611283623061769350835642507685971395430 s^{61} + 29161593207696952748858569497038738633864241945330333111920704065412119348848939985430075319832794334869879850023996708247535549059156491824231084935719206354971 s^{60} - 212327816319189028259724642016863359179101718891007890445995348684380591825989430217119720095336385217052124416275470698676505789265779131434375136267188640787400 s^{59} + 1504613685680622928062684851738852521699623217444410922505492926627316046384665914443298803561233484471565375893068194989164077819527388864682880176929665862607816 s^{58} - 10374495978653039224900586717322659598924065468581501058532564764516235154332510451370434730693018233787659539517300393791945591479343070702368534998999344108290480 s^{57} + 69586788632811273065880440857571673430182048904384069580581465300690971670881871386329863598347135979814888173911686210145901437773492986722572958949884938558673768 s^{56} - 453931434929098799036346880318614334808820527161473376101595395453715920619047285081975776494727202633682274132935404164912004820600753967118097473221941591899312192 s^{55} + 2878953845232776069724165163798080655771235146230635673716229488752549751883322682507475788177696477148785445008548650389088757505825693169364441722503533776930259824 s^{54} - 17747133211231081186020043092129133596568924129294197432762179849290339535804597487325286616800428132079269986488220677465538306132998828608842356427062957473637984000 s^{53} + 106298938781235005033783825206664909121067385334029443637725727453398860092346992400812014116921224424390977686250900639381653876848057898341352964873569319121274310608 s^{52} - 618421059987346562205331844645611295608666899524955336451646700445064855559315763556503838688528298133303431353919628274502518545840093615313610689191718690065428919200 s^{51} + 3493252598343138684725379280294529025663041268268663098124161429505696909077797297579937719576603106572019707346484357015315696094226476445765024188822166955175545946848 s^{50} - 19150898362939019687741456040149998560840101636132958830057082356317834890039642650905505045931231679884411907336272321812637948896868808109672988709192215389688565165968 s^{49} + 101852321984659845942602342923546106452477375909272461162717281232732400959184326943349939862470191499803085832595232246501855523840908519928878529070236316642292423788440 s^{48} - 525253884707094334372891427152678439636426903007308698568962983777355188441475650670423265286342547295205257609844840538357356477294911928074969513650229082774150320638048 s^{47} + 2625193203290356698262450205010371464079524903999964417865706032783944977742038009970099304395843707487398278449143291747574513077036910085455633836350034810273383713437748 s^{46} - 12708823750482493187188598190767476486252220480788869294291419641427489001825169262349605803034723573305682537014575285284562182157268595839133188599588427516690027607998228 s^{45} + 59557863972403655477291387606109743155567512394508495032584959434277016096739927427783607375573762884883712095405392675451610486590099230637925924995287778808445556113064986 s^{44} - 270008046062142760207710036247544746566989399937962965397599077559850441505065550747849379029923963172957834617405209369141198613136752538968430412127081258233026227567885360 s^{43} + 1183329783419536786380446086798792653054231697579009821534008929448031739826391709555907403457308310239450405563041283788559445414410569282162432883832594071409901980491026304 s^{42} - 5009377946382934492951891438657112520490134101510458720918660383762623707624353977944431021337053133786461830304904970841993886811209870141590878040728725157139984341542676864 s^{41} + 20466015682167777784482409754813891351475255134551784541766314633386295404855433659708371621362016365511924843115540450520158558390063757031486278410682427355900846056301959440 s^{40} - 80618802788478801812113956448801858169760062126852541696583824016816365631185486573237140510493483893431818218129521945858680534645077599882790721875793228869383195105938712864 s^{39} + 305863181510707020796864206645639951559230694567516088400307001865917921423814699201348924993927590766481670791584301723407430089406550759459868156719995234342132704958194591448 s^{38} - 1116305608449436359994426301682876058731371254632069497730924661756685966554126060860272548220325479142733201127016703776780327301788452348400564519240088875923462690145277753704 s^{37} + 3913921262824404932386379884293482829949414222433797156681053497348824096069659972991230862132072232542669262653591674098226456522738507109237187421722427027634264312767322722212 s^{36} - 13162447759860117579452276931166392949635427499280533053982657744727104904926517415600785364912363799929799564410480896775637977700272139782122047786558597026755032738914710946816 s^{35} + 42381028778832802130724925499140689504743999563640865068530866151776504897678074062413102747959378140886249688903771627165677627416903951331803764699931147706748166334767700290736 s^{34} - 130374244786365303533912603888323228860606865444899716466401096936872712035174753385389915034135664463044358857327447110663680301999701563098937294602682140364743488206920843598032 s^{33} + 382197032949394189178438367703984764850380199772849680095474612657148954062174074429597893167434425129918377877814316382657272098143426555605347319950669755099449717059564957096168 s^{32} - 1064369540939183336390645737213177541263692917551551568617483388375256081481556238368798331359255089005356720319340184400506576835709926921735722909417483309191587353758825109294976 s^{31} + 2804630538574925138948362578136124986676638416793243440041826000460440084268512821417365339428475318447675787601313928696781699895294756249615882175002369161850902914924364200301426 s^{30} - 