Values of the Tracy–Widom densities $f_\beta(s)$

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Numbers

$\beta$

$s$

$f_\beta(s)$

$1$, GOE

-6:

1.455451226520047363437251319463873190851962993994388857628362883011648892788145642745597322699224585e-5

$1$, GOE

-23/4:

4.936000762483685778726804573323902403190946149166366135457885830102034311978409941111006871599474718e-5

$1$, GOE

-17/3:

7.256504786905663818082325948488938962520272673357672877397820661497933177325804700320338929603982312e-5

$1$, GOE

-11/2:

1.519226917709834292869905307838183789878604877962413668273498873999840902835706302567132184466761700e-4

$1$, GOE

-16/3:

3.051361655749787603852334573664043153335253297273921673260971492292196955379562690364413310444476901e-4

$1$, GOE

-21/4:

4.259075221392643278848277378274186463872627454481578061840961504646304706479343642553473283396031422e-4

$1$, GOE

-5:

1.091485588001369051843323250644817058880338501928684009673305866756974622731344012392005645099478645e-3

$1$, GOE

-19/4:

2.566159819712311422041438030531883129550340698470164095122789315428703347563569642759762646490201677e-3

$1$, GOE

-14/3:

3.349547447646164920568421726585681346109893494368770299509359929141302041071412840834534393705132052e-3

$1$, GOE

-9/2:

5.554591301729867440769362186388184364769787218862943562522488163459690648439106204088100149096039060e-3

$1$, GOE

-13/3:

8.892866800403064217168218901132691990068669057444460469805517248022347613056474094322426148711644350e-3

$1$, GOE

-17/4:

1.110831218515906526525443143903109357655654837530429777338414369608447587235803634109322287847292231e-2

$1$, GOE

-4:

2.059585111570425007765902145970192553406892899575325672122297322425023052200191414752072608958095807e-2

$1$, GOE

-15/4:

3.552519646636897291367828884368648415934706703854829598443285852516653495588140319313741823757357427e-2

$1$, GOE

-11/3:

4.195054747309244338750089448812196391401215294734043962579253429933153402387382017365392570802045756e-2

$1$, GOE

-7/2:

5.719875864812356185483259724075582264428412258673228933626423776278569797436515053543474120790304514e-2

$1$, GOE

-10/3:

7.575064552376968957613990587131533796907271578935869458548659017675130269426644529858858999493037856e-2

$1$, GOE

-13/4:

8.625309947146916110704479152229828885606598302725420672523506296194099896002305983969763998473119374e-2

$1$, GOE

-3:

1.222134180212629925505522725249014603222856995689805782784318907814267076770178043801101307555882766e-1

$1$, GOE

-11/4:

1.632328435395710756839430710323285655203810214446361507916592068461699188084083672753452891762544302e-1

$1$, GOE

-8/3:

1.775161259550843173068287375019261924651164902253512888874157452550426772317155066021387837124919693e-1

$1$, GOE

-5/2:

2.061560086076730283945062957584731941292544532030696927026502772426269075015728972613243544209039244e-1

$1$, GOE

-7/3:

2.338519390034850395799706948871406594922160981807727994179099098579090214098278082634753285510896041e-1

$1$, GOE

-9/4:

2.469464892671878342877361726895974789585939955725855616662077064640728133290129649422739129504976543e-1

$1$, GOE

-2:

2.813884309026728262530992445624393336007726137352247801418401974156464966322573369728271817277298776e-1

$1$, GOE

-7/4:

3.058721128495300865707430740931058865040104636326489065862167011271765194500843371882861018106046602e-1

$1$, GOE

-5/3:

3.113689489820206201570128135418152392725745419583720988539069358103645478252375379176505914614275347e-1

$1$, GOE

-3/2:

3.180455427479075405187995578113789340067783574559237543690024131080630063639038425063148940520392862e-1

$1$, GOE

-4/3:

3.188875253480241505420134172644362828598440498502352923181617119510060412002615773412260096336139313e-1

$1$, GOE

-5/4:

3.171671067835501155381026848477971323764080117109750813563663389849598625763550584037491189293879699e-1

$1$, GOE

-1:

3.040987842301569656906845412083986501564192377973312705766917391823783110526976481554274418429363798e-1

$1$, GOE

-3/4:

2.809888292217111459472904712406196321577424446668156797209479365870996335427573432536536056816923391e-1

$1$, GOE

-2/3:

2.715486595957128954400728784554887711789782285099680049585817679221543129885670767896745835996175707e-1

