{
 "what": "Lane F of the openai/math audit: every finite core named by the census (corpus/openai-math/f-lane.json, fixed before any decision), decided by a clean-room exact decider written here from the paper's statement, reading the release's own bytes at the pin. CERTIFIED means the published object has the claimed property, decided by our code; each row's `decides` says whether that is the whole headline or a finite component of an argument whose analytic parts are not decided here.",
 "release": "fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb",
 "membership": {
  "rows": 66,
  "manuscripts": 65,
  "tiers": {
   "A": 47,
   "B": 4,
   "C": 15
  }
 },
 "counts": {
  "CERTIFIED": 44,
  "REFUSED": 7,
  "NEEDS DATA": 15
 },
 "rows": [
  {
   "id": "F-009",
   "index": 9,
   "family": "005",
   "title": "Catalan's constant is irrational",
   "tier": "A",
   "dir": "Catalans-constant-is-irrational-September-24-2026",
   "leanChallenges": [
    "Catalan"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the three finite components of the proof and their comparison, not the headline. (1) the real-place certificate — for the published trial sequences, the right side of eq:energy-dual is <= -2.290939875 (kappa=2) and -2.296789875 (kappa=1), with every stationary point of the two barrier functions isolated and every printed bracket, variation count and candidate value recomputed; (2) the three fixed 48x48 matrices B0, B0+B1, B0-B1 are invertible (exact determinants over Q, the mod-101 pivot table, the base matrix built two ways); (3) the finite-place constant -8609/4608 - (2809/4608) log 2 > -2.29084 (the loss function and its area 8609/2 from the counting formulas, 505/4608); (4) the strict comparison upper < lower. The analytic framework that makes these imply irrationality is not decided.",
   "decider": "instruments/openaimath/finite/f009_catalan.py",
   "decides": "a finite component: the three finite components of the proof and their comparison, not the headline. (1) the real-place certificate — for the published trial sequences, the right side of eq:energy-dual is <= -2.290939875 (kappa=2) and -2.296789875 (kappa=1), with every stationary point of the two barrier functions isolated and every printed bracket, variation count and candidate value recomputed; (2) the three fixed 48x48 matrices B0, B0+B1, B0-B1 are invertible (exact determinants over Q, the mod-101 pivot table, the base matrix built two ways); (3) the finite-place constant -8609/4608 - (2809/4608) log 2 > -2.29084 (the loss function and its area 8609/2 from the counting formulas, 505/4608); (4) the strict comparison upper < lower. The analytic framework that makes these imply irrationality is not decided.",
   "value": {
    "rhs_kappa2": "-2.291102507313857",
    "rhs_kappa1": "-2.297495441030621",
    "lower_constant": "-2.290809555163387",
    "margin": "2.929522e-04",
    "paper_margin": "9.987500e-05",
    "sup": {
     "X_2": "-0.984033385315694",
     "Y_2": "-1.608946411646794",
     "X_1": "-1.324655807046008",
     "Y_1": "-1.428286151250130"
    },
    "norm_upper": {
     "kappa=2": "0.778415976283593",
     "kappa=1": "0.931985203900479"
    },
    "log2_enclosure_width": "3.9e-56",
    "det_digits": {
     "B0": [
      1994,
      1770
     ],
     "B0+B1": [
      2008,
      1769
     ],
     "B0-B1": [
      2013,
      1767
     ]
    },
    "loss_area": "8609/2"
   },
   "checks": [
    {
     "check": "0. the paper states the theorem and the two constants (introduction.tex:16-18, 177-182; conclusion.tex:9-15)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. the energy dual and the barrier fields as used (energy.tex:10-16, 37-47)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. the rational enclosures of log and arctan pass their cross-checks",
     "pass": true,
     "detail": "log 2 in [0.69314718055994528623, +3.9e-56]; pi in [3.14159265358979311600, +3.2e-55]; both cross-checked by a second series"
    },
    {
     "check": "0. parameters alpha, beta, gamma, eta = 11/48, 7/48, 4/48, 2/48",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. trial tables: kappa=2 has d=10 coefficients for p and v, kappa=1 has d=8, lambda_2 = 0, lambda_1 >= 0",
     "pass": true,
     "detail": "lambda_1 = 247405979/100000000"
    },
    {
     "check": "1. tail counts with conjugates: p 10, v 13 (printed: 10, 13)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. base moduli at most .94 (p) and .984 (v), so every tail decays geometrically",
     "pass": true,
     "detail": "max |z|^2 = 2209/2500, 15129/15625"
    },
    {
     "check": "1. all |l_k| and |r_z| are less than 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. -1 + alpha + gamma = -11/16 (real-place.tex:7-10)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. printed bound for kappa||p||^2 + ||v||^2/2 (kappa=2): 194603994071/250000000000",
     "pass": true,
     "detail": "enclosure [0.778415976283593, 0.778415976283593]"
    },
    {
     "check": "1. printed bound for kappa||p||^2 + ||v||^2/2 (kappa=1): 931985203901/1000000000000",
     "pass": true,
     "detail": "enclosure [0.931985203900479, 0.931985203900479]"
    },
    {
