Channel Theory for Polynomial Continued Fractions: Asymptotic Channels, the ξ₀ = 2/√β₂ Identity, and a Bridge Conjecture

{"Version":[0,994,996],"notice":[1,128,536,945,1054],"(v1.7).":[2],"This":[3,177],"version":[4,954,1046,1062,1079,1442],"corrects":[5],"and":[6,73,90,117,151,165,199,224,277,341,375,475,534,569,626,645,681,695,713,738,756,771,782,822,886,897,920,943,953,965,977,1064,1087,1166,1290,1343,1360,1383,1435,1478,1502,1547,1568,1570,1580],"attributes;":[7],"it":[8,102,152,186,372,498,710,931,1138,1561],"alters":[9],"no":[10,69,487,820,1460,1576],"theorem,":[11],"proposition":[12],"or":[13,161,1160,1598],"numerical":[14],"value":[15],"of":[16,108,125,352,581,596,658,840,973,1229,1333,1347,1357,1399],"v1.6.":[17],"Concept":[18],"DOI":[19,997,1097],"is":[20,25,28,49,64,75,95,138,147,153,169,172,179,187,197,278,349,364,368,373,469,472,508,522,560,570,632,641,646,671,684,711,743,749,759,825,879,941,1075,1098,1353,1372,1463,1562,1573,1583],"preserved":[21],"(10.5281/zenodo.19941678);":[22],"v1.6":[23,51,140,159,731,966,1037,1089,1469],"(10.5281/zenodo.22782807)":[24],"superseded.":[26],"It":[27,432,471,559],"a":[29,65,98,132,350,370,390,451,610,615,648,855,869,1108,1143,1268,1329,1355,1388,1397,1420,1432,1441,1451,1475,1592],"description-only":[30,66],"version;":[31],"the":[32,41,50,87,91,111,118,139,148,154,158,180,202,233,269,299,330,353,366,376,456,463,492,505,575,582,588,593,599,603,633,662,672,677,688,696,744,747,757,768,772,823,841,874,898,906,928,970,991,1043,1068,1076,1084,1088,1095,1101,1169,1213,1221,1226,1230,1274,1278,1299,1304,1317,1321,1336,1339,1344,1350,1362,1368,1384,1409,1445,1486,1524,1536,1559,1581],"paragraph":[33],"immediately":[34],"below":[35,185],"states":[36],"what":[37,191],"that":[38,134,171,454,682,762,1423,1439,1571],"means":[39],"for":[40,228,253,545,629,803,958,962,1220,1277,1403,1549],"attached":[42,46],"file.":[43],"The":[44,54,80,325,361,409,512,526,637,651,669,720,784,844,914,1055,1114,1200,1453,1482],"file":[45],"to":[47,77,189,345,436,458,904,1061,1172,1192,1298,1428,1591],"v1.7":[48,55,63,1047,1074],"build,":[52,1476,1488],"unchanged.":[53],"corrections":[56,1457],"are":[57,84,201,623,848,967,985,1458],"recorded":[58],"in":[59,86,298,510,557,614,704,873,938,987,1093,1408,1443,1605],"this":[60,127,535,705,890,944,988,1053,1358],"description":[61,1291],"only.":[62,960,1281],"new":[67,566],"version:":[68],"PDF":[70,162,1484],"was":[71,691,726,1131,1218,1316,1448,1485],"rebuilt,":[72],"none":[74],"claimed":[76],"have":[78,900],"been.":[79],"v1.4–v1.6":[81],"LaTeX":[82],"sources":[83,160],"not":[85,144,323,369,477,500,538,555,571,647,715,733,735,881,901,947,986,1267,1354,1394,1449,1466,1511],"working":[88],"repository,":[89],"only":[92,1555],"source":[93],"held":[94],"v1.3.1":[96],"—":[97,232,248,281,307,328,348,687,732,737,982,995,998,1000,1003,1005,1006,1007,1010,1012,1013,1014,1017,1019,1021,1024,1026,1028,1031,1033,1035,1038,1040,1042,1048,1049,1050,1051,1052,1121,1130,1196,1287,1292,1328,1396,1471,1491,1508,1575],"build":[99,133,168,925],"made":[100,114],"from":[101,548,969,1083,1100,1107,1295],"would":[103],"silently":[104],"revert":[105],"two":[106,630,663,838,899,1197,1401,1425,1429],"versions":[107,984],"corrections,":[109],"including":[110],"Painlevé-target":[112],"correction":[113],"at":[115,122,337,343,384,395,516,578,598,643,693,700,723,