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History of Roots of the cuckoo hashing threshold equation

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compare when who what
2026-09-19 19:21 zeta3 state the extended T288 range and narrowed cross-checks current reviewed
2026-09-19 19:20 zeta3 with codex-c roots xi_{k,ell} of the cuckoo hashing threshold equation for 2 <= k <= 12 and 1 <= ell <= 8
2026-09-17 02:50 zeta3 table-repair@1.109+64cc4f29 move Poisson notation out of the definition
2026-09-17 02:49 zeta3 table-repair@1.109+64cc4f29 remove non-root convention and define Poisson notation
2026-09-17 02:33 zeta3 table-build@1.130+3450381c repair cuckoo root draft prose after audit
2026-09-17 02:30 zeta3 with codex-c table-build@1.130+3450381c roots xi_{k,ell} of the cuckoo hashing threshold equation for 2 <= k <= 7 and 1 <= ell <= 6
2026-09-17 02:25 zeta3 table-build@1.130+3450381c claim cuckoo hashing threshold-equation roots draft

What changed between 2026-09-19 19:20 and 2026-09-19 19:21

from line 88 (7 lines, 1 more than before) @@ -88,6 +88,7 @@ 
   - CITE{CainSandersWormald}   complete: 'no'-  complete-note: it holds $\xi_{k,\ell}$ for every $2\leq k\leq7$ and $1\leq\ell\leq6$-    except $(k,\ell)=(2,1)$, the case with no positive root+  complete-note: it holds $\xi_{k,\ell}$ for every $2\leq k\leq12$ and $1\leq\ell\leq8$+    except $(k,\ell)=(2,1)$, extending the $\ell=1$ column through the range of HREF{T287}[the+    XORSAT root table] and carrying the bucket-capacity range through $\ell=8$   rigour details: Each stored root is computed in ball arithmetic with 64 guard bits     beyond the 100 digits written. The root $\xi_{k,\ell}$ is enclosed by bisection
from line 97 (191 lines, 210 fewer than before) @@ -96,401 +97,191 @@ 
     stored ball. The generator also compares every stored root with an mpmath computation     at 120 digits that writes $Q$ as the regularised incomplete gamma function, checks-    the stored rows with $\ell=1$ and $k\geq3$ against HREF{T287}[the XORSAT root-    table], and compares every derived threshold with HREF{T274}[the cuckoo hashing-    threshold table] for every stored row.+    the stored rows with $\ell=1$ and $3\leq k\leq12$ against HREF{T287}[the XORSAT+    root table], and compares each derived threshold with HREF{T274}[the cuckoo hashing+    threshold table] for the rows that table also holds. Display properties:   number-header: $\xi_{k,\ell}$ Numbers:-- comment: The associated cuckoo hashing threshold is HREF{T274#2,2}[$c^*_{2,2}$].-  params:-    k: '2'-    ell: '2'-  number: '2.687999345499491341463796937790041489618073899549236397935209599334514422260144214209251869745312739'-- comment: The associated cuckoo hashing threshold is HREF{T274#2,3}[$c^*_{2,3}$].-  params:-    k: '2'-    ell: '3'-  number: '5.071469708158110598648643546349746111224536523134635041837351018367446479545269819843879943093996070'-- comment: The associated cuckoo hashing threshold is HREF{T274#2,4}[$c^*_{2,4}$].-  params:-    k: '2'-    ell: '4'-  number: '7.321703358983259836814496629523591100449318121409982457321275056792696725897271000383910518926918359'-- comment: The associated cuckoo hashing threshold is HREF{T274#2,5}[$c^*_{2,5}$].-  params:-    k: '2'-    ell: '5'-  number: '9.496116339631318678686127155202154774263071948634199710067456727225709958713336937514852230633003440'-- comment: The associated cuckoo hashing threshold is HREF{T274#2,6}[$c^*_{2,6}$].-  params:-    k: '2'-    ell: '6'-  number: '11.62201083475983117017772549639697906479047433129974127980306614609092022913414759542897782318128499'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,1}[$c^*_{3,1}$].-    This is also HREF{T287#3}[$\xi_{3}$].-  params:-    k: '3'-    ell: '1'-  number: '2.149125799907062542079080149425149925528828496714285459034660801183768221524370129087681616217046728'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,2}[$c^*_{3,2}$].-  params:-    k: '3'-    ell: '2'-  number: '5.656576349425156386746617941568588192659141691293506742008009681442341139286464459237741396129083249'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,3}[$c^*_{3,3}$].-  params:-    k: '3'-    ell: '3'-  number: '8.849851652473190362812087757716704933249004939448022043446615039577597636956590060258950889502471470'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,4}[$c^*_{3,4}$].-  params:-    k: '3'-    ell: '4'-  number: '11.93324063021001784506567092268224882890651379335616698876346069366242440532751486950076095207023045'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,5}[$c^*_{3,5}$].