     "check": "1. X_2: Q has no root in the open domain (Sturm count of Q/(x(1-x)) on [-1,1] is 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: the derivative terms agree with the value function (mean-value containment on [a, a+10^-8] at 7 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: degree of the numerator A = Q f' is 36 (printed 36)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: Descartes variation counts on the printed division points ['-1', '0', '1']",
     "pass": true,
     "detail": "recomputed [9, 9], printed [9, 9]"
    },
    {
     "check": "1. X_2: A is nonzero at the domain ends ['-1', '0', '1']",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: Sturm counts 9 distinct critical points in (-1, 0); 9 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: Sturm counts 9 distinct critical points in (0, 1); 9 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: all 18 printed brackets are disjoint, each has N_A(m) N_A(m+2) < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: brackets per interval [9, 9] = Sturm counts [9, 9], total = the printed variation total 18: every critical point lies in exactly one printed bracket",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2(-9601109148) B printed upper bound -39361361951/40000000000",
     "pass": true,
     "detail": "enclosure [-0.984034048775052, -0.984034048775052]"
    },
    {
     "check": "1. X_2(-8942317572) B printed upper bound -984375363807/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984375363807218, -0.984375363807218]"
    },
    {
     "check": "1. X_2(-7608305633) B printed upper bound -6150212829/6250000000",
     "pass": true,
     "detail": "enclosure [-0.984034052640499, -0.984034052640499]"
    },
    {
     "check": "1. X_2(-6503394794) B printed upper bound -196821104523/200000000000",
     "pass": true,
     "detail": "enclosure [-0.984105522615257, -0.984105522615257]"
    },
    {
     "check": "1. X_2(-5185864065) B printed upper bound -492017026707/500000000000",
     "pass": true,
     "detail": "enclosure [-0.984034053414638, -0.984034053414638]"
    },
    {
     "check": "1. X_2(-4015634158) B printed upper bound -7872736491/8000000000",
     "pass": true,
     "detail": "enclosure [-0.984092061375723, -0.984092061375723]"
    },
    {
     "check": "1. X_2(-3108806646) B printed upper bound -984034037901/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984034037901933, -0.984034037901933]"
    },
    {
     "check": "1. X_2(-2067921826) B printed upper bound -39374561243/40000000000",
     "pass": true,
     "detail": "enclosure [-0.984364031075363, -0.984364031075363]"
    },
    {
     "check": "1. X_2(-1589849496) B printed upper bound -19680680769/20000000000",
     "pass": true,
     "detail": "enclosure [-0.984034038450053, -0.984034038450053]"
    },
    {
     "check": "1. X_2(1531948062) B printed upper bound -196806677063/200000000000",
     "pass": true,
     "detail": "enclosure [-0.984033385315694, -0.984033385315694]"
    },
    {
     "check": "1. X_2(2072448179) B printed upper bound -984776982913/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984776982913570, -0.984776982913570]"
    },
    {
     "check": "1. X_2(3208186484) B printed upper bound -492017013449/500000000000",
     "pass": true,
     "detail": "enclosure [-0.984034026898745, -0.984034026898745]"
    },
    {
     "check": "1. X_2(4381119427) B printed upper bound -984264010107/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984264010107532, -0.984264010107532]"
    },
    {
     "check": "1. X_2(5851354199) B printed upper bound -984034029391/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984034029391120, -0.984034029391120]"
    },
    {
     "check": "1. X_2(7269030693) B printed upper bound -196849008901/200000000000",
     "pass": true,
     "detail": "enclosure [-0.984245044505194, -0.984245044505194]"
    },
    {
     "check": "1. X_2(8390277402) B printed upper bound -492017008147/500000000000",
     "pass": true,
     "detail": "enclosure [-0.984034016294865, -0.984034016294865]"
    },
    {
     "check": "1. X_2(9332614564) B printed upper bound -984567630027/1000000000000",
     "pass": true,
     "detail": "enclosure [-0.984567630027626, -0.984567630027626]"
    },
    {
     "check": "1. X_2(9709786219) B printed upper bound -246008481721/250000000000",
     "pass": true,
     "detail": "enclosure [-0.984033926884962, -0.984033926884962]"
    },
    {
     "check": "1. X_2(-10000000000) P printed upper bound -493363685773/500000000000",
     "pass": true,
     "detail": "enclosure [-0.986727371546962, -0.986727371546962]"
    },
    {
     "check": "1. X_2: printed derivative bound |f'| < 12852 on every printed bracket",
     "pass": true,
     "detail": "max enclosure of |f'| on the brackets: 0.000000"
    },
    {
     "check": "1. X_2: sup over the domain (this file: -0.984033385316, at x = 0.1531948062) <= max printed value + .000024 = -196801877063/200000000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: stronger cutoff -123/125 exceeds every printed-point value (true values, enclosed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_2: sup <= coarse cutoff -98399/100000 + .000024 (the value used in the final line)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: Q has no root in the open domain (Sturm count of Q/(x(1-x)) on [-1,1] is 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: the derivative terms agree with the value function (mean-value containment on [a, a+10^-8] at 5 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: degree of the numerator A = Q f' is 24 (printed 24)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: Descartes variation counts on the printed division points ['0', '1/4', '1/2', '3/4', '1']",