730,752,780,792,799,1134,1179,1184,1349,1381,1498,1506,1565],"v1.4":[116,694,1023,1194,1454],"analytic":[119,1584],"residual":[120,1582],"added":[121],"v1.6,":[123],"both":[124,1091],"which":[126,521,655,742,1122,1444],"relies":[129],"on.":[130],"Attaching":[131],"cannot":[135,529],"be":[136,478,530,716,905],"verified":[137,404,1183],"defect":[141],"(1)":[142,483,1199,1481],"repeated;":[143],"attaching":[145],"one":[146,155,549,607,627,777],"avoidable":[149],"choice,":[150],"taken.":[156],"If":[157],"become":[163],"available":[164],"an":[166,1405],"updated":[167],"wanted,":[170],"v1.8.":[173],"0.":[174],"Current":[175],"state":[176],"section":[178],"whole":[181],"current":[182,1044],"state.":[183],"Nothing":[184],"needed":[188],"know":[190],"Channel":[192,866,939,1314],"Theory":[193,940,1315],"now":[194,382,750,1563],"asserts;":[195],"§1":[196],"history":[198,959],"§2–§7":[200],"detail.":[203],"Proven.":[204],"For":[205,1204],"Q_n":[206,292,789],"=":[207,213,216,220,237,246,286,290,302,305,309,318,339,380,397,402,413,425,446,794,801,856,1119,1126,1136,1176,1181,1186,1189,1216,1241,1265,1310,1541],"b(n)Q_{n−1}":[208],"+":[209,314,320,1258,1260,1543],"Q_{n−2}":[210],"with":[211,242,316,357,434,450,830,865,1124,1251,1411,1521,1554],"Q_{−1}":[212],"0,":[214,857],"Q_0":[215],"1,":[217,835,858],"deg":[218,854],"b":[219],"d":[221,338,386,396,401,445,754,793,1125,1135,1180,1185,1188,1264,1551],"≥":[222,230,256,387,1211,1552],"1":[223,257,259,388,871,1542],"b(k)":[225,239],"≠":[226,273,1312],"0":[227,252,274,802,1209],"k":[229],"1:":[231],"normalised":[234],"ratio":[235,1342],"r_n":[236,245],"Q_n/∏_{k≤n}":[238],"converges":[240],"unconditionally,":[241],"r_∞":[243,261,272,414,484],"−":[244,421,429,466,1244],"O(n^{−(2d−1)});":[247],"if":[249],"b(n)":[250,424,1207],">":[251,1208],"all":[254,1550],"n":[255],"then":[258],"≤":[260,262,459],"exp":[263],"Σ_{k≥1}":[264],"1/(b(k)b(k+1))":[265],"<":[266],"∞,":[267],"so":[268,808,889,1073],"hypothesis":[270],"(H):":[271],"holds":[275,383],"automatically":[276],"two-sidedly":[279],"bounded;":[280],"under":[282,563,573],"(H),":[283],"lim":[284],"|Q_n/(dn)!|^{1/n}":[285],"|β_d|/d^d,":[287],"hence":[288],"B(w)":[289],"Σ_n":[291,788],"w^n/Γ(dn+1)":[293],"has":[294,486],"radius":[295,326,748],"d^d/|β_d|":[296],"and,":[297],"uniformiser":[300],"ζ":[301],"w^{1/d},":[303],"ξ₀":[304,379,1202,1215],"d/|β_d|^{1/d};":[306],"Q_n/Γ(dn+1)":[308],"K":[310,412,490],"(β_d/d^d)^n":[311],"n^s":[312],"(1":[313],"O(1/n))":[315],"s":[317,507],"β_{d−1}/β_d":[319],"(d−1)/2.":[321],"Attributed,":[322],"claimed.":[324],"identity":[327,1214],"i.e.":[329],"statement":[331,367,392],"carried":[332,561],"since":[333],"v1.0":[334],"as":[335,389,480,728,1418,1516,1602],"prop:xi0":[336],"2":[340,398,859,1127],"promoted":[342],"v1.3":[344,1016,1152,1507],"Conjecture":[346,1173,1377,1566],"3.3.A★":[347,1567],"corollary":[351],"Perron–Kreuser":[354],"theorem":[355,1356],"together":[356,829],"Pincherle's":[358],"minimal/dominant":[359],"dichotomy.":[360],"conjecture":[362],"label":[363],"retired:":[365],"conjecture,":[371,1594],"classical,":[374],"cross-degree":[377,1116,1170],"formula":[378,1171],"d/|β_d|^{1/d}":[381],"every":[385,656,753,804],"proved":[391,751,1219],"rather":[393,1585],"than":[394,1586],"proven":[399,1178],"/":[400,416,1103],"4":[403],"(§2).":[405],"Verified,":[406],"never":[407