$1$, GOE

-1/2:

2.507672720191265963843228431739114616443135066379187150989702220760900900658624811690750635462613891e-1

$1$, GOE

-1/3:

2.282372043741205860821774480080562331368794506452738469713121153272719137232757014689235800705856278e-1

$1$, GOE

-1/4:

2.165995879614159915727261066401097016112659190849217058952880452821955578713797846726837894804307044e-1

$1$, GOE

1.814195712213347428688072553938930727564310049044414173857271904848354394273617395688182461294799243e-1

$1$, GOE

1/4:

1.476123082453306936865303079974377725738278761173003313012139511492737389569603493217877743404706959e-1

$1$, GOE

1/3:

1.369606913847026751398761449569625871408430735658523640787402414935784680174977536983913388968994854e-1

$1$, GOE

1/2:

1.168647716591365902284651558037912665877605428927604623739857007041214634987366856526671058724230295e-1

$1$, GOE

2/3:

9.857868020973742847987513039145740340708585760090571766021301201747432047951022314116675930769044357e-2

$1$, GOE

3/4:

9.016092643843775420372548753684211729380276781875990705210196167174192613502984897681920504239682786e-2

$1$, GOE

6.787674295730346335305800111322683455976915511121104183322285525716633301879695250583949824934840999e-2

$1$, GOE

5/4:

4.992660063117014861579251265172792663277128032483740685840375336471327006355666809337224242126652605e-2

$1$, GOE

4/3:

4.484484819706389600878488854995578868013284151553747227852164419186473741790758795443482498937717997e-2

$1$, GOE

3/2:

3.592045928681086472999271594412168700117566106100543236355549180013980787219229473187176676463146790e-2

$1$, GOE

5/3:

2.850377950172828496811503743610991161429044141114804398442844788631623149741369995729793703908747962e-2

$1$, GOE

7/4:

2.530434650552356102271049188764590160602939080095488658456358001964046728300406110919249609777754518e-2

$1$, GOE

1.747005419767500403493079881299346456356465339319924390373940136269407921637578485707153436645566228e-2

$1$, GOE

9/4:

1.183052074594540443536224162882879274675322232791895839652156844269729532078410298150071265502324026e-2

$1$, GOE

7/3:

1.034588006432722980519088990480202158106289134916409214147552935702543380929282220129453875077718636e-2

$1$, GOE

5/2:

7.864199713254346931952935946137770572746096420453738704212407516271359736736399885045823817384304982e-3

$1$, GOE

8/3:

5.930534939405330122264878419118576922206336655234459516722861387015733181088796524083173233463321234e-3

$1$, GOE

11/4:

5.135071570766526305773408847838838040268866593251024374494295716812206152652234951285097828979058314e-3

$1$, GOE

3.295740689215113209177324602298087810238818119104982426139857243398872690035521842686454228638378684e-3

$2$, GUE

-6:

9.582544316852528850997891973427566253916114126499482785831619256231855822159486364505478972079405457e-8

$2$, GUE

-23/4:

7.655777508137332373426670734338735379626974009770218620245242665295206534194209265697345607895385710e-7

$2$, GUE

-17/3:

1.468881520637197664658757867884018369866233975573437966341030811778740228105113154188813571616502659e-6

$2$, GUE

-11/2:

5.092710769694626251578860107142167630225640022861000936567038633244424758690741349834173801211186767e-6

$2$, GUE

-16/3:

1.633067747617382647182158225725288372242103071344383668751872504175725998253218261969327001215599794e-5

$2$, GUE

-21/4:

2.841953656365569899956529148519558404156099087213389845949458713005653118376127394454915511738182158e-5

$2$, GUE

-5:

1.340391722347746436271845958504624891908203083962147474490996094505514368608334706338503085193615879e-4

$2$, GUE

-19/4:

5.382820603138880839792956785777944816684648272152663222167954635571452814754802249997010412154189505e-4

$2$, GUE

-14/3:

8.266133046115754108868762360501282240674280252563889354614777329655862026099406529670406521414034105e-4

$2$, GUE

-9/2:

1.854139235485122303358689700095421388981613648148802024252279886508975247589526600638382233389587191e-3

$2$, GUE

-13/3:

3.897330870464179746192627965136344519310461778290810759349150133995023675400034006343171872660813357e-3

$2$, GUE

-17/4:

5.518096791321953576252846685800941543412202281991619462769573689855074891140168368795201326463887131e-3

$2$, GUE

-4:

