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Values of Bessel functions of the second kind $Y_\nu(x)$
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Numbers
$\nu$
$x$ 
$Y_\nu(x)$
0
$c_{1}$:
0.5207864124022675111071507369263396714572660066858921169526398843640010337623107538794386760422069248
0
$c_{2}$:
-0.3403180455234405582259362168709347946967269256026429820200262838695682655967293281729934815818126363
0
$c_{3}$:
0.2714598773115335450267041314241600691934592931454078578428236148081385268511020818684243395480089482
0
$c_{4}$:
-0.2324617660170387411323253554219948358925029583320133630902301155172691641036764976827441887966793624
0
$c_{5}$:
0.2065471103565925999158706335351226872637773649762553069887606017496779325944656292058868160096366678
0
$c_{6}$:
-0.1877290919149096989978120097133176120366702012888013090803990196135382988667511769382068178948536707
0
1:
0.08825696421567695798292676602351516282781752309067554671104384761199978932351337130107720035921993680
0
1/2:
-0.4445187335067065571483984750683319103735651244015110204148911793882396879314172860004064331153411199
0
2:
0.5103756726497451195966065927271578732681392270858461355718392719327313950418217977114130750041427489
0
3:
0.3768500100127903819671101923966164112168621357559481922077199666275649850093665066759546004203897137
0
5:
-0.3085176252490337800736489842120466113863470616273440437958514158982579156428808518901793164635184073
0
8:
0.2235214893875662205273234004986203592748149307814235775783335878367342592733136198060825932232203629
1/2
$c_{1}$:
0.4561860309838110003508886886926532731665892972455404342972017960863105401616259947586676781213173618
1/2
$c_{2}$:
-0.3193313063394751713947173696057065946272233988113236789565444411075616406769694314692200790525921936
1/2
$c_{3}$:
0.2602668366998830732444902705130485458751407440077780052313692754768763714346299488836354646993441612
1/2
$c_{4}$:
-0.2252578357657631893436953151454142915711533569042908722024588881312152842685796026331073568063734064
1/2
$c_{5}$:
0.2014190526758447647214604125514271313692832868157085275680831553453395443793536253057411902338636848
1/2
$c_{6}$:
-0.1838410477310629449113664628254685048026504798482581349127285925323105766304890777011731986882414959
1/2
1:
-0.4310988680183760795205209672985334000880560106886169046523715983528791955453662210509259188692898198
1/2
1/2:
-0.9902458802434048800233519554234875575625827321421758180980931412985159357847978913328591707821021911
1/2
2:
0.2347857104062484691740346837934112099540265092069508239774801395653838442208863325760070350252990004
1/2
3:
0.4560488207946331788468332602116153459178924408186227801023081929551139438122201708149972799652288015
1/2
5:
-0.1012177091851083995650601769914447317097584188839025053121735094240500899218850700633866808519171236
1
$c_{1}$:
0.4167299281064513764986063032908415114912240441418635657331632291174131345110628984379389908415218655
1
$c_{2}$:
-0.3031737401374894388754383777874664537734934193954843526625798669343993991620977454309915394712187424
1
$c_{3}$:
0.2509125362778126165018081683837006814220240545605380665097324148896091430376107464321185811331513311
1
$c_{4}$:
-0.2189747879090994999451326974858669806732162699758145824429664229358027276411447984443576362934256596
1
$c_{5}$:
0.1968276056838549375198278081342310409991081556547724381026879178014432554261771118914381504469859000
1
$c_{6}$:
-0.1802974126299862154110095659980709144657739576809327750759556269119081640486659576187314964017670468
1
1:
-0.7812128213002887165471500000479648205499063907164446078438332461277843915385602167276292380048056346
1
1/2:
-1.471472392670243069188584635323297453241088055435748322955922383406694090952357067886863470598097451
1
2:
-0.1070324315409375468883707722774766366874808982350538605257945572313199774994906724960750929874561422
1
3:
0.3246744247917999784370128392879532396692751433723549568387176631861363203951083524407391647131702866
1
5:
0.1478631433912268448010506754880372053734652184104756133006976595086882659262171067506925531074677045
1
8:
-0.1580604617312474942555552661874835503273440495267057376512626493603737016721985369589523236942339861