,1106],"proven.":[408],"closed":[410,488,543,612],"form":[411,544,613],"(2π)^{(d−1)/2}":[415],"(√d":[417],"·":[418],"∏_i":[419,427,464],"Γ(1":[420,465],"r_i)),":[422],"where":[423],"β_d":[426,1245,1556],"(n":[428,1210],"r_i)":[430,467],"(§4).":[431],"agrees":[433,1123],"computation":[435,1578],"109–115":[437],"decimal":[438],"digits":[439,461,1430],"over":[440],"eight":[441],"test":[442],"polynomials":[443],"spanning":[444],"2,":[447,795,862,1182,1553],"3,":[448],"4,":[449,896,1145,1187],"vacuity":[452],"control":[453],"collapses":[455],"agreement":[457],"0.75":[460],"when":[462],"factor":[468,1398],"removed.":[470],"labelled":[473],"VERIFIED":[474],"must":[476,714],"written":[479],"PROVEN.":[481],"Open.":[482],"itself":[485,685,1373],"form.":[489],"isolates":[491],"single":[493],"transcendental":[494],"input":[495],"into":[496,1431],"r_∞;":[497],"does":[499,537,946,1465],"evaluate":[501],"it.":[502,540],"(2)":[503,1282,1513],"Whether":[504,767],"exponent":[506],"already":[509],"print.":[511],"literature":[513],"route":[514],"ends":[515],"Wong":[517],"&":[518],"Li":[519],"II,":[520],"NOT":[523,1492],"OBTAINED":[524],"(§3).":[525],"priority":[527],"position":[528],"settled":[531],"without":[532],"it,":[533],"settle":[539],"(3)":[541,1528],"A":[542,876,1529],"S_{ζ}.":[546],"Excluded":[547],"explicitly":[550],"printed":[551,842],"search":[552],"space":[553],"(§6.4);":[554],"excluded":[556,827],"general.":[558],"here":[562],"its":[564,809,1390,1494,1503],"own":[565,868,1391],"tag,":[567],"op:cc-szeta-closed-form,":[568],"filed":[572],"op:cc-stokes-closed-form:":[574],"latter,":[576],"created":[577],"v1.5":[579,964,1030,1326,1415,1483],"out":[580],"demoted":[583],"Theorem":[584,1334],"3.7(iii),":[585],"asks":[586,605],"whether":[587,606],"CC-channel":[589,1340],"connection-coefficient":[590,1341],"*ratio":[591],"matches":[592],"Riemann–Hilbert":[594,1345],"datum":[595,775,1346],"P_V/D₅^(1)":[597,1348],"V_quad":[600,652,773,1351],"τ-parameter,":[601,1352],"whereas":[602],"former":[604],"number":[608],"admits":[609],"multiplicative":[611],"stated":[616,642],"basis.":[617],"Both":[618],"mention":[619],"Stokes":[620],"data;":[621],"they":[622],"different":[624],"questions,":[625],"name":[628],"objects":[631],"trap":[634],"§6.0":[635],"forbids.":[636],"relation":[638],"between":[639,1338],"them":[640],"§6.4":[644],"merger.":[649],"(4)":[650,1588],"normalisation":[653],"map,":[654],"comparison":[657,1389],"Borel-plane":[659,774],"constants":[660],"across":[661],"normalisations":[664],"still":[665,707],"waits":[666],"on":[667,676,980,1303,1558],"(§6.5).":[668],"target":[670],"Painlevé":[673,1296,1300],"V":[674,1301],"transcendent":[675,1302],"Sakai":[678,1305],"surface":[679,1306],"D₅^(1),":[680],"identification":[683,697],"conjectural":[686,699,1374],"P-III(D₆)":[689,1323],"naming":[690],"corrected":[692,1294,1436],"marked":[698],"v1.5.":[701,1468],"Working":[702],"material":[703],"repository":[706,989],"says":[708],"P-III(D₆);":[709],"stale":[712,1322],"copied":[717],"forward.":[718],"(5)":[719],"singularity":[721],"type":[722,758],"ξ₀.":[724],"op:xi0-d3-direct":[725,1514],"withdrawn":[727,1515],"vacuous":[729],"closed,":[734],"answered":[736],"replaced":[739,1518],"by":[740,761,837,853,1128,1225,1404,1519],"op:xi0-singularity-type,":[741,1520],"live":[745],"problem:":[746],"(§2),":[755],"untouched":