1.429140060174864377742242721154322333081023526781070031114213589919960507206948088365330255596694867e-2

$2$, GUE

-15/4:

3.243991532992658451630374554115140119986365483910890257992262645719555118425307975448423860109690246e-2

$2$, GUE

-11/3:

4.145189017001896316907580037306523199113890302662368868499497062314517073902764238567747679944485557e-2

$2$, GUE

-7/2:

6.498812807892189335582402793330425516927310062214347940562803323116597485005438187489251794734750359e-2

$2$, GUE

-10/3:

9.667934722479323793938922643204481808686352174588603535864034105229993167407739267531080184412128769e-2

$2$, GUE

-13/4:

1.156931297503026598507192324026984312908821189412639537943068212721417805341812480260169282292688189e-1

$2$, GUE

-3:

1.842466838283594695788981956153168168967432563328895807750280159733075809363669972022366665964429441e-1

$2$, GUE

-11/4:

2.641969351555450753605030691114298577173217112994061588885809201729888423286374302844566912521371685e-1

$2$, GUE

-8/3:

2.913748823144237740401690216089797721033481758140805471981997888185697952361432075556979673129229166e-1

$2$, GUE

-5/2:

3.432536102928316045109630003014922755126923364286894337651908260810122680405614042866806249103793418e-1

$2$, GUE

-7/3:

3.880637928088471825833965584374155767876395277885164704862772307288872414002870133466386389345360463e-1

$2$, GUE

-9/4:

4.065165205247322799084763610048161276991691058166893964616309416511913574689831689425804999807457015e-1

$2$, GUE

-2:

4.413818018617784019278943743248877347457575865229842371140964942792571096683667752745433775026497422e-1

$2$, GUE

-7/4:

4.417643322663529844798436783431307619054347452716392182077163481694264161588653775887542696882934990e-1

$2$, GUE

-5/3:

4.343574985983663135038747170749455353652365011398946168532844893490186648487164538539515343221784318e-1

$2$, GUE

-3/2:

4.096693790107232682905305342360056779194277114707318900774907774238954147616072921318857987677446166e-1

$2$, GUE

-4/3:

3.743009384296151894564433598552871022168794032073468415408245319628594424508195269240417527198202967e-1

$2$, GUE

-5/4:

3.536894304078159326335005757473026345344818373572498323697393591883077218455481317121518286605907819e-1

$2$, GUE

-1:

2.855509382361543181688652420763469608511334429577292700066556736837451205479354377736140990194103298e-1

$2$, GUE

-3/4:

2.164672817267012509966367732913393415198477169429537791987506154125986377502146797219265304878237736e-1

$2$, GUE

-2/3:

1.947608209755108250805559841026581602182627982619771873214072713287217895104204694285535321525449837e-1

$2$, GUE

-1/2:

1.546576887212504830359790009214866573215291153792948442738056519865678559870829569454424282546302860e-1

$2$, GUE

-1/3:

1.198024666893683533801214917167908004399644286862433390770812459774817834710873911748134177169302939e-1

$2$, GUE

-1/4:

1.044958634165485264780943955326917445265686051462987377330612721087071761004706698690267327683031006e-1

$2$, GUE

6.697530713277931168006697499081782218644682922935036229026473722270983305728277396695958654683514498e-2

$2$, GUE

1/4:

4.083470703528962609425730745731527811609912188487345917776313883997492603027013760201165504478146242e-2

$2$, GUE

1/3:

3.425868659266257392631123227766154266848930209440238761074863228809275104582923626568122657664313107e-2

$2$, GUE

1/2:

2.374303924587727272913891195060649668884382151699689637772347231460990640935031314454445045872363962e-2

$2$, GUE

2/3:

1.612809425334540592644934590556704839649530655653041803348994696252965325331155611634187029942140828e-2

$2$, GUE

3/4:

1.319536168513140252316933125213818893754058935446853479965924876677990538722365006575034998576962903e-2

$2$, GUE

7.023835292213994416407629969704029497827337438389548597663967340532015942486321027303850147556922348e-3

$2$, GUE

5/4:

3.587564236339516123197516979393297185378447415755726229784581690029660244375025848129818772720551159e-3

$2$, GUE

4/3:

2.842401795254536479860044986759395335209717435497983455785548974842834226109022578878262697032044024e-3

$2$, GUE

3/2:

1.761269678833633713731132857992260153066997666450632133582371337218812690350156970691258657715941107e-3

$2$, GUE

5/3:

1.073043561617211940346028236283041679481810630757106454902969319952520655807212858723430448705624983e-3

$2$, GUE

7/4:

8.323659108960344092793133051427455532250774904888337904841371833527731355123082962739080851127385319e-4

$2$, GUE

3.791991116936172629316229049706757913345597988771721583690353647322329843000824250388847960458670323e-4

$2$, GUE

9/4:

1.667382169656828695019285728113616599622161273162984219256036712045299510209761345313902644877914672e-4

$2$, GUE

7/3:

1.258231447753189068067528254396997787213084187246175854316754532494997012475813234896122580679929473e-4

$2$, GUE

5/2:

7.084708892978735640096300966755485540877334370156907443027743522528718588945980966420072025361652664e-5

$2$, GUE

8/3:

3.930759336051370492353308693261476919500524122214391107937363748052131409989323641309656940610301040e-5

$2$, GUE

11/4:

2.912000081632168803288074355538889284231450805280554605818449706292953466311308119155615438707151819e-5

$2$, GUE

1.158965989354614939677236327814643095434423038445009754924150878624311547734860818839920016849912605e-5

$4$, GSE

-6:

1.637921233786379064235153006376786689095005393126650618622184471954543925151812116010117899329648162e-8

$4$, GSE

-23/4:

2.205608092541086705714857254710753113769522868445117079916584112310840857469057003472320633461820813e-7

$4$, GSE

-17/3:

4.959990253384327553323098680402860888664711592608151657571916673563271007316723825087566708010851547e-7

$4$, GSE

-11/2:

2.310639935108501107938597593839933009972895571547537896932995858674086522942376228483949408497198776e-6

$4$, GSE

-16/3:

9.673671161221073267972017301214380569062640550518933654585519822874684982674280324239573714195857433e-6

$4$, GSE

-21/4:

1.903494934555961307213271657886031314133558219311168744836055224224508345173210753050568728222278627e-5

$4$, GSE

-5:

1.246234271609432326565203496017923761588022779915391726499036699891300358171991499598370852364956446e-4

$4$, GSE

-19/4:

6.553108432920911697994224089656458484215917213788892212701300334070064577322126529762098873360747496e-4

$4$, GSE

-14/3:

1.087319495290610571384775103346611869069902319778471237923283714950746454238954797821169888777537312e-3

$4$, GSE

-9/2:

2.796497935092554604202627942420267894574352697587308110041757895977155322647003265678088332370550765e-3

$4$, GSE

-13/3:

6.584939407882537039592250964237602664234532546227847475871756600910030730093346656600512092737771391e-3

$4$, GSE

-17/4:

9.784936596545379701954288787129964962115436875144174158739293932830097029850790230874810914627133935e-3

$4$, GSE

-4:

2.835687246681453901001168706017815921119144696393766284312994055469171243217214423780000694348499829e-2

$4$, GSE

-15/4:

6.873911017180101975734862006468473068474334736094265404461909554211382109126338363366606240256119478e-2

$4$, GSE

-11/3:

8.889497385758399415707500759373959155079406381410386628594126503978198897310631951588615245944825497e-2

$4$, GSE

-7/2:

1.407258654637519845936062793646880347454704719927905983642598038815350852168024960164261121014158332e-1

$4$, GSE

-10/3:

2.075214089244446899052513567018603147277162174257389937622660223362521797956580527802852277380045683e-1

$4$, GSE

-13/4:

2.455951142232876875570666108867575461162742751298732524793813698102567580463136910921942686636722743e-1

$4$, GSE

-3:

3.686766554697288653758400368578954848690977464675260356569967522762072894961139631060709144920781332e-1

$4$, GSE

-11/4:

4.801648672576526469677456739535025851605489879881767751562478383130699402692439299333362037512891083e-1

$4$, GSE

-8/3:

5.088055846576929158173586557908462114038107477103691611663309377929797386476612983151489191743872246e-1

$4$, GSE

-5/2:

5.470284509264640142452074496524454289699113007348526023866358291028757252286346618698491037656278807e-1

$4$, GSE

-7/3:

5.560727932310002801632466982102239756389258913675786970817568838962507531321731761108336696035734168e-1

$4$, GSE

-9/4:

5.493676750370439079908912002351082563916768956693912985597271809146487725286925962171402686420230463e-1

$4$, GSE

-2:

4.898927686851133745096404677048438881267188191468939643866490129250792556532251995439434382642090822e-1

$4$, GSE

-7/4:

3.905356906477763377978112665348329284255330593267043260297186655843117125539419435692973024904029063e-1