-  params:-    k: '3'-    ell: '5'-  number: '14.97035798828812027443562587311940396807103878054315654978741750621682886826142355899362774443783528'-- comment: The associated cuckoo hashing threshold is HREF{T274#3,6}[$c^*_{3,6}$].-  params:-    k: '3'-    ell: '6'-  number: '17.98693229671281992725192590653386429471971042695500411009746368493209696294996668806395227602097357'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,1}[$c^*_{4,1}$].-    This is also HREF{T287#4}[$\xi_{4}$].-  params:-    k: '4'-    ell: '1'-  number: '3.593511969447426082272169884555315798860946340951301392760865045230967927293167350976509334752600022'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,2}[$c^*_{4,2}$].-  params:-    k: '4'-    ell: '2'-  number: '7.907674019123758850790566789844419922995108412022415004802350765389662552216371593154538151656605121'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,3}[$c^*_{4,3}$].-  params:-    k: '4'-    ell: '3'-  number: '11.97840754548836518417183962921129418652192059426167092339410984246940990181779382295840160771130815'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,4}[$c^*_{4,4}$].-  params:-    k: '4'-    ell: '4'-  number: '15.99506456917725118329956899417052588708356806076354172978753823547047795118689779120004162634038769'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,5}[$c^*_{4,5}$].-  params:-    k: '4'-    ell: '5'-  number: '19.99889979196351691674775775685275577102351928181638457324870251227217088836762137643885395177578993'-- comment: The associated cuckoo hashing threshold is HREF{T274#4,6}[$c^*_{4,6}$].-  params:-    k: '4'-    ell: '6'-  number: '23.99975947604794575691841460248553012829750258139642040533729634692718122214453372206287303646511694'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,1}[$c^*_{5,1}$].-    This is also HREF{T287#5}[$\xi_{5}$].-  params:-    k: '5'-    ell: '1'-  number: '4.801007549722517843402554184936876206309580045670987737307642940448096932959992199826971793616174495'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,2}[$c^*_{5,2}$].-  params:-    k: '5'-    ell: '2'-  number: '9.976864303760496064518075863793575319292786047595390211066663963569991540942269856056674145453186667'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,3}[$c^*_{5,3}$].-  params:-    k: '5'-    ell: '3'-  number: '14.99741350114707424427436529215687505875158666319175852037547329980901221594064290374107937980231339'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,4}[$c^*_{5,4}$].-  params:-    k: '5'-    ell: '4'-  number: '19.99972511819509434997149501618076756231262856417056906108861824236177909822971068355118859476119916'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,5}[$c^*_{5,5}$].-  params:-    k: '5'-    ell: '5'-  number: '24.99997174425962980488681204119343855489405198561699586682058895464280942938737404120353105180800265'-- comment: The associated cuckoo hashing threshold is HREF{T274#5,6}[$c^*_{5,6}$].-  params:-    k: '5'-    ell: '6'-  number: '29.99999715761549573994778634148325126494830338224240111896361395105934488672352627555997411293020102'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,1}[$c^*_{6,1}$].-    This is also HREF{T287#6}[$\xi_{6}$].-  params:-    k: '6'-    ell: '1'-  number: '5.903000058948943810458175385422799724242893481515842146827627134583507853576375177519654679441566068'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,2}[$c^*_{6,2}$].-  params:-    k: '6'-    ell: '2'-  number: '11.99466733019710920890189533055350295714134739551676260854413064936109585534197475883076176045683468'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,3}[$c^*_{6,3}$].-  params:-    k: '6'-    ell: '3'-  number: '17.99973347635061616803411249948014909629259883939838752321236504416573051808206817338186269012884600'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,4}[$c^*_{6,4}$].-  params:-    k: '6'-    ell: '4'-  number: '23.99998747487764539641451256705490343509005188952388808136197333588086708276694929140893017510583906'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,5}[$c^*_{6,5}$].-  params:-    k: '6'-    ell: '5'-  number: '29.99999943152413327868681837784491101013451684606105198098041990967162124172322669914399673273473956'-- comment: The associated cuckoo hashing threshold is HREF{T274#6,6}[$c^*_{6,6}$].-  params:-    k: '6'-    ell: '6'-  number: '35.99999997475451784136341288345456857119692422196366712451810528566280859222197757374080216380199990'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,1}[$c^*_{7,1}$].