numerical-evidence.json

back to table · edit · history · where entries came from · files · download

19506 bytes, as of the version from 2026-11-09 09:52 (current). Recorded here, not run.

{
  "control_cases": [
    "Arb and mpmath integral of t^2 = 1/3",
    "Clausen log-sine integral at pi/3",
    "conditional mean at t=1 = log(2)",
    "conditional mean versus integrated cotangent at 1/4,1/2,3/4",
    "BS equations (34),(35),(36),(43),(44) with (24) and (10)"
  ],
  "elapsed_seconds": 17.654815196990967,
  "entries": [
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.32306594721945051409363651072380639407224184078058701613086847036101511280726984208378760908937139207348729175596243546188767",
      "independent_75d": "0.323065947219450514093636510723806394072241840780587016130868470361015112807",
      "independent_absolute_difference_upper": "2.2615674062230331425771244715385514466918418650934555450948887712111171807135985068498470400912576558079819388752174695883255223031973146129946677961412422057492673248423486e-126",
      "k": 1,
      "n": 3,
      "number": "0.3230659472194505140936365107238063940722418407805870161308684703610151128072698420837876090893713921",
      "radius": "2.9437103e-133"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.18277045187202515960804612962733613213543887791186649308172869215222305226702231931484765562441763392266714655137079896440319",
      "independent_75d": "0.182770451872025159608046129627336132135438877911866493081728692152223052267",
      "independent_absolute_difference_upper": "1.6318894453097705669940367629196479703677075291888642143784953602710977568326559001174350717074933159444988961075857375353387149194988756247297113139234676639998724069159108e-126",
      "k": 2,
      "n": 3,
      "number": "0.1827704518720251596080461296273361321354388779118664930817286921522230522670223193148476556244176339",
      "radius": "1.5316962e-133"
    },
    {
      "accuracy_bits": 437,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.10979712636522801374998476018863043821853059796767791103765042695264286134745664567627611105163646486546353720056005798240193",
      "independent_75d": "0.109797126365228013749984760188630438218530597967677911037650426952642861347",
      "independent_absolute_difference_upper": "3.4313650675368545983487253393339655603738329544852957554316851425861560134756334470043295489961322463840110768427871928369891754291918800998133141571674794548695542434393911e-127",
      "k": 3,
      "n": 3,
      "number": "0.1097971263652280137499847601886304382185305979676779110376504269526428613474566456762761110516364649",
      "radius": "9.8602123e-134"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 571,
      "independent_125d": "0.068126354024067243912499029147879731393227687576368805988751516590927551682788411354418169805909266707857023772666811143412512",
      "independent_75d": "0.0681263540240672439124990291478797313932276875763688059887515165909275516828",
      "independent_absolute_difference_upper": "4.6904017380206959150002095046959526347058852273472094706298730415700338928780404257095602091433886843048152268113572487599171532515894057692939608512162349648643688684820386e-127",
      "k": 4,
      "n": 3,
      "number": "0.06812635402406724391249902914787973139322768757636880598875151659092755168278841135441816980590926671",
      "radius": "6.3869461e-134"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.043137773756968400487278436692585205325072484606566573950217601069686416665460156873950251308776221281741790370031211038622933",
      "independent_75d": "0.0431377737569684004872784366925852053250724846065665739502176010696864166655",
      "independent_absolute_difference_upper": "2.9847761126908691983407245127363634766671662458491702271391421324769991365414377718617763904480676989724055430374661261385113812448770329741916392031349483211983501911382327e-127",
      "k": 5,
      "n": 3,
      "number": "0.04313777375696840048727843669258520532507248460656657395021760106968641666546015687395025130877622128",
      "radius": "3.5956293e-134"
    },
    {
      "accuracy_bits": 437,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.027699198059591683453646996746852589725068557154935366019122796168027082474626052889842280296788682859183890545016418302310622",
      "independent_75d": "0.0276991980595916834536469967468525897250685571549353660191227961680270824746",
      "independent_absolute_difference_upper": "4.7040480386704936135107790166819364704700099322680849671988083056664328569968461407571747002153132335905120110332537127733624768343805161871273782169620228063770645263588276e-127",
      "k": 6,
      "n": 3,
      "number": "0.02769919805959168345364699674685258972506855715493536601912279616802708247462605288984228029678868286",