$4$, GSE

-5/3:

3.535952744884792334724275950474781800479621207195460702739212737560462793924401725322361576323785637e-1

$4$, GSE

-3/2:

2.800680519586465006089616180841392029152548837192196228107568304731348564491906248353953133512579607e-1

$4$, GSE

-4/3:

2.121849581192852424467305355016089500054412710832619936979597654298902091529716114711515733956372119e-1

$4$, GSE

-5/4:

1.817264518095749889812313692379700896023713002273104719771965144775710206068395395321507160493518130e-1

$4$, GSE

-1:

1.072573759405398896352813230841123912029187361651947440517713505552875061813390831090849824216313174e-1

$4$, GSE

-3/4:

5.786254566733305776314551012683153534527072502037339987779242373663661062912555038266825588418868132e-2

$4$, GSE

-2/3:

4.620793350178690118534957037381930000047302629270831471027461243749349851586461412173027666861086336e-2

$4$, GSE

-1/2:

2.865845181088246744918995432087690864090398288565712029696542214600534782088583918124042951092811246e-2

$4$, GSE

-1/3:

1.714256989492552435927438018455952756870475265199081274556856851990124706839668292319335812989734586e-2

$4$, GSE

-1/4:

1.308418057263671718080524069606530026648751334776757054668380763805554460923092409212146890460458232e-2

$4$, GSE

5.526846546652940683921753051873789434125136079832941557751155873664755485021831868109418943759231292e-3

$4$, GSE

1/4:

2.167218774055046908813273815526101090771255770617872315619162375952466922870922958472179134936040364e-3

$4$, GSE

1/3:

1.561105191518405983305762906789546651205915483043042549871315409984800796605900736072581049268743354e-3

$4$, GSE

1/2:

7.913209213998946474432287866026447555752903549510674312935286682621674690601261451843082701937310356e-4

$4$, GSE

2/3:

3.890887336919953937390808888408912297899382046286771122146606875448348696913181983012657701219745288e-4

$4$, GSE

3/4:

2.698004920334698271743379569275165049747936612165956179474319210583594820873948568595900292006386764e-4

$4$, GSE

8.611645012220977127292906458487462466478707981035360561959247200692390081421272613190769777760513295e-5

$4$, GSE

5/4:

2.579313421937065928021298293669184075572888465518766855694379209407865056657431767319248130537921633e-5

$4$, GSE

4/3:

1.702167685972919692535275397445305475150346199477091893877212229979778312393844062735487334458948480e-5

$4$, GSE

3/2:

7.265005961357420122528081826809725829369624397981361697803743074494239869591423357595825029558704646e-6

$4$, GSE

5/3:

3.019918701362024295501439725281611890266870924938290514338148999807160968200736788899614974792267510e-6

$4$, GSE

7/4:

1.928179623070533628380534908012063304712334414892694833473451849172008165198258043939739223387227628e-6

$4$, GSE

4.831018366413092477578606145900117502968605535948556402930865616530973610087276115807456578791008448e-7

$4$, GSE

9/4:

1.144591771920964197342874893457974143839472865660906181666867517951023856845993435567862327751027079e-7

$4$, GSE

7/3:

6.997016237021903580741357167891329662743743534532149022116830878262598356347540590993373086168214159e-8

$4$, GSE

5/2:

2.568451321992474817382754989847050874554460174996505518721066192693321736450118560130824623130329572e-8

$4$, GSE

8/3:

9.209378067871076003738038495402533796208799102650616525380374683940619984551245669046572988759355297e-9

$4$, GSE

11/4:

5.466900970694305764050997893120550374127645191811837798463673446347659376992003844439240878825838022e-9

$4$, GSE

1.105242442238244967919738815707452815825694068544543495658848519539338726954979703993024177905313242e-9

Definition

For $\beta\in\{1,2,4\}$, let $F_\beta$ be the Tracy–Widom distribution function [2] in the soft-edge scaling used by [1]. The table gives its density $f_\beta(s)=dF_\beta(s)/ds$.

Parameters

$\beta$

— Dyson index

$s$

— argument

Formulas

(1)

Let $T_t$ be the operator on $L^2(0,\infty)$ with kernel $T_t(x,y)=\operatorname{Ai}(x+y+t)$.

(2)

$F_1(s)=\det(I-T_s)$, $F_2(s)=\det(I-T_s)\det(I+T_s)$, and $F_4(s)=\frac12(\det(I-T_{\sqrt2s})+\det(I+T_{\sqrt2s}))$ [1].