     "pass": true,
     "detail": "recomputed [5, 2, 2, 6], printed [5, 2, 2, 6]"
    },
    {
     "check": "1. Y_2: A is nonzero at the domain ends ['0', '1']",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: Sturm counts 15 distinct critical points in (0, 1); 15 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: all 15 printed brackets are disjoint, each has N_A(m) N_A(m+2) < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: brackets per interval [15] = Sturm counts [15], total = the printed variation total 15: every critical point lies in exactly one printed bracket",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2(176402802) B printed upper bound -804473205823/500000000000",
     "pass": true,
     "detail": "enclosure [-1.608946411646794, -1.608946411646794]"
    },
    {
     "check": "1. Y_2(330649406) B printed upper bound -10062184407/6250000000",
     "pass": true,
     "detail": "enclosure [-1.609949505120562, -1.609949505120562]"
    },
    {
     "check": "1. Y_2(764952882) B printed upper bound -402240073957/250000000000",
     "pass": true,
     "detail": "enclosure [-1.608960295828112, -1.608960295828112]"
    },
    {
     "check": "1. Y_2(1227250753) B printed upper bound -804547298927/500000000000",
     "pass": true,
     "detail": "enclosure [-1.609094597854745, -1.609094597854745]"
    },
    {
     "check": "1. Y_2(2149465998) B printed upper bound -3217920853/2000000000",
     "pass": true,
     "detail": "enclosure [-1.608960426500573, -1.608960426500573]"
    },
    {
     "check": "1. Y_2(3048189112) B printed upper bound -201124024209/125000000000",
     "pass": true,
     "detail": "enclosure [-1.608992193672299, -1.608992193672299]"
    },
    {
     "check": "1. Y_2(4322699096) B printed upper bound -804480214243/500000000000",
     "pass": true,
     "detail": "enclosure [-1.608960428486447, -1.608960428486447]"
    },
    {
     "check": "1. Y_2(5564757994) B printed upper bound -1608979509517/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.608979509517787, -1.608979509517787]"
    },
    {
     "check": "1. Y_2(6801929373) B printed upper bound -804480215239/500000000000",
     "pass": true,
     "detail": "enclosure [-1.608960430478166, -1.608960430478166]"
    },
    {
     "check": "1. Y_2(8031988371) B printed upper bound -402247637203/250000000000",
     "pass": true,
     "detail": "enclosure [-1.608990548812704, -1.608990548812704]"
    },
    {
     "check": "1. Y_2(8877037851) B printed upper bound -1608960429811/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.608960429811238, -1.608960429811238]"
    },
    {
     "check": "1. Y_2(9577761832) B printed upper bound -321811160579/200000000000",
     "pass": true,
     "detail": "enclosure [-1.609055802895772, -1.609055802895772]"
    },
    {
     "check": "1. Y_2(9838463999) B printed upper bound -321792082567/200000000000",
     "pass": true,
     "detail": "enclosure [-1.608960412835494, -1.608960412835494]"
    },
    {
     "check": "1. Y_2(9972727815) B printed upper bound -1609694899071/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.609694899071999, -1.609694899071999]"
    },
    {
     "check": "1. Y_2(9992037196) B printed upper bound -1608958123669/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.608958123669472, -1.608958123669472]"
    },
    {
     "check": "1. Y_2(2500000000) P printed upper bound -160897361703/100000000000",
     "pass": true,
     "detail": "enclosure [-1.608973617030970, -1.608973617030970]"
    },
    {
     "check": "1. Y_2(5000000000) P printed upper bound -804485850949/500000000000",
     "pass": true,
     "detail": "enclosure [-1.608971701898138, -1.608971701898138]"
    },
    {
     "check": "1. Y_2(7500000000) P printed upper bound -402244227013/250000000000",
     "pass": true,
     "detail": "enclosure [-1.608976908052155, -1.608976908052155]"
    },
    {
     "check": "1. Y_2: printed derivative bound |f'| < 102001 on every printed bracket",
     "pass": true,
     "detail": "max enclosure of |f'| on the brackets: 0.000028"
    },
    {
     "check": "1. Y_2: sup over the domain (this file: -1.608946411647, at x = 0.0176402803) <= max printed value + .000024 = -804461205823/500000000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: stronger cutoff -160891/100000 exceeds every printed-point value (true values, enclosed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_2: sup <= coarse cutoff -16089/10000 + .000024 (the value used in the final line)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. kappa=2: norm term below the stronger cutoff 77843/100000 and the coarse cutoff 19461/25000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: Q has no root in the open domain (Sturm count of Q/(x(1-x)) on [-1,1] is 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: the derivative terms agree with the value function (mean-value containment on [a, a+10^-8] at 7 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: degree of the numerator A = Q f' is 13 (printed 13)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: Descartes variation counts on the printed division points ['-1', '-1/2', '0', '1/2', '1']",