[760],"proof":[763],"(§5,":[764],"§8).":[765],"(6)":[766],"generating-function":[769,1223],"operator":[770,785],"determine":[776],"another.":[778],"New":[779],"v1.7,":[781],"untagged.":[783],"L_d":[786],"annihilating":[787],"z^n":[790],"has,":[791],"Katz":[796],"invariant":[797],"1/2":[798],"z":[800],"admissible":[805],"coefficient":[806,1239],"vector,":[807],"multiset":[810,878],"always":[811],"contains":[812],"1/2;":[813],"van":[814],"der":[815],"Put–Saito's":[816],"D̃₅/P_V":[817],"row":[818,824],"prints":[819],"1/2,":[821],"therefore":[826],"outright,":[828],"six":[831,1069],"others":[832],"(§7.2,":[833],"Tier":[834,861],"confirmed":[836],"readings":[839],"page).":[843],"three":[845],"surviving":[846],"rows":[847,885],"PIII(D₈),":[849],"PIII(D₇),":[850],"PIII(D₆),":[851],"selected":[852],"(§7.3,":[860],"self-derived":[863],"only),":[864],"Theory's":[867],"≡":[870],"falling":[872],"first.":[875],"matching":[877],"necessary,":[880],"sufficient:":[882],"§7":[883],"excludes":[884],"assigns":[887],"none,":[888],"neither":[891],"supports":[892],"nor":[893],"disputes":[894],"item":[895],"been":[902],"shown":[903],"same":[907],"object":[908,1224,1447],"(§7.0,":[909],"§7.5,":[910],"§7.6).":[911],"Not":[912],"formalised.":[913],"Lean":[915],"development":[916],"carries":[917],"seven":[918],"sorrys":[919],"zero":[921],"discharged":[922],"obligations.":[923],"lake":[924],"succeeding":[926],"shows":[927,932],"statements":[929],"typecheck;":[930],"nothing":[933,1158],"about":[934,1400],"provability.":[935],"No":[936],"result":[937],"machine-checked,":[942],"claim":[948,1462],"formal":[949,1270],"verification.":[950],"1.":[951],"Erratum":[952],"stack":[955],"Read":[956],"downward":[957],"Entries":[961],"v1.4,":[963],"quoted":[968],"deposited":[971],"descriptions":[972],"records":[974],"20585624,":[975],"22717792":[976],"22782807,":[978],"fetched":[979],"2026-09-19":[981,1082],"those":[983],"(see":[990,1526],"box":[992],"below).":[993],"Published":[999],"Disposition":[1001],"v1.1":[1002,1111,1115],"10.5281/zenodo.19941679":[1004],"superseded":[1008,1015,1022,1029,1036,1273],"v1.2":[1009],"10.5281/zenodo.19951331":[1011],"10.5281/zenodo.19972394":[1018],"2026-05-02":[1020],"10.5281/zenodo.20585624":[1025],"2026-06-07":[1027],"10.5281/zenodo.22717792":[1032],"2026-09-12":[1034],"10.5281/zenodo.22782807":[1039],"2026-09-16":[1041],"published":[1045],"concept":[1056,1085,1096],"record":[1057,1086,1109],"10.5281/zenodo.19941678":[1058],"currently":[1059],"resolves":[1060],"1.6,":[1063],"DataCite":[1065],"lists":[1066],"exactly":[1067],"HasVersion":[1070],"identifiers":[1071],"above,":[1072],"correct":[1077],"next":[1078],"number.":[1080],"(Read":[1081],"record,":[1090],"listed":[1092],"§10;":[1094],"taken":[1099],"conceptdoi":[1102],"parent":[1104],"field,":[1105],"number.)":[1110],"→":[1112],"v1.2.":[1113],"candidate":[1117],"c(d)":[1118],"2√((d−1)!)":[1120],"coincidence":[1129],"empirically":[1132],"falsified":[1133],"4:":[1137],"predicts":[1139],"≈":[1140,1146],"4.899":[1141],"against":[1142],"measured":[1144],"22":[1147],"%":[1148,1413],"disagreement":[1149],"(Remark":[1150],"3.3.E).":[1151],"(2026-05-02).":[1153],"Purely":[1154],"additive":[1155],"framing":[1156],"revision;":[1157],"rewritten":[1159],"retracted.":[1161],"Added":[1162],"§3.5":[1163],"(rem:h4-median),":[1164],"§3.6,":[1165],"op:cc-median