(3)

Let $q$ be the Hastings–McLeod solution of $q''(s)=sq(s)+2q(s)^3$ with $q(s)\sim\operatorname{Ai}(s)$ as $s\to+\infty$, and put $u(s)=\int_s^\infty q(x)\,dx$ and $J(s)=\int_s^\infty q(x)^2\,dx$. Then $f_1(s)=F_1(s)(J(s)+q(s))/2$, $f_2(s)=F_2(s)J(s)$, and $f_4(s)=2^{-1/2}F_2(\sqrt2s)^{1/2} (J(\sqrt2s)\cosh(u(\sqrt2s)/2)-q(\sqrt2s)\sinh(u(\sqrt2s)/2))$ [1].

Comments

(4)

The table uses the Tracy–Widom scaling in which $F_2(s)$ is the Airy-kernel Fredholm determinant. For $\beta=4$ this is the $N\times N$ quaternionic GSE convention: $F_4(s)$ is expressed through the operators $T_{\sqrt2s}$ in (2). In the alternative $2N\times2N$ complex convention the $\sqrt2$ is omitted.

(5)

The Airy function $\operatorname{Ai}$ appears in the Fredholm determinant kernel and in the boundary condition for the Hastings–McLeod solution. It is tabulated in the table of Airy function values.

(6)

It retains all 165 previously computed values for $\beta=1,2,4$ and $-6\leq s\leq3$ with reduced denominator at most $4$; $s=0$ is the soft-edge reference, while the relevance of the nonzero sampling family remains unresolved.

(7)

The argument $s=0$ is the unshifted soft spectral edge in the scaling of [1], equation (2.4). The nonzero rational arguments are retained reference samples. A bound on their denominators, or agreement with a companion table, does not establish separate mathematical or application-specific significance for each sample. The references cited here do not justify that whole selection.

Programs

(P1)

Python

from generate import TracyWidomDensities

generator = TracyWidomDensities()
print(generator.value({'beta': '2', 's': '0'}, 100)['number'])

References

[1]

Folkmar Bornemann, On the numerical evaluation of distributions in random matrix theory: a review, Markov Processes and Related Fields 16 (2010), 803-866. (arXiv)

Links

[2]

Wikipedia: Tracy-Widom distribution

Similar tables

Values of the Tracy–Widom distribution functions $F_\beta(s)$ — gives the distribution functions whose derivatives are tabulated here

Values of the Hastings–McLeod solution $q(s)$ of Painlevé II — gives the Painlevé-II solution used in the density formulas

Cumulants $\kappa_n$ of the Tracy–Widom distributions — gives moment information for the same three soft-edge laws

Skewness and excess kurtosis of the Tracy-Widom distributions — gives standardised third and fourth cumulants for the same three soft-edge laws

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — gives the Airy function in the Fredholm determinant kernel and in the Hastings-McLeod boundary condition

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it retains all 165 previously computed values for $\beta=1,2,4$ and $-6\leq s\leq3$ with reduced denominator at most $4$; $s=0$ is the soft-edge reference, while the relevance of the nonzero sampling family remains unresolved)

How they were obtained:

The attached generator evaluates (3) using multiprecision Taylor integration of the Hastings–McLeod solution and its three tail integrals. All 165 entries carry 100 significant decimal digits.

Two runs use respectively 230 and 260 working decimal digits, 180 and 220 Taylor terms, maximum steps $1/8$ and $1/10$, and right boundaries $40$ and $44$. The right-boundary data use the Airy function, its derivative and its tail integrals; the integrals are not set to zero. The identity $J(s)=q^{\prime}(s)^2-sq(s)^2-q(s)^4$ is checked at every target. Every density must agree between the runs to relative $10^{-110}$ before 100 digits are written. The observed worst relative difference is less than $7.1\cdot10^{-145}$, at $\beta=4,s=-6$. Independent derivatives of the Fredholm determinants (2) agree beyond 140 significant digits for all three laws at zero and for $\beta=4$ at both endpoints; the zero controls also vary the quadrature order and interval length. The existing parameter selection and normalisation, including the $\sqrt2$ argument and chain-rule factor for $\beta=4$, are unchanged. Arb provides multiprecision arithmetic, but Taylor step results are replaced by their midpoints, and neither the truncation error nor the nonlinear boundary error is rigorously enclosed. The stated rigour remains heuristic (agreement-checked), not proven. The Python example uses the attached generate.py file and its checked computation.