      "radius": "2.7549983e-134"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 571,
      "independent_125d": "0.017968476193891927194959730821894657774619647273559377230197648911189691628184798357534058533837429672072846553355761732827415",
      "independent_75d": "0.0179684761938919271949597308218946577746196472735593772301976489111896916282",
      "independent_absolute_difference_upper": "7.9818112544346916782534457289271176885505616790855537013238578857931176669737414899784277632541358418554896960510437464840468333938057882000078974476685651534441758019660841e-128",
      "k": 7,
      "n": 3,
      "number": "0.01796847619389192719495973082189465777461964727355937723019764891118969162818479835753405853383742967",
      "radius": "1.6726333e-134"
    },
    {
      "accuracy_bits": 437,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.011747366627279797441511409494695097299874361120110187216586010039461992677868334891921571864765470312300567826258333086537631",
      "independent_75d": "0.0117473666272797974415114094946950972998743611201101872165860100394619926779",
      "independent_absolute_difference_upper": "2.9966556922266418377136828158085893141283789836575699515376312278604712679375534404665112706871379803287429109221877773535685055010619606221521509856046333348686512759536430e-127",
      "k": 8,
      "n": 3,
      "number": "0.01174736662727979744151140949469509729987436112011018721658601003946199267786833489192157186476547031",
      "radius": "1.1696037e-134"
    },
    {
      "accuracy_bits": 437,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.0077275145219876661216271076083216684007873370254290541403771710212999115875209204507191910824109430206138871471319066444374065",
      "independent_75d": "0.00772751452198766612162710760832166840078733702542905414037717102129991158752",
      "independent_absolute_difference_upper": "5.1763689773696555934548459207045334305556618587929202282559816534529178128206062634561274732896384300829236182938178379002531583121682215357183831945773341373455057889959126e-128",
      "k": 9,
      "n": 3,
      "number": "0.007727514521987666121627107608321668400787337025429054140377171021299911587520920450719191082410943021",
      "radius": "7.7659445e-135"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.0051085724349028166842256205893773766587011148123530814253857942979998011940057322298535489057437218602134381835619540358608772",
      "independent_75d": "0.00510857243490281668422562058937737665870111481235308142538579429799980119401",
      "independent_absolute_difference_upper": "5.0325604295724122924651176896363242624765908925275599903472766617974260847861682380174499654688947179409928301276374412583353793424916991944657400224891554902065932678453079e-128",
      "k": 10,
      "n": 3,
      "number": "0.005108572434902816684225620589377376658701114812353081425385794297999801194005732229853548905743721860",
      "radius": "4.8042247e-135"
    },
    {
      "accuracy_bits": 437,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.0033911234797760832215845555260572185674327798983856055962857745436453046139065639496063666106314326722200860759772104691248491",
      "independent_75d": "0.00339112347977608322158455552605721856743277989838560559628577454364530461391",
      "independent_absolute_difference_upper": "2.2194454459477190492713453873348469398631565852885294926192565288340529701013480157052104708110341442092261140434547491599614519453215889030859903686962076185656213690513826e-128",
      "k": 11,
      "n": 3,
      "number": "0.003391123479776083221584555526057218567432779898385605596285774543645304613906563949606366610631432672",
      "radius": "3.5963057e-135"
    },
    {
      "accuracy_bits": 438,
      "higher_accuracy_bits": 570,
      "independent_125d": "0.0022588449231027152553058632586661567534982964895649278548473889404124307785461284284627195484802033748565027524308026223973161",
      "independent_75d": "0.00225884492310271525530586325866615675349829648956492785484738894041243077855",
      "independent_absolute_difference_upper": "2.7046015078834781964108733747508453880758197079050183971605940831745146726452866533927617616780038870662640667627455822316670143023762404390964729905837151016878932072030710e-128",
      "k": 12,
      "n": 3,
      "number": "0.002258844923102715255305863258666156753498296489564927854847388940412430778546128428462719548480203375",
      "radius": "2.5615612e-135"
    },
    {
      "accuracy_bits": 436,
      "higher_accuracy_bits": 568,
      "independent_125d": "0.42627839881750579092352142659616687305800676962963510754160645802652945122911658148912418833224294358504803799987307848190447",
      "independent_75d": "0.426278398817505790923521426596166873058006769629635107541606458026529451229",