-    This is also HREF{T287#7}[$\xi_{7}$].-  params:-    k: '7'-    ell: '1'-  number: '6.953455713353474571757750459668776777157703675694193843415968941936946472738479953900380174362202268'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,2}[$c^*_{7,2}$].-  params:-    k: '7'-    ell: '2'-  number: '13.99885801107353040821026786739382027366723850806829945798703162739514390358101431148202137692247181'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,3}[$c^*_{7,3}$].-  params:-    k: '7'-    ell: '3'-  number: '20.99997542174545116865984920724717820919492653448481645717520596822045513748851642102679476382231719'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,4}[$c^*_{7,4}$].-  params:-    k: '7'-    ell: '4'-  number: '27.99999950416907812559837246489148384069148256876607111079228415555129553225681370701807267652350603'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,5}[$c^*_{7,5}$].-  params:-    k: '7'-    ell: '5'-  number: '34.99999999034126708478328429759287181356009191496219282981587565655646617784379490803056882062904127'-- comment: The associated cuckoo hashing threshold is HREF{T274#7,6}[$c^*_{7,6}$].-  params:-    k: '7'-    ell: '6'-  number: '41.99999999981590402367849092443096959604956068083312186280042756840973653364740468966946712440564761'-- params:-    k: '2'-    ell: '7'-  number: '13.71478188066531187419592663386781404667256909748779267691699334062513889841987887243770904218501949'-- params:-    k: '2'-    ell: '8'-  number: '15.78402951575241749514337877292627806948039669526522457278218532643843899662344975085249937108013643'-- params:-    k: '3'-    ell: '7'-  number: '20.99428721963569996390356939647821605540269988019492846111677264277639899563812999474020204223244644'-- params:-    k: '3'-    ell: '8'-  number: '23.99752228639355428601762208898877869365731075533340009984939339959751647474711325899830231566627929'-- params:-    k: '4'-    ell: '7'-  number: '27.99994816711800378712866883725114403620723445488544625643687745996149027919522284745023750300229475'-- params:-    k: '4'-    ell: '8'-  number: '31.99998894879824441417761862037158295979322617563944053294099209179496582008077917131009191700694329'-- params:-    k: '5'-    ell: '7'-  number: '34.99999971828689495073763165197240517066200863512164923396388731330097929439240393661829966466206228'-- params:-    k: '5'-    ell: '8'-  number: '39.99999997237895330621788962981778883738297516823892960097598945470307781177261883554921351313680319'-- params:-    k: '6'-    ell: '7'-  number: '41.99999999889542414122366233303485143320654681741093711561127695504527103766420273066250232144793028'-- params:-    k: '6'-    ell: '8'-  number: '47.99999999995219035911121073083261956776388908486262174893059213350857113390914160633563926285687229'-- params:-    k: '7'-    ell: '7'-  number: '48.99999999999654292747548744500679934181770500628007276070831287944725977561733647959273498745564802'-- params:-    k: '7'-    ell: '8'-  number: '55.99999999999993577845740933717347678095367153698328337540863073043813488092281488750741802432689143'-- params:-    k: '8'-    ell: '1'-  number: '7.978107736977070325801043211904953879262698393828180241246253405412112875926606187603354958634672340'-  comment: This is also HREF{T287#8}[$\xi_{8}$].-- params:-    k: '8'-    ell: '2'-  number: '15.99976948102927827986542124541488713267127746399729766072612805128795102265768110760940453279424376'-- params:-    k: '8'-    ell: '3'-  number: '23.99999791249776477526583476080506216234102397159993337514112341684209297104091256244852792327457149'-- params:-    k: '8'-    ell: '4'-  number: '31.99999998229421298460254326745152221550730148833673533729524296426543935216880675395508263187643527'-- params:-    k: '8'-    ell: '5'-  number: '39.99999999985498950806757475620303349980850606530853798231452429691164904454209053823396541671456058'-- params:-    k: '8'-    ell: '6'-  number: '47.99999999999883796011733170868052057137582534802368310535910608098338588451248045651064858878746216'-- params:-    k: '8'-    ell: '7'-  number: '55.99999999999999082549391562002909328701108769779675895602517556708034266361241800416573464377913288'-- params:-    k: '8'-    ell: '8'-  number: '63.99999999999999992834402649754375760190095303136801320458170026394134384645740733675987984543584298'-- params:-    k: '9'-    ell: '1'-  number: '8.989912519269479737859486962678439859820876867181214191191759058921700750894868987723157685967200046'-  comment: This is also HREF{T287#9}[$\xi_{9}$].