     "pass": true,
     "detail": "recomputed [1, 1, 2, 1], printed [1, 1, 2, 1]"
    },
    {
     "check": "1. X_1: A is nonzero at the domain ends ['-1', '0', '1']",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: Sturm counts 2 distinct critical points in (-1, 0); 2 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: Sturm counts 3 distinct critical points in (0, 1); 3 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: all 5 printed brackets are disjoint, each has N_A(m) N_A(m+2) < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: brackets per interval [2, 3] = Sturm counts [2, 3], total = the printed variation total 5: every critical point lies in exactly one printed bracket",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1(-9917299785) B printed upper bound -1389245765787/500000000000",
     "pass": true,
     "detail": "enclosure [-2.778491531574792, -2.778491531574792]"
    },
    {
     "check": "1. X_1(-2259572153) B printed upper bound -331166682987/250000000000",
     "pass": true,
     "detail": "enclosure [-1.324666731948684, -1.324666731948684]"
    },
    {
     "check": "1. X_1(2543808026) B printed upper bound -662327903523/500000000000",
     "pass": true,
     "detail": "enclosure [-1.324655807046008, -1.324655807046008]"
    },
    {
     "check": "1. X_1(4437270259) B printed upper bound -1352236629957/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.352236629957685, -1.352236629957685]"
    },
    {
     "check": "1. X_1(6348298970) B printed upper bound -52986653177/40000000000",
     "pass": true,
     "detail": "enclosure [-1.324666329425421, -1.324666329425421]"
    },
    {
     "check": "1. X_1(-10000000000) P printed upper bound -1387909038763/500000000000",
     "pass": true,
     "detail": "enclosure [-2.775818077526863, -2.775818077526863]"
    },
    {
     "check": "1. X_1(-5000000000) P printed upper bound -308811563981/200000000000",
     "pass": true,
     "detail": "enclosure [-1.544057819905265, -1.544057819905265]"
    },
    {
     "check": "1. X_1(5000000000) P printed upper bound -336889420219/250000000000",
     "pass": true,
     "detail": "enclosure [-1.347557680876120, -1.347557680876120]"
    },
    {
     "check": "1. X_1: printed derivative bound |f'| < 12852 on every printed bracket",
     "pass": true,
     "detail": "max enclosure of |f'| on the brackets: 0.000000"
    },
    {
     "check": "1. X_1: sup over the domain (this file: -1.324655807046, at x = 0.2543808027) <= max printed value + .000024 = -662315903523/500000000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: stronger cutoff -66221/50000 exceeds every printed-point value (true values, enclosed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. X_1: sup <= coarse cutoff -3311/2500 + .000024 (the value used in the final line)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: Q has no root in the open domain (Sturm count of Q/(x(1-x)) on [-1,1] is 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: the derivative terms agree with the value function (mean-value containment on [a, a+10^-8] at 5 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: degree of the numerator A = Q f' is 9 (printed 9)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: Descartes variation counts on the printed division points ['0', '1']",
     "pass": true,
     "detail": "recomputed [5], printed [5]"
    },
    {
     "check": "1. Y_1: A is nonzero at the domain ends ['0', '1']",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: Sturm counts 5 distinct critical points in (0, 1); 5 isolated here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: all 5 printed brackets are disjoint, each has N_A(m) N_A(m+2) < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: brackets per interval [5] = Sturm counts [5], total = the printed variation total 5: every critical point lies in exactly one printed bracket",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1(532669786) B printed upper bound -1142628921/800000000",
     "pass": true,
     "detail": "enclosure [-1.428286151250130, -1.428286151250130]"
    },
    {
     "check": "1. Y_1(2504239325) B printed upper bound -757801323681/500000000000",
     "pass": true,
     "detail": "enclosure [-1.515602647362797, -1.515602647362797]"
    },
    {
     "check": "1. Y_1(5701738806) B printed upper bound -357083933093/250000000000",
     "pass": true,
     "detail": "enclosure [-1.428335732372779, -1.428335732372779]"
    },
    {
     "check": "1. Y_1(7966939383) B printed upper bound -22901346323/15625000000",
     "pass": true,
     "detail": "enclosure [-1.465686164672404, -1.465686164672404]"
    },
    {
     "check": "1. Y_1(9454490138) B printed upper bound -1428335358167/1000000000000",
     "pass": true,
     "detail": "enclosure [-1.428335358167221, -1.428335358167221]"
    },
    {
     "check": "1. Y_1: printed derivative bound |f'| < 102001 on every printed bracket",