      "independent_absolute_difference_upper": "4.0995660383547044685365896860740662973970970973411722863587776010647578752283839054365773317194011756783709519845841240463599513245424178482556852566705181854751923748937056e-126",
      "k": 1,
      "n": 4,
      "number": "0.4262783988175057909235214265961668730580067696296351075416064580265294512291165814891241883322429436",
      "radius": "1.2946396e-132"
    },
    {
      "accuracy_bits": 435,
      "higher_accuracy_bits": 567,
      "independent_125d": "0.22748716706632111183518571166853687393131299075275462761598113173999166356157158926936142513804667919165788610081811842210287",
      "independent_75d": "0.227487167066321111835185711668536873931312990752754627615981131739991663562",
      "independent_absolute_difference_upper": "4.5275242146922042739082038098250089906981772786187747577648980367004226329411267593836625167857376804037658884547100636958743711051812010383916475885933229517778836643908501e-126",
      "k": 2,
      "n": 4,
      "number": "0.2274871670663211118351857116685368739313129907527546276159811317399916635615715892693614251380466792",
      "radius": "1.3027775e-132"
    },
    {
      "accuracy_bits": 434,
      "higher_accuracy_bits": 567,
      "independent_125d": "0.13206530382166951825723931745585854314863920732140722348180370160250136630676057208666390002786488848317054167583283846536336",
      "independent_75d": "0.132065303821669518257239317455858543148639207321407223481803701602501366307",
      "independent_absolute_difference_upper": "1.4479216035907395485570360936227977315418634281092539689048158716049746522942985711004950002484414612119964124945084648969693915695096969472065212712662476761612203496008617e-126",
      "k": 3,
      "n": 4,
      "number": "0.1320653038216695182572393174558585431486392073214072234818037016025013663067605720866639000278648885",
      "radius": "1.9386980e-132"
    },
    {
      "accuracy_bits": 434,
      "higher_accuracy_bits": 566,
      "independent_125d": "0.080093838178677611230794050764843464618610105329715022922914385630726054486880294514597130171804014614147084811702121784626735",
      "independent_75d": "0.0800938381786776112307940507648434646186101053297150229229143856307260544869",
      "independent_absolute_difference_upper": "3.8179378577608891459881522574286802965772884052981167064418683193163596716752545732239286219772784671890866165589656230091978321460781529230479921506020631429340673712007318e-127",
      "k": 4,
      "n": 4,
      "number": "0.08009383817867761123079405076484346461861010532971502292291438563072605448688029451459713017180401461",
      "radius": "1.3014439e-132"
    },
    {
      "accuracy_bits": 433,
      "higher_accuracy_bits": 566,
      "independent_125d": "0.049885663385794977377926902614979065050808320500715623988516989526040739457759704385487295606550857965781684332127601789232207",
      "independent_75d": "0.0498856633857949773779269026149790650508083205007156239885169895260407394578",
      "independent_absolute_difference_upper": "2.2199065642202201588079540145119618884546835359409150309264933210124722892076129643699375526842543765207400818955125800948564251269790908083795904465684120203662011096231445e-127",
      "k": 5,
      "n": 4,
      "number": "0.04988566338579497737792690261497906505080832050071562398851698952604073945775970438548729560655085797",
      "radius": "8.2251367e-133"
    },
    {
      "accuracy_bits": 433,
      "higher_accuracy_bits": 566,
      "independent_125d": "0.031631486033870237531800121225644023369788085295628184283448925646446301265653617361764701729297120738355938131722278412203056",
      "independent_75d": "0.0316314860338702375318001212256440233697880852956281842834489256464463012657",
      "independent_absolute_difference_upper": "3.6512381180773095503764178001176275808287029247629142411278560450695007931633547446131779024699380358460815038458707839034531901665156868792285528270389077122538844194869271e-127",
      "k": 6,
      "n": 4,
      "number": "0.03163148603387023753180012122564402336978808529562818428344892564644630126565361736176470172929712074",
      "radius": "9.8682767e-133"
    },
    {
      "accuracy_bits": 433,
      "higher_accuracy_bits": 565,
      "independent_125d": "0.020315646524097777453536505019186293484886314445792974770596434928229733126373226330477038028711377148936519245577066194402126",
      "independent_75d": "0.0203156465240977774535365050191862934848863144457929747705964349282297331264",
      "independent_absolute_difference_upper": "2.3135427962258814306303868951471765355749207956777457939839247486839678724189858915217615911489404137477289573622398430415519074841027189449875686664466963323313062107457687e-127",
      "k": 7,
      "n": 4,
      "number": "0.02031564652409777745353650501918629348488631444579297477059643492822973312637322633047703802871137715",