-- params:-    k: '9'-    ell: '2'-  number: '17.99995558761294977966818068289267088363733341183405815612510976231904637506443052969637856628689981'-- params:-    k: '9'-    ell: '3'-  number: '26.99999983352352961042941034339633686304220206518353674195280425530683901186238491867268566768515694'-- params:-    k: '9'-    ell: '4'-  number: '35.99999999941561385098893372788096996561690281820301749212947615290516804614045183422834008690123778'-- params:-    k: '9'-    ell: '5'-  number: '44.99999999999801919305082234770327569019046986214351730820829668360886856474366257676088334662390077'-- params:-    k: '9'-    ell: '6'-  number: '53.99999999999999343065909329240090609003008758610857626415549884626998582815644147538220664394674172'-- params:-    k: '9'-    ell: '7'-  number: '62.99999999999999997853447576929085008477958958894807023008980265237111984961160627932241566256965976'-- params:-    k: '9'-    ell: '8'-  number: '71.99999999999999999993061441609145926492092198540629260762095728903122120396676581175298500932542075'-- params:-    k: '10'-    ell: '1'-  number: '9.995441133814842741400932592133052045659869599058257396947103091116896080275781481170753765750751443'-  comment: This is also HREF{T287#10}[$\xi_{10}$].-- params:-    k: '10'-    ell: '2'-  number: '19.99999175532397639510723938539197172685876190125580078940097278567402872373190369462770109320805875'-- params:-    k: '10'-    ell: '3'-  number: '29.99999998736720884786849000606127949609681146401724642592851062919967298195042923078301953657939331'-- params:-    k: '10'-    ell: '4'-  number: '39.99999999998187368851045929393106882058385746209414203789099520628733836767523116000236057505306730'-- params:-    k: '10'-    ell: '5'-  number: '49.99999999999997488606968796925152644784476238575890384234977335355924395199220785928036687273938648'-- params:-    k: '10'-    ell: '6'-  number: '59.99999999999999996595468615463592787867667500988890573654324673171421401025800846179121604786771409'-- params:-    k: '10'-    ell: '7'-  number: '69.99999999999999999995452841941861924002553783985688095997335889540455229361435728186723724816936075'-- params:-    k: '10'-    ell: '8'-  number: '79.99999999999999999999993991987781392399810404873673798479132940346038560854294804041994268606612031'-- params:-    k: '11'-    ell: '1'-  number: '10.99797533724523459964809795243574469075450093882819090308597431902540854000078754311674399450335085'-  comment: This is also HREF{T287#11}[$\xi_{11}$].-- params:-    k: '11'-    ell: '2'-  number: '21.99999851488517276973356296887971416297310448187286037590198521442657586161954157013882052989047286'-- params:-    k: '11'-    ell: '3'-  number: '32.99999999907915484654545491315215449163745431990297375840265137828907644347735690959548374034591679'-- params:-    k: '11'-    ell: '4'-  number: '43.99999999999946531877181122147256072419894574950910293217574266605946316139268086367466348320254086'-- params:-    k: '11'-    ell: '5'-  number: '54.99999999999999970022294676201591390239148458347613314279899585265004830296739456794500897466038155'-- params:-    k: '11'-    ell: '6'-  number: '65.99999999999999999983554796698832443426458379839525580909098650734782934863450405163700046159950972'-- params:-    k: '11'-    ell: '7'-  number: '76.99999999999999999999991111670246507512745952320140207088988905626058824490013908561046140475302416'-- params:-    k: '11'-    ell: '8'-  number: '87.99999999999999999999999995247642335132558037025826968825608144272689315958189086407055151646688758'-- params:-    k: '12'-    ell: '1'-  number: '11.99911450952184187994188030867116719218749935503081985334373686869595954724308587188802621404555431'-  comment: This is also HREF{T287#12}[$\xi_{12}$].-- params:-    k: '12'-    ell: '2'-  number: '23.99999973906263763888635703659509263599693504819018910906283928752947053662071847875940174056266984'-- params:-    k: '12'-    ell: '3'-  number: '35.99999999993506820569541850604743043381727068503034310488356895861522915810675023393886249474191992'-- params:-    k: '12'-    ell: '4'-  number: '47.99999999999998486927236110518216962947226737596170690457984108010802546017790713704709072849131913'-- params:-    k: '12'-    ell: '5'-  number: '59.99999999999999999659546861546359288175342248216531481598929226516094870792739885326898926339202027'-- params:-    k: '12'-    ell: '6'-  number: '71.99999999999999999999925046437135835625690761305989053541447887800803389833608802049978599676410160'-- params:-    k: '12'-    ell: '7'-  number: '83.99999999999999999999999983741994273900471654555240956479446650018331675054632253001815588891446266'-- params:-    k: '12'-    ell: '8'-  number: '95.99999999999999999999999999996511406950363288604986811024745437877209428743378103551374565332127081'+  '2':+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#2,2}[$c^*_{2,2}$].