3/2
$c_{1}$:
0.3888906299108018433456282458028436665359202630291939352305061973518866974664175878449528180377763818
3/2
$c_{2}$:
-0.2901380830053771331008630502095978819421908044183806119442057818560631687178931213382296016074115377
3/2
$c_{3}$:
0.2429099565546167454208256471970549092990126967048430352365115625967313578029163914953401796049934421
3/2
$c_{4}$:
-0.2134174552052284597492803862664750416453870842401978169617625331226112179190229398411403442907234316
3/2
$c_{5}$:
0.1926779632307628211946179994047669299669609346397308021407881934620751725324828912839535053022688216
3/2
$c_{6}$:
-0.1770460128408025903373339292437991199226121647577510038600633943400103670977123753080938362349753010
3/2
1:
-1.102495575160179169936884979339000480633696827457962817911750415845326432910554084585082419606286466
3/2
3/2:
-0.6805601853491455442607639916158849484316723776076307502663218427987540879388648985845298950211912334
3/2
1/2:
-2.521465550421337851377617045511091528615767697159027615037526796053150929260871491006970567468127086
3/2
2:
-0.3956232813587035170786745067305788173777946058481933378927159384954471339741728928175069996223453175
3/2
3:
0.08700809072083528150160639033257616891001716007958109465722770869461860486835873784608266165752808736
3/2
5:
0.3219244429611401298466580005703958391065991891768639148041134822453668156142313921439130568047761539
2
$c_{1}$:
0.3676639131282223519629722385305014155999338318974851494544735811899076921938929414743341050021794084
2
$c_{2}$:
-0.2792752634407206619063932357154395061918869363167416368089514708597251600711657090174005911744203052
2
$c_{3}$:
0.2359400948052101819197603689361488440148830878704399456940148311155387516482912399374423453020104961
2
$c_{4}$:
-0.2084457653801302682165151478466594826059342067707443748648541997140077903109166163112543718967312630
2
$c_{5}$:
0.1888979723015989217714345821122031126426346233830530804275659219614002145474021885001565960182507029
2
$c_{6}$:
-0.1740454823233254436550387955976964589958254952891715457709394797593818018550186863162254485460985935
2
1:
-1.650682606816254391077226766119444803927630304523564762398710339867568572400633804756335676368831206
2
2:
-0.6174081041906826664849773650046345099556201253208999960976338291640513725413124702074881679915988911
5/2
$c_{1}$:
0.3506689717520420835783583214624734833536226829005396832894046121255772838221138774901664996506029955
5/2
$c_{2}$:
-0.2700055959710845681490806987176216427004938752614013589343955358748046693031606202943582242335719657
5/2
$c_{3}$:
0.2297831259878666006860245194302836396116689215599268282533895934359163523015807876278503384427436039
5/2
$c_{4}$:
-0.2039559290354760944122377838865560273842238189461098892317216876395715897363033587004275756322813559
5/2
$c_{5}$:
0.1854315012004992624466764312258912612190000937387731987300433490335799875285047463488039488162960687
5/2
$c_{6}$:
-0.1712624934027550086208590905603828218252225431086875939789209464790942688529222663737998122091199018
5/2
1:
-2.876387857462161430290133970718468041813034471685271549082879649183100103186296032704321339949569579
5/2
5/2:
-0.5726306044391483917061794816735240697481712614843573456332472108143263962845074101055228527826755192
3
$c_{1}$:
0.3365957689602984097876163395996148704098290447470893870719031629807500434317238240353302101620567079
3
$c_{2}$:
-0.2619502631981886274182556350919093061241130586866499401848240498588548168573249497138132028271590999
3
$c_{3}$:
0.2242814317842638586607303051475136251585021496677055365366134238118610626427758761329058654113008144
3
$c_{4}$:
-0.1998689965515693241743914139075845040333762628119357994514127925348336246451370747777411390835382635
3
$c_{5}$:
0.1822340597390260175447950774536249538057926450919393985967620863868625744200519989017206071958542363
3
$c_{6}$:
-0.1686698174214293569224484136336520820507988322471014787956874403573116045096961363303235493703862775
3
1:
-5.821517605964728847761757064429814395160614827377814441751008113342489898063975002297713467470519189
3
3:
-0.5385416161050316180047039053385944638079578636048592514812619180796573489452784925057993744175385389
7/2
$c_{1}$:
0.3246505610827119068955027895563628207861359782431387277496333653034068974004735117007344658282298108
7/2
$c_{2}$:
-0.2548487643165488125497799832344936067364831968755889187446380621374139312024851582143128173671472804
7/2
$c_{3}$:
0.2193183272801903166989114369986093169801202537850847722106534621361045724475757914942835910734062847
7/2
$c_{4}$:
-0.1961235791552523546272814541160128015961979035373456063485371790534697705433800759340541616905023194
7/2
$c_{5}$:
0.1792698142451000508116715437782436780464732635545041700768561804627174093306143287291797517469694901
7/2
$c_{6}$:
-0.1662449345026217460983303344091986131768519607777008528342393715729359484397479634707923125817637269
7/2
1:
-13.27944371215062798151378487425333972843147553096839492750264783007017408302092607893652428014156143
7/2
7/2:
-0.5113512606614967896776535863851915073445302559358363186282726880814728000775545832931841100453419490
4
$c_{1}$:
0.3143181009295216963802214874467479766542386633784934935911795347070138335758535894083092020984103200
4
$c_{2}$:
-0.2485146692484686933806596631493469573056258398116434458964738677737352619957352490691805223274106432
4
$c_{3}$:
0.2148051298563767479218581281771556283917960972089837566439397318060777797712438737251759282328188688
4
$c_{4}$:
-0.1926710432788665216818206983426806411111421091923735920315568199790656752875936709948221879490074254
4
$c_{5}$:
0.1765094975722225857645601313408130913518759897676633273895409276335359318492634767281725258280249114
4
$c_{6}$:
-0.1639690163313631570180273003147554161338807907712940302303997600096260513691424473304625730525231660
4
1:
-33.27842302897211869549331562045944156703605865974332188810733834018737081598321620902994512845428393
4
4:
-0.4889367685338425106156573983399132062187401820796279747372669215691248724787807185125061050377555887
9/2
$c_{1}$:
0.3052460944221655662427990560987048588910520733208513478408700977698899658087787405240747969451198545
9/2
$c_{2}$:
-0.2428102043717993470087628951925841858014223996687240268872688238499713505299877033033890930055196127
9/2
$c_{3}$:
0.2106729391732862459000779841032063765747236783197346799812082130325942936804919679197035198406400802
9/2
$c_{4}$:
-0.1894722363831533876674078992795456112628320949069595834378278464354919972443932102631120315422246106
9/2
$c_{5}$:
0.1739289111335098694175081042985918013923502203024713135899184485942393687232973370548211171027341816
9/2
$c_{6}$:
-0.1618261682870875056357166887222756267699777354667877046938310164509125871170798979068315465989889073
9/2
1:
-90.07971812759223444030636014905491005720729424509349294343565516130811847796018651985134862104136044
9/2
9/2:
-0.4700001059968980128298226861384806821821226685928655921059436464522709573970129154479820723092901809
5
$c_{1}$:
0.2971836833506379714695715431196780170583925876024100867602649024781114616926520592030830244032080398
5
$c_{2}$:
-0.2376308289490977440110229361620207521416798932129264854810964854364859454683398997593988443697682859
5
$c_{3}$:
0.2068672136273109330392249939173568241284701530385429213558706818594807595616268366577111389128834143
5
$c_{4}$:
-0.1864951951995172198010205075357270942979232183283927182691511725607193241461003000551948027006264786
5
$c_{5}$:
0.1715078290499361930662089282914854966907158262936970662976976694731178309690362859768787364457862678
5
$c_{6}$:
-0.1598028556474235756348946266610788134012255780046014617985514451452802069818443670810854954213767085
5
1:
-260.4058666258122207161847678992457181411278544505687606631076986081564766298017546699418475601637523
5
5:
-0.4536948224911018807638424572568681593584614416663231900329785812102907084716694263367914246372910307
11/2
$c_{1}$:
0.2899463104212699283075881089731704977337947082838332689669266634813563966351348705311004669147863511
11/2
$c_{2}$:
-0.2328954450613602441366883544284425311159744394783965857102324923307356285245340567394985901798623324
11/2
$c_{3}$:
0.2033440774119746366587137948758443731320365346981098966562630836988560813925922964113200747563269103
11/2
$c_{4}$:
-0.1837135036544322246337459101101656167944386319549698462651253754404419977977425369509636800640447706
11/2
$c_{5}$:
0.1692291818430427610182264542973572192243013534852471944352990692189459062129548132996976197394773368
11/2
$c_{6}$:
-0.1578874628528849553462381538570455202055471572868952982013472262772535645640211067045831896087243759
11/2
1:
-797.4380194361794819812434564672408507864341726748730415634182486217028922186207525997256133092306825
11/2
11/2:
-0.4394410733908565555217520592933474410854489974531503776988109769557496257351566817888845301402254104
6
1:
-2570.780243229150088466354363371997739844242485845944284742969647741377395482034330490388530473183239
6
6:
-0.4268258568917561073743063340665631591818406103744957475632758467875254186559809126146117696916215926
13/2
1:
-8681.738495670382067353371660990594448593568605178509964254165079677423695926868092077130397780496147
13/2
13/2:
-0.4155455509692096546598557427855819631262364614564153259447699519167109196610148476754899284601917344
7
1:
-30588.95705212398884088006759256472715998978197570076265625252807428837226915461021121472051811803511
7
7:
-0.4053710186067680479258554293041477007677031844951928236566704192184659642173554509650068096771999797
15/2
1:
-112065.1624242787873936125881364104869809299576946457564937407277871848051548306644444029695578372192
15/2
15/2:
-0.3961254767162161289870942397379344498577518333365544295888232018271317252670385435472129470209132585
8
1:
-425674.6184865066936838545919325341825000127051739647329027924233922958343726825086265156987231793083
8
8:
-0.3876699399771849685416415027070800968568908405078429514505171487131854574671075768385006542631665283
17/2
1:
-1672295.697868511428836835450385166710265355796814507837441856751728094653626533098573967412969777792
17/2
17/2:
-0.3798933461129564288427644225465662772358051612484693552548849950668031614759157398684293756272963693
Definition
This table gives the real principal values of the Bessel function of the second kind $Y_\nu(x)$, Weber\'s function, [1] [2], for nonnegative integer or positive half-integer order $\nu$ and positive real argument $x$.
Parameters
$\nu$
—   order ($\nu\in\tfrac12\mathbb{Z}_{\geq0}$)
$x$
—   argument ($x>0$)
Formulas
(1)
If $C_\nu$ is either $J_\nu$ or $Y_\nu$, then $C_{\nu-1}(x)+C_{\nu+1}(x)=2\nu C_\nu(x)/x$ [4].
(2)
For an integer $n$, $Y_{-n}(x)=(-1)^nY_n(x)$ [3].
(3)
$Y_{1/2}(x)=-\sqrt{2/(\pi x)}\cos x$.
Comments
(4)
The principal branch is that of [2]. For real $\nu$ and $x>0$, $Y_\nu(x)$ is real. The point $x=0$ is excluded because $Y_\nu$ is singular there, which is what distinguishes it from the table of $J_\nu$.
(5)
Negative integer orders are not listed separately; they reduce to positive integer orders by (2). For nonintegral negative orders, the connection formulas in [3] express the values through $J_\nu(x)$ and $Y_\nu(x)$.
(6)
These are the ordinary cylindrical Bessel functions. The spherical Bessel function of the second kind is $\mathrm{y}_n(x)=\sqrt{\pi/(2x)}Y_{n+1/2}(x)$ [5].
(7)
Half-integer orders are elementary. The Bessel polynomials occur in explicit formulas for the associated spherical and modified spherical functions [6].
(8)
It preserves the existing isolated extremal heights and adds unit-argument and order-scale diagonal references for cylindrical orders $0,\ldots,8$ and spherical radial orders $1/2,3/2,\ldots,17/2$. Orders $0,1$ additionally use $x=1/2,2,3,5,8$, and orders $1/2,3/2$ use $x=1/2,2,3,5$. These sparse dimensionless radii are exploratory references, not interpolation nodes or independently documented applications; the diagonal $x=\nu$ compares argument and order scales, not extrema. Integer and half-integer order families come from cylindrical and spherical separation, while the finite cutoffs are practical choices. [12] [13] [14]
(9)
The reference inputs and finite cutoffs are explicit choices rather than a completeness or application-demand claim. The attached balanced-selection.json specifies every argument exactly. Existing entries retain their identifiers, decimal strings and annotations.