      "radius": "5.8537709e-133"
    },
    {
      "accuracy_bits": 433,
      "higher_accuracy_bits": 565,
      "independent_125d": "0.013174173592430744090492191876720445206617262766298774495764404526366682164037166532351758204532757359543070279206228038939116",
      "independent_75d": "0.013174173592430744090492191876720445206617262766298774495764404526366682164",
      "independent_absolute_difference_upper": "3.7313496257413952566102525685814957551651677711006772389557953302948938015288516180817012515400923886307033478510552594578900697991221214835186291094041466702547715125985229e-128",
      "k": 8,
      "n": 4,
      "number": "0.01317417359243074409049219187672044520661726276629877449576440452636668216403716653235175820453275736",
      "radius": "3.5117948e-133"
    },
    {
      "accuracy_bits": 432,
      "higher_accuracy_bits": 565,
      "independent_125d": "0.0086073593762074020041129267069517412324365803035930165101599499945658236323689806347051188352995760757986053967921894802643146",
      "independent_75d": "0.00860735937620740200411292670695174123243658030359301651015994999456582363237",
      "independent_absolute_difference_upper": "3.9114951123208338503240370641828073804439324679355874872587581299481454027651193953152740470223033500821237828948279208156337638986296495975091920029432106360088222431065889e-128",
      "k": 9,
      "n": 4,
      "number": "0.008607359376207402004112926706951741232436580303593016510159949994565823632368980634705118835299576076",
      "radius": "4.0222249e-133"
    },
    {
      "accuracy_bits": 432,
      "higher_accuracy_bits": 565,
      "independent_125d": "0.0056574284969931491116295120840069947696728158784639729717274910487848949547941854158517076821554559436813175080108263421505736",
      "independent_75d": "0.00565742849699314911162951208400699476967281587846397297172749104878489495479",
      "independent_absolute_difference_upper": "1.9320584609937366208247076690585909876233373957683142492556319224059149898371705539976710890410562961204746402563740854998897482578743617731058889814288002129715393013579015e-128",
      "k": 10,
      "n": 4,
      "number": "0.005657428496993149111629512084006994769672815878463972971727491048784894954794185415851707682155455944",
      "radius": "2.5124467e-133"
    },
    {
      "accuracy_bits": 432,
      "higher_accuracy_bits": 564,
      "independent_125d": "0.0037367706766937032372276531533787082561346587572421874022484261772807638696660962114312670280960345502808659664663781754149291",
      "independent_75d": "0.00373677067669370323722765315337870825613465875724218740224842617728076386967",
      "independent_absolute_difference_upper": "2.0436314353377145413606769180236539450586112433605520919573197561285054835046691345250677067050612119276780291691904215973651100508926446287133545260516350457570202111848445e-128",
      "k": 11,
      "n": 4,
      "number": "0.003736770676693703237227653153378708256134658757242187402248426177280763869666096211431267028096034550",
      "radius": "1.6936760e-133"
    },
    {
      "accuracy_bits": 431,
      "higher_accuracy_bits": 564,
      "independent_125d": "0.0024782551319130610903002523059504741787696275506553194504794666984101739454832083995814651395160851270616116643187321146398419",
      "independent_75d": "0.00247825513191306109030025230595047417876962755065531945047946669841017394548",
      "independent_absolute_difference_upper": "1.3047203850013557000940002819279332234621834564319589445454007514979779045534527954436645914700903760906626217868564914752229587703089680146368082383923279620524785716220851e-128",
      "k": 12,
      "n": 4,
      "number": "0.002478255131913061090300252305950474178769627550655319450479466698410173945483208399581465139516085127",
      "radius": "1.9932852e-133"
    }
  ],
  "independent_check": "All 24 entries by mpmath log-sine/Clausen quadrature at 75 and 125 decimal digits; discrepancy from 140-digit ball less than 1e-115.",
  "minimum_accuracy_bits": 431,
  "properties": "Positive, decreasing after multiplication bound log(2), strictly log-convex on every applicable triple.",
  "series_N": 235,
  "series_uniform_error_bound": "[9.246548113172635812784869391063010826245405819683520327027382497261351938380775547125883117159907281771144374893571081923504110611234e-145 +/- 3.88e-278]",
  "skill_manifest": "agents/runs/20261109T080742Z-build-skills/manifest.json",
  "skill_versions": {
    "live": "886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b",
    "local": "886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b"
  },
  "table": "T685",
  "versions": {
    "mpmath": "1.4.1",
    "numberdb": "0.1.10",
    "sage": "10.9"
  },
  "working_bits": 445,
  "source_files": {
    "logsin1.pdf": "473010bb0eb09e9fd2b1295206e111687c80bf8f1f132437468406840ce65f7c"
  }
}