     "pass": true,
     "detail": "max enclosure of |f'| on the brackets: 0.000000"
    },
    {
     "check": "1. Y_1: sup over the domain (this file: -1.428286151250, at x = 0.0532669786) <= max printed value + .000024 = -1142609721/800000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: stronger cutoff -28561/20000 exceeds every printed-point value (true values, enclosed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Y_1: sup <= coarse cutoff -357/250 + .000024 (the value used in the final line)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. kappa=1: norm term below the stronger cutoff 18641/20000 and the coarse cutoff 9321/10000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. every bracket point is more than .0007 from 0 and 1; the bracket nearest 1 ends at 9992037198/10^10, distance 3981401/5000000000",
     "pass": true,
     "detail": "min distance 3981401/5000000000"
    },
    {
     "check": "1. 120000 * 2/10^10 = .000024 (the mean-value allowance)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. .693146 < log 2 < .693149",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. kappa=2 final line: -11/16(346573/500000) + 19461/25000 - 98399/100000 - 16089/10000 + 3/62500 = -18327519/8000000 exactly",
     "pass": true,
     "detail": "-18327519/8000000"
    },
    {
     "check": "1. kappa=1 final line: -11/16(346573/500000) + 9321/10000 - 3311/2500 - 357/250 + 3/62500 = -18374319/8000000 exactly",
     "pass": true,
     "detail": "-18374319/8000000"
    },
    {
     "check": "1. Prop prop:real-upper: the larger case constant is the printed bound -18327519/8000000 < -22909/10000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. kappa=2: this file's right side of eq:energy-dual, -2.291102507314, is <= the printed case bound -18327519/8000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. kappa=1: this file's right side of eq:energy-dual, -2.297495441031, is <= the printed case bound -18374319/8000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. N = 1 parameters: n0=48, b0=7, g0=4, h0=2, L0=59, C0=63, H0=65 (foundations.tex:12-15 at N=1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. the 49 row polynomials (P_r, D_r): explicit Chebyshev sums = the recipe recursion",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. supports: P_r in [19, 64], D_r in [20, 64] for r < 48; the auxiliary row has min deg P_48 = 18, D_48 = 19 (eq:nonvanishing-auxiliary-degrees)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. contact: t P_r / f - D_r has t-order exactly 59 + r, leading coefficient 2^(4-r), for r = 0..48 (eq:contact; O(t^107) for the auxiliary row)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "2. boundary arrays k-_d (d <= 64), k+_d (d <= 58): explicit H* sums = the recurrences",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. M^0(i, j), 0 <= i <= 64, 0 <= j <= 58: diagonal-reduction closed form = the Y recipe",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. the small reduced values printed at odd-primes.tex:279-289 (M^0(u,0), M^0(u,1), Z^0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. the base matrix B (49 x 48): every entry equal by the two routes",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. every entry of B has a 65-smooth denominator (so it reduces mod 101)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. det(B0) != 0, exactly over Q",
     "pass": true,
     "detail": "numerator 1994 digits, denominator 1770 digits"
    },
    {
     "check": "2. sigma=0 pivots mod 101, printed line 1 (entries 1-16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=0 pivots mod 101, printed line 2 (entries 17-32)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=0 pivots mod 101, printed line 3 (entries 33-48)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=0: row swaps of the elimination rule [] (printed [])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=0: all 48 pivots nonzero mod 101, and det mod 101 = (sign) prod(pivots) agrees with the exact det",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. det(B0 + B1) != 0, exactly over Q",
     "pass": true,
     "detail": "numerator 2008 digits, denominator 1769 digits"
    },
    {
     "check": "2. sigma=1 pivots mod 101, printed line 1 (entries 1-16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=1 pivots mod 101, printed line 2 (entries 17-32)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=1 pivots mod 101, printed line 3 (entries 33-48)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=1: row swaps of the elimination rule [(30, 31), (45, 46)] (printed [(30, 31), (45, 46)])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=1: all 48 pivots nonzero mod 101, and det mod 101 = (sign) prod(pivots) agrees with the exact det",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. det(B0 - B1) != 0, exactly over Q",
     "pass": true,
     "detail": "numerator 2013 digits, denominator 1767 digits"
    },
    {