6955912056712104674655746067095315245583491997704302852851092361217738104668925140443867201627487726305438713352075838980489370052594514752884038957148318924447385927488650193895010 s^{29} + 16119840293025713917214412293931597320418894416753327349949267344330058931398191398798517937642761255823057091236402329733864246582020109686974984056761485928624856223291099951221625 s^{28} - 34529662771230707335219951547004030585080610222905335091328080194061372395061136082086053833985158036194324503848361810477393426760591759650818525227917806075879443445531848093739576 s^{27} + 67169055368257864319128414339356775300043063578236431356960405551833718297330790976406869923860562949880809788167714167900627484688138315465217490175154691109684703543933115769160872 s^{26} - 114778685778620223839086865928090354816098469519749211320936107538530928492179950424220874680209210464114735532032563820421019696128479408069793809733151116471117578741376954495301952 s^{25} + 159235638455706689877934267032866081190823324973675149152360435155053371178290375883137334363677967575709371378860921248123897871953929083286155301492172315187729412827269231244920624 s^{24} - 131542323925706316629128197557594269618805403538317928284971716553722090873226868534046188609181859059611451286240113088194202100678856792961282312998011830262863606202175281267180800 s^{23} - 142660249894157894409972547913798523203158037238125212155845532630379649209441180149564447436919739560050209063465586079852375921314888047469260238605951130174591305485507436203042048 s^{22} + 1008661865115849958513767321241593592329412385166707806598124421362676357752483162036782643640297262506647050448859943255512975777860113697822690588875400866031110228874621330164183040 s^{21} - 3007261213018720506800235133849679103019573922530862607321844255364670256181166803348956200055137542739821136381031406313829275096154428424719024921520421435546012508207646640577249536 s^{20} + 6741638045458340935940448099222774160509926696413703180829587751343608319231759125592470337149663506502459084854434019653405345835626394104294886318874569582009463074369738693559470080 s^{19} - 12415194868454379694624803347053847857354696324576099034444888432624109282684391487813867525328074007123884726429845549620434601441717700273644831110306764005641929836405982289518495744 s^{18} + 18912327531434227968126301426775111664049753587327662484883479348512608440038529839791826133269537869529951889295998328607931440201869505365908891623384751196908590964191348087266992128 s^{17} - 22555220467685618549001626132531046324462643663098399408174221404939811651699166939188993416780626226433562655222891238694375355180975084305370036243118488892446654567328239374760964096 s^{16} + 16190209997928458889610778308307298943345961079357980070663761335953465993151777234418801948404188403314534553041996296126673832235534697542840499139280655615545363850249014350700478464 s^{15} + 10186786573107656635876002348891288810726574278460373991353535193537084148170957344605089998502581888699965247359086622894813846359008311981746376767960255767618996796074622965734539264 s^{14} - 66055289154901041839305855845398190514578106498990526240315687057829584649207467787151011269577701196580392100600933895501160386571678026919404634971582161711267651582900968996317691904 s^{13} + 154272332902414417625527793867509418022597948893456044924207523590474448029230607827485471212309953858487419681176785646045767311505748119432956931357786859367431403877927646576869769216 s^{12} - 264750300193460123625509966522497698916626540275649122972508156757570068188342070968135491559071178211170578364022282645766071110157758717580607352294181171444696237363255168567226662912 s^{11} + 372637045702809818019956695230799386980616880115292869147974648188413024505309040219603956884496837527618330971224129070588663215624985292037426916385542119372199217592922265722466861056 s^{10} - 445108205902035490389068934556525278806445559820918024734079261318268060814828194019048408531378412479896192164101065503321521852109767391237208075897015787989933574203535072703262228480 s^{9} + 455941502826910846456883319724016322865566494574327308049951589651087650403134579068551051026723061850292025841478508936933382940660206794837925264732267526917515274771492564700210659328 s^{8} - 400174736065040893587953890168534492743140794357657510562749858373138362000240223027932761443670764566760571041631548571904984931589754683308903830370578760751418861289721870348588154880 s^{7} + 298395443616340795736231002707282968895056803093837090634648703116759694141097416687300005971623239493424647188212450846843989319706271077488145887124040889317215192104456493331801702400 s^{6} - 186077384087780498454065214443252934106396073880797597959161268786366223950285624616497833365846218360268014302757513586899417029472601612265275184207841304780464883310755378857246720000 s^{5} + 94670606710498192779138970159330451351125073315327460391504581428936560180160205930365988135687718677952046240678814024966805930082376978355299988028794268001122987013203931567226880000 s^{4} - 37820958812012669251005314092161320438332248209788004413338033377757118963927661731636995619967532267657631052780606592883598271642325054552800391554554762759904322188627983466496000000 s^{3} + 11144498831733182755522712247376107124239295733222214585318072764848537778093840266827624919495047571793505943540962033070488140464264470696139346705455241289383409660649648685056000000 s^{2} - 2156302551432108707595489719112839655993263985628799748070841471621176447110342622734176723933151197647769367339743337662119808690412416276021148597236129232783235683060138639360000000 s + 205719605048717722894599553843731199566757178785380699723860342870623038754205750722783581387888597281294249872534381957173931243028773292519535461024555523620754511671643340800000000 = 0$