+      number: '2.687999345499491341463796937790041489618073899549236397935209599334514422260144214209251869745312739'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#2,3}[$c^*_{2,3}$].+      number: '5.071469708158110598648643546349746111224536523134635041837351018367446479545269819843879943093996070'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#2,4}[$c^*_{2,4}$].+      number: '7.321703358983259836814496629523591100449318121409982457321275056792696725897271000383910518926918359'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#2,5}[$c^*_{2,5}$].+      number: '9.496116339631318678686127155202154774263071948634199710067456727225709958713336937514852230633003440'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#2,6}[$c^*_{2,6}$].+      number: '11.62201083475983117017772549639697906479047433129974127980306614609092022913414759542897782318128499'+    '7': '13.71478188066531187419592663386781404667256909748779267691699334062513889841987887243770904218501949'+    '8': '15.78402951575241749514337877292627806948039669526522457278218532643843899662344975085249937108013643'+  '3':+    '1':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,1}[$c^*_{3,1}$].+        This is also HREF{T287#3}[$\xi_{3}$].+      number: '2.149125799907062542079080149425149925528828496714285459034660801183768221524370129087681616217046728'+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,2}[$c^*_{3,2}$].+      number: '5.656576349425156386746617941568588192659141691293506742008009681442341139286464459237741396129083249'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,3}[$c^*_{3,3}$].+      number: '8.849851652473190362812087757716704933249004939448022043446615039577597636956590060258950889502471470'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,4}[$c^*_{3,4}$].+      number: '11.93324063021001784506567092268224882890651379335616698876346069366242440532751486950076095207023045'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,5}[$c^*_{3,5}$].+      number: '14.97035798828812027443562587311940396807103878054315654978741750621682886826142355899362774443783528'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#3,6}[$c^*_{3,6}$].+      number: '17.98693229671281992725192590653386429471971042695500411009746368493209696294996668806395227602097357'+    '7': '20.99428721963569996390356939647821605540269988019492846111677264277639899563812999474020204223244644'+    '8': '23.99752228639355428601762208898877869365731075533340009984939339959751647474711325899830231566627929'+  '4':+    '1':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,1}[$c^*_{4,1}$].+        This is also HREF{T287#4}[$\xi_{4}$].+      number: '3.593511969447426082272169884555315798860946340951301392760865045230967927293167350976509334752600022'+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,2}[$c^*_{4,2}$].+      number: '7.907674019123758850790566789844419922995108412022415004802350765389662552216371593154538151656605121'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,3}[$c^*_{4,3}$].+      number: '11.97840754548836518417183962921129418652192059426167092339410984246940990181779382295840160771130815'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,4}[$c^*_{4,4}$].+      number: '15.99506456917725118329956899417052588708356806076354172978753823547047795118689779120004162634038769'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,5}[$c^*_{4,5}$].+      number: '19.99889979196351691674775775685275577102351928181638457324870251227217088836762137643885395177578993'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#4,6}[$c^*_{4,6}$].+      number: '23.99975947604794575691841460248553012829750258139642040533729634692718122214453372206287303646511694'+    '7': '27.99994816711800378712866883725114403620723445488544625643687745996149027919522284745023750300229475'+    '8': '31.99998894879824441417761862037158295979322617563944053294099209179496582008077917131009191700694329'+  '5':+    '1':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,1}[$c^*_{5,1}$].+        This is also HREF{T287#5}[$\xi_{5}$].