     "check": "2. sigma=-1 pivots mod 101, printed line 1 (entries 1-16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=-1 pivots mod 101, printed line 2 (entries 17-32)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=-1 pivots mod 101, printed line 3 (entries 33-48)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=-1: row swaps of the elimination rule [] (printed [])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. sigma=-1: all 48 pivots nonzero mod 101, and det mod 101 = (sign) prod(pivots) agrees with the exact det",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. d0/N = 0 on [0,23], 2x-46 on [23,29], 12 on [29,65/2] (odd-primes.tex:529-536)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. R/N, S/N on [65/2,40] = (105 + -2 x, 12 + 0 x) (odd-primes.tex:542-549)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. R/N, S/N on [40,52] = (65 + -1 x, 52 + -1 x) (odd-primes.tex:542-549)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. R/N, S/N on [52,58] = (117 + -2 x, -52 + 1 x) (odd-primes.tex:542-549)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. R/N, S/N on [58,59] = (59 + -1 x, 6 + 0 x) (odd-primes.tex:542-549)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. d is continuous at 65/2 (both counting regimes give 88)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. loss table row [0,25]: d = 96 + 0 x, endpoint values 96,96, integral 2400",
     "pass": true,
     "detail": "recomputed integral 2400"
    },
    {
     "check": "3. loss table row [25,29]: d = 146 + -2 x, endpoint values 96,88, integral 368",
     "pass": true,
     "detail": "recomputed integral 368"
    },
    {
     "check": "3. loss table row [29,65/2]: d = 88 + 0 x, endpoint values 88,88, integral 308",
     "pass": true,
     "detail": "recomputed integral 308"
    },
    {
     "check": "3. loss table row [65/2,69/2]: d = 153 + -2 x, endpoint values 88,84, integral 172",
     "pass": true,
     "detail": "recomputed integral 172"
    },
    {
     "check": "3. loss table row [69/2,40]: d = 222 + -4 x, endpoint values 84,62, integral 803/2",
     "pass": true,
     "detail": "recomputed integral 803/2"
    },
    {
     "check": "3. loss table row [40,58]: d = 182 + -3 x, endpoint values 62,8, integral 630",
     "pass": true,
     "detail": "recomputed integral 630"
    },
    {
     "check": "3. loss table row [58,59]: d = 124 + -2 x, endpoint values 8,6, integral 7",
     "pass": true,
     "detail": "recomputed integral 7"
    },
    {
     "check": "3. loss table row [59,65]: d = 65 + -1 x, endpoint values 6,0, integral 18",
     "pass": true,
     "detail": "recomputed integral 18"
    },
    {
     "check": "3. the rows tile [0,65] and the area is int_0^65 d = 8609/2 (printed 8609/2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. odd-prime constant: (8609/2) / 2304 = 8609/4608 (n^2 = 2304 N^2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. 2-adic constant: delta = (q+g+h)/n = 10/48, delta/2 + delta^2/8 = 505/4608 = the quadratic minimum / n^2 (at m = 53/96 n, l = n - m)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the log 2 coefficient: 1/2 + 505/4608 = 2809/4608",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the printed rational bound 2 sum_{j<=5} 3^-(2j+1)/(2j+1) + 2 3^-13/(13(1-3^-2)) < 693149/10^6",
     "pass": true,
     "detail": "322318903/177324147000000"
    },
    {
     "check": "3. log 2 < 693149/10^6 (this file's enclosure)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. -8609/4608 - (2809/4608)(693149/10^6) = -10556055541/4608000000 > -2.29084",
     "pass": true,
     "detail": "-10556055541/4608000000"
    },
    {
     "check": "4. FINAL: this file's upper constant max(RHS_2, RHS_1) = -2.291102507314 < lower constant -8609/4608 - (2809/4608) log 2 = -2.290809555163 (margin 2.930e-04)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. the paper's chain: printed upper -2.290939875 < printed lower -2.29084",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/certificate.tex": "113f8438138e239f2f39820692aa2180b97e0729463799e40f7100a304e73c9a",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/real-place.tex": "761db4596cc260f8f62ea5480c59ce1c0b2cfac0f5b4e1ca75d9bc0cbbfd3173",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/energy.tex": "046baf5cfb1c774d0a398f47958bfbf2d4e7551e66f9bdd361347e1af51cc9eb",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/foundations.tex": "66e91b9f49c71b4453e447c371e05a43251ef6be436c84cebe9970392eab7939",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/nonvanishing.tex": "5eb530e75f876ba18b90bfe9979bdcd2336443d3f100471ca797d7b5711c29b8",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/odd-primes.tex": "81fcbfa0b6db03d727949ea7f6de286f73c0d88ada423ccb1b3943733ec450dc",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/introduction.tex": "dfef42bb80b1d6cf8caadac31e39ae8d4bfc41c9ec14f4e555cb789095349e80",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/sections/conclusion.tex": "f7d8a388041f6b1229f1d4a6f42fbbe61c81f65b68a652be270969cc6c80f1fa",
    "preprints/Catalans-constant-is-irrational-September-24-2026/build/main.tex": "bd842d0cf4310faf566ab3e68d53c1caf0db1588e8a5b79ff353d600e827334f"
   },
   "forges": [
    {
     "forge": "one kappa=2 trial coefficient changed by 10^-8 (p_1 45559127 -> 45559128): printed values no longer reproduce",
     "verdict": "REFUTED"
    },
    {
     "forge": "one root bracket moved by 2/10^10 (kappa=2, X, -9601109148 -> -9601109146)",
     "verdict": "REFUTED"
    },
    {