Verified upper bound

$\frac{27790682139953}{2000000000000}$

13.89534106997650

The reported value, verified here.

Evidence: E-evand-exact-ceilings-2026-10-05-exact-replay, E-evand-exact-ceilings-2026-10-05-source-replay

Reported lower bound

13.40967364…

13.40967364599

Proved by: Hiroshi Nagamochi 2005
Kind: Nagamochi
Note: General closed form: $s (N)$ >= min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1).
Source: [Nagamochi 2005]
Evidence: E-nagamochi-lower

Verified lower bound

13.39379695…

13.3937969582

Corrects: Nagamochi 2005 (T-007)
Evidence: E-karakus-strip-lower, E-karakus-strip-measure-interval

Gap

0.50154411…

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; chelokot · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$
T-101 V3 C3 Daniel after Couzo, de Winter, Ellsworth, Levy · 2026-10-06 · 77 cases
Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

Verification

upper: replayed here; lower: external proof (read here), audited here

formal lower differs from report; reported lower: defect recorded

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 143 of the retained witness (witness id 144) translates 0.895341 along (0, 1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 10 of its 179 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Evidence and sources

9 evidence entries

E-evand-exact-ceilings-2026-10-05-report, E-evand-exact-ceilings-2026-10-05-exact-replay, E-evand-exact-ceilings-2026-10-05-source-replay, E-kingbird-upper-register, E-nagamochi-lower, E-basic-grid-upper, E-karakus-strip-lower, E-karakus-strip-measure-interval, E-nagamochi-lemma1-counterexample