(10)
For $\nu=0$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(2249/1024,1125/512)$, $(5559/1024,695/128)$, $(4401/512,8803/1024)$, $(12031/1024,47/4)$, $(7627/512,15255/1024)$, $(4619/256,18477/1024)$, respectively. For $\nu=1/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(1523/512,3047/1024)$, $(6351/1024,397/64)$, $(2399/256,9597/1024)$, $(12827/1024,3207/256)$, $(4013/256,16053/1024)$, $(9637/512,19275/1024)$, respectively. For $\nu=1$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(3771/1024,943/256)$, $(1777/256,7109/1024)$, $(5183/512,10367/1024)$, $(3401/256,13605/1024)$, $(8417/512,16835/1024)$, $(5015/256,20061/1024)$, respectively. For $\nu=3/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(2229/512,4459/1024)$, $(7839/1024,245/32)$, $(11117/1024,5559/512)$, $(7183/512,14367/1024)$, $(17603/1024,4401/256)$, $(10417/512,20835/1024)$, respectively. For $\nu=2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(2561/512,5123/1024)$, $(8551/1024,1069/128)$, $(11851/1024,2963/256)$, $(15115/1024,3779/256)$, $(18361/1024,9181/512)$, $(21599/1024,675/32)$, respectively. For $\nu=5/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(5769/1024,2885/512)$, $(9247/1024,289/32)$, $(12573/1024,6287/512)$, $(3963/256,15853/1024)$, $(19109/1024,9555/512)$, $(11177/512,22355/1024)$, respectively. For $\nu=3$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(6403/1024,1601/256)$, $(9931/1024,2483/256)$, $(13283/1024,3321/256)$, $(16579/1024,4145/256)$, $(19847/1024,2481/128)$, $(23101/1024,11551/512)$, respectively. For $\nu=7/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(7027/1024,1757/256)$, $(2651/256,10605/1024)$, $(437/32,13985/1024)$, $(1081/64,17297/1024)$, $(20577/1024,10289/512)$, $(745/32,23841/1024)$, respectively. For $\nu=4$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(1911/256,7645/1024)$, $(11269/1024,5635/512)$, $(14675/1024,3669/256)$, $(9003/512,18007/1024)$, $(5325/256,21301/1024)$, $(6143/256,24573/1024)$, respectively. For $\nu=9/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(8253/1024,4127/512)$, $(11925/1024,5963/512)$, $(15359/1024,15)$, $(4677/256,18709/1024)$, $(43/2,22017/1024)$, $(12649/512,25299/1024)$, respectively. For $\nu=5$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(8857/1024,4429/512)$, $(12575/1024,393/32)$, $(4009/256,16037/1024)$, $(4851/256,19405/1024)$, $(22725/1024,11363/512)$, $(13009/512,26019/1024)$, respectively. For $\nu=11/2$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(9455/1024,591/64)$, $(13219/1024,3305/256)$, $(16707/1024,4177/256)$, $(10047/512,20095/1024)$, $(23429/1024,11715/512)$, $(26733/1024,13367/512)$, respectively. These labels identify arguments, not the values at them.
(11)
The retained pre-extension subfamily has the following argument definitions: Integer orders $\nu=m$ label cylindrical Fourier modes $e^{im\varphi}$. Positive half-integer orders $\nu=\ell+1/2$ correspond to spherical harmonic radial orders through the standard relation with spherical Bessel functions. These are structural order families, not a rational order mesh. The prefixes $m,\ell=0,\ldots,5$ and six positive critical arguments per order are explicit finite lookup depths, not canonical cutoffs. Each argument $c_k$ is the unique zero of the derivative of the tabulated cylindrical Bessel function in its stated rational interval. The entries are extremal heights, not zero locations. The labels enumerate the isolating intervals; a global root-count proof is not claimed. At order zero the zero of $J_0^{\prime}$ at the origin is excluded. For half-integer order these are extrema of $J_\nu$ or $Y_\nu$, not of the differently scaled spherical radial functions. [7]. [8] [9] [10] [11]
(12)
The attached generate.py forwards to balanced_generate.py, the standalone generator for the entire current selection. Its default offline check recomputes both new and retained values. The previous entry point is preserved byte-for-byte as pre-balanced-generator.py and in revision history. Earlier supporting sources and checks are preserved as provenance for their original subfamilies; balanced-provenance.json identifies their source hashes and the retained website revision.