     "forge": "kappa=2 trial coefficient p_1 raised by 10^-3: this file's own bound is ABOVE the lower constant (-2.286528786750258)",
     "verdict": "REFUTED"
    },
    {
     "forge": "one printed modular pivot changed (sigma=0, first pivot 60 -> 61)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the auxiliary row 48 truncated (its degree-18 / degree-19 terms dropped, the slip nonvanishing.tex:168-174 warns of); the three determinants stay nonzero",
     "verdict": "REFUTED"
    },
    {
     "forge": "the log 2 cutoff of the lower constant 693149/10^6 -> 693147/10^6",
     "verdict": "REFUTED"
    },
    {
     "forge": "loss table: the [69/2,40] integral and the area each lowered by 1 (8609/2 -> 8607/2)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 10.5
  },
  {
   "id": "F-045",
   "index": 45,
   "family": "020",
   "title": "Squarefree values of quartics and power-free values of polynomials",
   "tier": "A",
   "dir": "Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026",
   "leanChallenges": [
    "PowerFreeValues"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the exact parameter certificate of Proposition prop:parameter-certificate (all 3,989 cells for d = 4..8, Table tab:parameter-certificate, the 150 endpoint derivative inequalities), the printed exponent margins, and the enlarged quartic cell of Proposition prop:cyclotomic-central; not the density theorem, whose other steps are proved in prose",
   "decider": "instruments/openaimath/finite/f045_squarefree_quartics.py",
   "decides": "a finite component: the exact parameter certificate of Proposition prop:parameter-certificate (all 3,989 cells for d = 4..8, Table tab:parameter-certificate, the 150 endpoint derivative inequalities), the printed exponent margins, and the enlarged quartic cell of Proposition prop:cyclotomic-central; not the density theorem, whose other steps are proved in prose",
   "value": {
    "4": {
     "z_d": 5124,
     "cells": 2124,
     "gap": "191/100000",
     "distinct_t*": 323,
     "argmin_z": 3555
    },
    "5": {
     "z_d": 4084,
     "cells": 1084,
     "gap": "293/50000",
     "distinct_t*": 160,
     "argmin_z": 3000
    },
    "6": {
     "z_d": 3552,
     "cells": 552,
     "gap": "321/2500",
     "distinct_t*": 81,
     "argmin_z": 3000
    },
    "7": {
     "z_d": 3226,
     "cells": 226,
     "gap": "23177/100000",
     "distinct_t*": 34,
     "argmin_z": 3000
    },
    "8": {
     "z_d": 3003,
     "cells": 3,
     "gap": "31683/100000",
     "distinct_t*": 1,
     "argmin_z": 3000
    }
   },
   "checks": [
    {
     "check": "the prescription constants as printed (K0, G0, tau, 2495/10000, grid i/50 with 1 <= i < 500, 999/1000, 3/20..5)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R(A,B) as printed in eq:hilbert-polynomial",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(H,H_a,H_b)=(10,5,10) at t*=1/2 as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:parameter-certificate parsed: five rows d = 4..8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. R = leading form of the Hilbert function, and H, H_a, H_b = the closed forms eq:parameter-polynomials (d = 4..8)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. d=4: stopping index z_d = 5124 (first z >= 3000 with eta-(1 - k eta-/d) < 2495/10000)",
     "pass": true,
     "detail": "printed 5124"
    },
    {
     "check": "1. d=4: 2124 intervals, z_d/K0 = 427/250",
     "pass": true,
     "detail": "printed 2124, 427/250"
    },
    {
     "check": "2. d=4: all 2124 cells pass (X<1, 0<m<1, min(eta-,b-)>m, both cut inequalities, 3/20<=v/u<=5, m+sum E_h<999/1000, m+2u<999/1000 in mode Q)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "3. d=4: certified gap min(1 - m - sum Ebar_h) = 191/100000",
     "pass": true,
     "detail": "printed 191/100000; attained at z=3555 t*=47/50 w=5967/25000"
    },
    {
     "check": "3. d=4: gap > d tau and > 1/1000 (so m + sum E_h + d tau < 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. d=5: stopping index z_d = 4084 (first z >= 3000 with eta-(1 - k eta-/d) < 2495/10000)",
     "pass": true,
     "detail": "printed 4084"
    },
    {
     "check": "1. d=5: 1084 intervals, z_d/K0 = 1021/750",
     "pass": true,
     "detail": "printed 1084, 1021/750"
    },
    {
     "check": "2. d=5: all 1084 cells pass (X<1, 0<m<1, min(eta-,b-)>m, both cut inequalities, 3/20<=v/u<=5, m+sum E_h<999/1000)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "3. d=5: certified gap min(1 - m - sum Ebar_h) = 293/50000",
     "pass": true,
     "detail": "printed 293/50000; attained at z=3000 t*=17/25 w=2769/20000"
    },
    {
     "check": "3. d=5: gap > d tau and > 1/1000 (so m + sum E_h + d tau < 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. d=6: stopping index z_d = 3552 (first z >= 3000 with eta-(1 - k eta-/d) < 2495/10000)",
     "pass": true,
     "detail": "printed 3552"
    },
    {
     "check": "1. d=6: 552 intervals, z_d/K0 = 148/125",
     "pass": true,
     "detail": "printed 552, 148/125"
    },
    {
     "check": "2. d=6: all 552 cells pass (X<1, 0<m<1, min(eta-,b-)>m, both cut inequalities, 3/20<=v/u<=5, m+sum E_h<999/1000)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "3. d=6: certified gap min(1 - m - sum Ebar_h) = 321/2500",
     "pass": true,
     "detail": "printed 321/2500; attained at z=3000 t*=43/50 w=274/3125"