+      number: '4.801007549722517843402554184936876206309580045670987737307642940448096932959992199826971793616174495'+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,2}[$c^*_{5,2}$].+      number: '9.976864303760496064518075863793575319292786047595390211066663963569991540942269856056674145453186667'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,3}[$c^*_{5,3}$].+      number: '14.99741350114707424427436529215687505875158666319175852037547329980901221594064290374107937980231339'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,4}[$c^*_{5,4}$].+      number: '19.99972511819509434997149501618076756231262856417056906108861824236177909822971068355118859476119916'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,5}[$c^*_{5,5}$].+      number: '24.99997174425962980488681204119343855489405198561699586682058895464280942938737404120353105180800265'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#5,6}[$c^*_{5,6}$].+      number: '29.99999715761549573994778634148325126494830338224240111896361395105934488672352627555997411293020102'+    '7': '34.99999971828689495073763165197240517066200863512164923396388731330097929439240393661829966466206228'+    '8': '39.99999997237895330621788962981778883738297516823892960097598945470307781177261883554921351313680319'+  '6':+    '1':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,1}[$c^*_{6,1}$].+        This is also HREF{T287#6}[$\xi_{6}$].+      number: '5.903000058948943810458175385422799724242893481515842146827627134583507853576375177519654679441566068'+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,2}[$c^*_{6,2}$].+      number: '11.99466733019710920890189533055350295714134739551676260854413064936109585534197475883076176045683468'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,3}[$c^*_{6,3}$].+      number: '17.99973347635061616803411249948014909629259883939838752321236504416573051808206817338186269012884600'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,4}[$c^*_{6,4}$].+      number: '23.99998747487764539641451256705490343509005188952388808136197333588086708276694929140893017510583906'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,5}[$c^*_{6,5}$].+      number: '29.99999943152413327868681837784491101013451684606105198098041990967162124172322669914399673273473956'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#6,6}[$c^*_{6,6}$].+      number: '35.99999997475451784136341288345456857119692422196366712451810528566280859222197757374080216380199990'+    '7': '41.99999999889542414122366233303485143320654681741093711561127695504527103766420273066250232144793028'+    '8': '47.99999999995219035911121073083261956776388908486262174893059213350857113390914160633563926285687229'+  '7':+    '1':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,1}[$c^*_{7,1}$].+        This is also HREF{T287#7}[$\xi_{7}$].+      number: '6.953455713353474571757750459668776777157703675694193843415968941936946472738479953900380174362202268'+    '2':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,2}[$c^*_{7,2}$].+      number: '13.99885801107353040821026786739382027366723850806829945798703162739514390358101431148202137692247181'+    '3':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,3}[$c^*_{7,3}$].+      number: '20.99997542174545116865984920724717820919492653448481645717520596822045513748851642102679476382231719'+    '4':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,4}[$c^*_{7,4}$].+      number: '27.99999950416907812559837246489148384069148256876607111079228415555129553225681370701807267652350603'+    '5':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,5}[$c^*_{7,5}$].+      number: '34.99999999034126708478328429759287181356009191496219282981587565655646617784379490803056882062904127'+    '6':+      comment: The associated cuckoo hashing threshold is HREF{T274#7,6}[$c^*_{7,6}$].+      number: '41.99999999981590402367849092443096959604956068083312186280042756840973653364740468966946712440564761'+    '7': '48.99999999999654292747548744500679934181770500628007276070831287944725977561733647959273498745564802'+    '8': '55.99999999999993577845740933717347678095367153698328337540863073043813488092281488750741802432689143'+  '8':+    '1':+      number: '7.978107736977070325801043211904953879262698393828180241246253405412112875926606187603354958634672340'+      comment: This is also HREF{T287#8}[$\xi_{8}$].