-resurgence-execute.":[1167],"Promoted":[1168],"3.3.A★,":[1174],"ξ₀(b)":[1175],"d/β_d^{1/d},":[1177],"3":[1190],"deferred":[1191],"op:xi0-d3-direct.":[1193],"(2026-06-07)":[1195],"corrections.":[1198],"EBR":[1201,1232],"upgrade.":[1203],"positivity":[1205],"families":[1206],"1),":[1212],"d/β_d^{1/d}":[1217],"physical":[1222,1255],"holonomic/positivity":[1227],"argument":[1228],"companion":[1231],"preprint":[1233],"10.5281/zenodo.20564079":[1234],"§3":[1235],"(D-finite":[1236],"G;":[1237],"leading":[1238],"a_{2d}(s)":[1240],"d^d":[1242],"s^{2d}(d^d":[1243],"s),":[1246],"roots":[1247],"{0,":[1248],"R};":[1249],"Pringsheim),":[1250],"scope":[1252],"positive-b":[1253,1279],"families,":[1254],"object,":[1256],"symbolic":[1257],"verified-to-d6":[1259],"Lean-checked":[1261],"algebraic":[1262],"layer":[1263],"2–5,":[1266],"symbolic-degree":[1269],"proof.":[1271],"That":[1272],"Conjecture-3.3.A★":[1275],"status":[1276],"case":[1280],"Surface-type":[1283],"correction.":[1284],"All":[1285],"references":[1286],"body,":[1288],"keywords":[1289],"were":[1293],"III(D₆)":[1297],"D₅^(1)":[1307],"(W(A₃^(1)),":[1308],"δ":[1309],"−1/2":[1311],"0).":[1313],"last":[1318],"deposit":[1319],"carrying":[1320],"root":[1324],"claim.":[1325],"(2026-09-12)":[1327],"demotion.":[1330],"Clause":[1331],"(iii)":[1332],"3.7,":[1335],"match":[1337],"paper":[1359,1376],"became":[1361],"open":[1363,1599],"problem":[1364,1572,1600],"op:cc-stokes-closed-form.":[1365],"Two":[1366],"grounds:":[1367],"imported":[1369],"surface-type":[1370],"premise":[1371],"(companion":[1375],"4.1,":[1378],"10.5281/zenodo.20455089,":[1379],"read":[1380],"v1.2),":[1382],"cited":[1385,1446,1564],"run":[1386],"stores":[1387],"verdict":[1392],"did":[1393],"incorporate":[1395],"accounted":[1402],"explicit":[1406],"halving":[1407],"source,":[1410],"~1.7":[1412],"unexplained.":[1414],"also":[1416],"withdrew":[1417],"non-evidential":[1419],"Domb–Sykes":[1421],"sentence":[1422,1455],"rounded":[1424],"endpoints":[1426],"agreeing":[1427],"three-digit":[1433],"\\"range\\",":[1434],"five":[1437,1472],"citations":[1438],"pinned":[1440],"yet":[1450],"conjecture.":[1452],"\\"both":[1456],"additive/clarifying;":[1459],"…":[1461],"retracted\\"":[1464],"describe":[1467],"(2026-09-16)":[1470],"defects":[1473],"repaired;":[1474],"labelling":[1477],"open-problem":[1479],"erratum.":[1480],"draft":[1487],"watermarked":[1489],"\\"DRAFT":[1490],"DEPOSITED\\",":[1493],"date":[1495],"line":[1496],"stalled":[1497],"\\"2026-06-07":[1499],"(v1.4,":[1500],"draft)\\"":[1501],"changelog":[1504],"ending":[1505],"front":[1509],"matter,":[1510],"text.":[1512],"vacuous,":[1517],"rem:d3-vacuous":[1522],"recording":[1523],"withdrawal":[1525],"§5).":[1527],"citation":[1530],"gap:":[1531],"10.5281/zenodo.20542162":[1532],"(deposited":[1533],"2026-06-04)":[1534],"derives":[1535],"slope-1/d":[1537],"edge":[1538],"polynomial":[1539],"χ_d(c)":[1540],"(−1)^{d+1}(β_d/d^d)":[1544],"c^d":[1545],"symbolically":[1546],"d-free":[1548],"appearing":[1557],"edge;":[1560],"op:xi0-degree-d,":[1569],"restated":[1574],"per-degree":[1577],"remains,":[1579],"algebraic.":[1587],"Every":[1589],"cross-reference":[1590],"proposition,":[1593],"remark,":[1595],"definition,":[1596],"example":[1597],"rendered":[1601],"\\"Theorem":[1603],"N\\"":[1604],"v1.0–v1.5":[1606],"(110":[1607],"sites),":[1608],"because":[1609]}