[evand exact optima 2026-10-05] formal certificate
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey
[Karakuş 2026] lower bound proof

In the frontier survey On GitHub

$s (179)$ — open

Exact certificate, 2026-10-06. Evan Daniel’s square-packing published on 5 October 2026 an exact rational certificate of this packing, decided here over $ℚ$ by two exact checkers that share no code with each other or with the source, and by the source’s own two run here: it proves $s (179) \leq 13.89534106997650$ , the verified upper bound (T-101). Until then the verified upper bound was the trivial grid bound $14$ .

Open. The best known packing gives $s (179) \leq 13.89534107$ , and the strongest lower bound independently verified here is $13.393796$ from Karakuş’s general theorem, leaving a gap of $0.5015$ . General bound: s(N) >= 1/2 + sqrt(N - floor(sqrt(N)) + 1/4).

The exact certificate

Evan Daniel’s square-packing published on 5 October 2026 exact rational certificates of the packings this register lists as best known, solved from its own witnesses, and offered those that do not lower a printed side as a replay of the existing bounds, asking for nothing to be registered from them. The certificate for this count holds the same 179 squares, each a rational centre and a rational $t = \tan (θ / 2)$ , in a square of side $13.8953410699764907318 \dots$ , $7.3 \times 10^{- 16}$ above the side the Kingbird catalogue prints, $13.89534106997649$ . Rounded to binary64, square for square, its pose is the one the atlas pictures for this count. Evan Daniel’s solver moves the binary64 pose to a nearby exact KKT point of the problem of minimizing the side under non-overlap, computed at 80 digits, and rounds it outward to rationals.

This repository decides the certificate exactly. Converted without rounding, every pair and every wall is decided over $ℚ$ twice, by sqpack’s exact separating-axis test and by an independent checker that shares no code with it, and the source’s own two checkers, run here as retained, accept it as well (receipts). That proves $s (179) \leq 13.89534106997650$ , the verified upper bound: the certificate’s side rounded up at the fourteen decimals the catalogue prints, as the record writes any certificate of a printed side. It says nothing about optimality. On jlevy/squares#375 its author wrote that the solver, its checkers and the batch “were written with Claude (Anthropic) as a coding and research agent, directed and reviewed by me.”

The verified upper bound and the printed side now agree to one unit of the printed side’s last place, the precision at which the record compares them. The printed side itself is not certified here: the certificate’s side lies above it.

The packing

Found by David Ellsworth in 2025. Improved by David Ellsworth in January 2026, via simulated annealing. In the catalogue’s words: “Optimized by Tej Stead in June 2026, working with Claude Fable 5.” Its side length is algebraic of degree 158 over $ℚ$ .

The lower bound

The strongest lower bound independently verified in this record is Karakuş’s general bound, which applies to every nonsquare $N \geq 8$ :

s (N) \geq \frac{1}{2} + \sqrt{N - ⌊ \sqrt{N} ⌋ + \frac{1}{4}}

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, $s (N) \geq min (⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1)$ , which is stronger at this $n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

Source-reported bounds are recorded separately from this independently verified theorem.

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

bound	evidence	run	code	programs
verified lower	`E-karakus-strip-lower`	a published proof	no code	no verification code
verified lower	`E-karakus-strip-measure-interval`	audited here	independent	`V-check-karakus-strip-measure` (first-party)
verified upper	`E-evand-exact-ceilings-2026-10-05-exact-replay`	replayed here	independent	`V-sqpack-verify`, `V-check-rational-witness-independent` (first-party); `V-evand-exact-certificates` (first-party, premises)
verified upper	`E-evand-exact-ceilings-2026-10-05-source-replay`	replayed here	producer’s code	`V-evand-verify-cert-py`, `V-evand-verify-cert2-py` (external); `V-evand-exact-certificates` (first-party, premises)