+    '2': '15.99976948102927827986542124541488713267127746399729766072612805128795102265768110760940453279424376'+    '3': '23.99999791249776477526583476080506216234102397159993337514112341684209297104091256244852792327457149'+    '4': '31.99999998229421298460254326745152221550730148833673533729524296426543935216880675395508263187643527'+    '5': '39.99999999985498950806757475620303349980850606530853798231452429691164904454209053823396541671456058'+    '6': '47.99999999999883796011733170868052057137582534802368310535910608098338588451248045651064858878746216'+    '7': '55.99999999999999082549391562002909328701108769779675895602517556708034266361241800416573464377913288'+    '8': '63.99999999999999992834402649754375760190095303136801320458170026394134384645740733675987984543584298'+  '9':+    '1':+      number: '8.989912519269479737859486962678439859820876867181214191191759058921700750894868987723157685967200046'+      comment: This is also HREF{T287#9}[$\xi_{9}$].+    '2': '17.99995558761294977966818068289267088363733341183405815612510976231904637506443052969637856628689981'+    '3': '26.99999983352352961042941034339633686304220206518353674195280425530683901186238491867268566768515694'+    '4': '35.99999999941561385098893372788096996561690281820301749212947615290516804614045183422834008690123778'+    '5': '44.99999999999801919305082234770327569019046986214351730820829668360886856474366257676088334662390077'+    '6': '53.99999999999999343065909329240090609003008758610857626415549884626998582815644147538220664394674172'+    '7': '62.99999999999999997853447576929085008477958958894807023008980265237111984961160627932241566256965976'+    '8': '71.99999999999999999993061441609145926492092198540629260762095728903122120396676581175298500932542075'+  '10':+    '1':+      number: '9.995441133814842741400932592133052045659869599058257396947103091116896080275781481170753765750751443'+      comment: This is also HREF{T287#10}[$\xi_{10}$].+    '2': '19.99999175532397639510723938539197172685876190125580078940097278567402872373190369462770109320805875'+    '3': '29.99999998736720884786849000606127949609681146401724642592851062919967298195042923078301953657939331'+    '4': '39.99999999998187368851045929393106882058385746209414203789099520628733836767523116000236057505306730'+    '5': '49.99999999999997488606968796925152644784476238575890384234977335355924395199220785928036687273938648'+    '6': '59.99999999999999996595468615463592787867667500988890573654324673171421401025800846179121604786771409'+    '7': '69.99999999999999999995452841941861924002553783985688095997335889540455229361435728186723724816936075'+    '8': '79.99999999999999999999993991987781392399810404873673798479132940346038560854294804041994268606612031'+  '11':+    '1':+      number: '10.99797533724523459964809795243574469075450093882819090308597431902540854000078754311674399450335085'+      comment: This is also HREF{T287#11}[$\xi_{11}$].+    '2': '21.99999851488517276973356296887971416297310448187286037590198521442657586161954157013882052989047286'+    '3': '32.99999999907915484654545491315215449163745431990297375840265137828907644347735690959548374034591679'+    '4': '43.99999999999946531877181122147256072419894574950910293217574266605946316139268086367466348320254086'+    '5': '54.99999999999999970022294676201591390239148458347613314279899585265004830296739456794500897466038155'+    '6': '65.99999999999999999983554796698832443426458379839525580909098650734782934863450405163700046159950972'+    '7': '76.99999999999999999999991111670246507512745952320140207088988905626058824490013908561046140475302416'+    '8': '87.99999999999999999999999995247642335132558037025826968825608144272689315958189086407055151646688758'+  '12':+    '1':+      number: '11.99911450952184187994188030867116719218749935503081985334373686869595954724308587188802621404555431'+      comment: This is also HREF{T287#12}[$\xi_{12}$].+    '2': '23.99999973906263763888635703659509263599693504819018910906283928752947053662071847875940174056266984'+    '3': '35.99999999993506820569541850604743043381727068503034310488356895861522915810675023393886249474191992'+    '4': '47.99999999999998486927236110518216962947226737596170690457984108010802546017790713704709072849131913'+    '5': '59.99999999999999999659546861546359288175342248216531481598929226516094870792739885326898926339202027'+    '6': '71.99999999999999999999925046437135835625690761305989053541447887800803389833608802049978599676410160'+    '7': '83.99999999999999999999999983741994273900471654555240956479446650018331675054632253001815588891446266'+    '8': '95.99999999999999999999999999996511406950363288604986811024745437877209428743378103551374565332127081'

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