Programs
(P1)
Python
import mpmath as mp
mp.mp.dps = 150
left = mp.mpf('2249') / 1024
right = mp.mpf('1125') / 512
argument = mp.findroot(lambda point: mp.diff(
    lambda value: mp.bessely(0, value), point), (left, right))
print(mp.nstr(mp.bessely(0, argument), 100))
Links
[12]
DLMF 10.2
[13]
DLMF 10.47
[14]
DLMF 10.73
Similar tables
Values of Bessel functions of the first kind $J_\nu(x)$ —   the other standard solution, at the same structural orders but at its own stationary arguments
Zeros of Bessel functions of the first kind $J_\alpha$ —   stores zeros of $J_\alpha$ for real order $\alpha$
Zeros of Bessel functions of the second kind $Y_\alpha$ —   stores zeros of $Y_\alpha$ for real order $\alpha$
Local extrema of Bessel functions of the first kind $J_\alpha$ —   stores the locations of extrema rather than their heights of $J_\alpha$ for real order $\alpha$
Local extrema of Bessel functions of the second kind $Y_\alpha$ —   stores the locations of extrema rather than their heights of $Y_\alpha$ for real order $\alpha$
Bessel polynomials $y_n$ —   occur in explicit formulas for half-integer spherical and modified spherical Bessel functions
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it preserves the existing isolated extremal heights and adds unit-argument and order-scale diagonal references for cylindrical orders $0,\ldots,8$ and spherical radial orders $1/2,3/2,\ldots,17/2$. Orders $0,1$ additionally use $x=1/2,2,3,5,8$, and orders $1/2,3/2$ use $x=1/2,2,3,5$. These sparse dimensionless radii are exploratory references, not interpolation nodes or independently documented applications; the diagonal $x=\nu$ compares argument and order scales, not extrema. Integer and half-integer order families come from cylindrical and spherical separation, while the finite cutoffs are practical choices)
How they were obtained:

Retained-entry provenance (unchanged numerical data): Each of the 72 extremal heights is written to 100 significant digits. Arb evaluates the cylindrical Bessel function and its derivative with exact rational order at 773 working bits. Opposite derivative signs at the rational endpoints of each defining interval establish a critical point. A Gronwall enclosure for the pair (function, derivative), combined with the Bessel differential equation, proves a fixed nonzero second-derivative sign throughout that interval, hence uniqueness and a genuine local extremum. A separate narrow rational bracket is certified and its uncertainty propagated through the function evaluation. mpmath only proposes starting approximations and is not the source of the rigorous error bound. The interval labels are not a global ordinal certificate.

more

Every written interval contains a recomputed 900-bit enclosure and independently agrees with mpmath at 160 and 210 decimal digits. The Bessel Wronskian is checked at every argument. All 36 half-integer-order values additionally contain an independent Arb enclosure derived from elementary sine/cosine functions and the spherical three-term recurrence. Finite-width tests, known initial values, and a deliberately displaced-value control guard against vacuous comparisons. See the attached source, selection.json and checks-evidence.json for the exact argument definitions and per-row certificates. Earlier grid-specific sources remain historical evidence, not checks for this selection.

Balanced extension: balanced_generate.py recomputes the entire attached selection, including retained values. New values use Arb ball arithmetic with exact rational inputs or enclosed algebraic and hyperbolic inputs at 773 working bits for 100 significant digits. Exact origin anchors are stored as integers. The attached critical-argument source is preserved and hash-identified; its certified argument enclosures are propagated through the evaluation. All old numerical strings, higher precision and manual entry annotations are preserved. balanced-checks.json records full independent higher-precision Arb recomputation and independent mpmath evaluations at 160 and 210 digits for every new value, with finite-width controls and family-specific identities or integral checks. The generator does not claim authorship of retained data; their prior supporting attachments remain unchanged, and the previous entry point is preserved as pre-balanced-generator.py.

digits: 100 (Unknown key)


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