    },
    {
     "check": "3. d=6: gap > d tau and > 1/1000 (so m + sum E_h + d tau < 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. d=7: stopping index z_d = 3226 (first z >= 3000 with eta-(1 - k eta-/d) < 2495/10000)",
     "pass": true,
     "detail": "printed 3226"
    },
    {
     "check": "1. d=7: 226 intervals, z_d/K0 = 1613/1500",
     "pass": true,
     "detail": "printed 226, 1613/1500"
    },
    {
     "check": "2. d=7: all 226 cells pass (X<1, 0<m<1, min(eta-,b-)>m, both cut inequalities, 3/20<=v/u<=5, m+sum E_h<999/1000)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "3. d=7: certified gap min(1 - m - sum Ebar_h) = 23177/100000",
     "pass": true,
     "detail": "printed 23177/100000; attained at z=3000 t*=26/25 w=6031/100000"
    },
    {
     "check": "3. d=7: gap > d tau and > 1/1000 (so m + sum E_h + d tau < 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. d=8: stopping index z_d = 3003 (first z >= 3000 with eta-(1 - k eta-/d) < 2495/10000)",
     "pass": true,
     "detail": "printed 3003"
    },
    {
     "check": "1. d=8: 3 intervals, z_d/K0 = 1001/1000",
     "pass": true,
     "detail": "printed 3, 1001/1000"
    },
    {
     "check": "2. d=8: all 3 cells pass (X<1, 0<m<1, min(eta-,b-)>m, both cut inequalities, 3/20<=v/u<=5, m+sum E_h<999/1000)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "3. d=8: certified gap min(1 - m - sum Ebar_h) = 31683/100000",
     "pass": true,
     "detail": "printed 31683/100000; attained at z=3000 t*=31/25 w=879/20000"
    },
    {
     "check": "3. d=8: gap > d tau and > 1/1000 (so m + sum E_h + d tau < 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. the 150 endpoint inequalities N >= 0, N^h < (k+t)^(h-1)(a+bt) at t = 3/20, 5; and N = D/g as an identity",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "5. printed margins: 24979/25000, 3122/3125, 40001/100000000, the large-eta weights, 9999/10000, the last Q interval ends at 4/3",
     "pass": true,
     "detail": "[True, True, True, True, True, True, True, True, True]"
    },
    {
     "check": "6. the enlarged-cell parameters as printed (13-cyclotomic.tex:61-102)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. enlarged cell: b- = 4 - 2 eta+ = 2999/1500 and b+ = 2 + 2 eps0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. (H, H_a, H_b)(1/2) = (10, 5, 10)",
     "pass": true,
     "detail": "(Fraction(10, 1), Fraction(5, 1), Fraction(10, 1))"
    },
    {
     "check": "6. enlarged cell: both strict cut inequalities with w = 24809/100000",
     "pass": true,
     "detail": "w = up_3(X): True"
    },
    {
     "check": "6. enlarged cell: m, u, v equal the printed values",
     "pass": true,
     "detail": "74427/100000 76819/1200000 313933/1000000"
    },
    {
     "check": "6. enlarged cell: 0 < m < min(eta-, b-) and 3/20 <= v/u <= 5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. enlarged cell: each printed threshold bound is a strict upper bound (dbar^h > delta^h) and delta_1 exact",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. enlarged cell: the printed bounds equal up_h of the thresholds",
     "pass": true,
     "detail": "['4729/100000', '5097/100000', '6899/100000', '76819/1200000']"
    },
    {
     "check": "6. enlarged cell: 1 - m - sum Ebar_h - 4 tau = 1941/100000 > 0",
     "pass": true,
     "detail": "1941/100000"
    },
    {
     "check": "6. enlarged cell: 1 - m - 2u - 2 tau = 15319/120000 > 0",
     "pass": true,
     "detail": "15319/120000"
    }
   ],
   "sources": {
    "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/build/sections/10-parameters.tex": "644e568819ec27886c4d3768938e31a0abaaf8579b365bc496b0b66be133c4de",
    "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/build/sections/05-determinant-cuts.tex": "8a747728f3664bed59f4d57756297b006c58ec44912f1cba0db60b736897792c",
    "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/build/sections/13-cyclotomic.tex": "b77437cc702254334be5cc77bb5355fc09c5300cb19bf79f699d5a018467ee17",
    "preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/build/sections/12-certificate.tex": "4f68ce23eb53d57c9d4a09ea0391670e11552794ed4c282b7b7d6a3d098b83c0"
   },
   "forges": [
    {
     "forge": "Table: d=4 certified gap printed 192/100000 instead of 191/100000",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table: d=6 stopping index printed 3553 instead of 3552",
     "verdict": "REFUTED"
    },
    {
     "forge": "enlarged cell: w moved one grid step down, 24808/100000 (m, u, v recomputed consistently)",
     "verdict": "REFUTED"
    },
    {
     "forge": "enlarged cell: printed delta_4 bound lowered to 4728/100000",
     "verdict": "REFUTED"
    },
    {
     "forge": "d=4, z=3000 with t* forced to 1/50 (the cell must fail)",
     "verdict": "REFUTED: m not in (0,1), min(eta-,b-) <= m, second cut inequality, v/u outside [3/20,5]"
    }
   ],
   "seconds": 5.0
  },
  {
   "id": "F-103",
   "index": 103,
   "family": "047",
   "title": "An explicit failure of complex affine-space cancellation",
   "tier": "A",
   "dir": "An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026",
   "leanChallenges": [
    "ComplexCancellation"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the cylinder isomorphism A[w] = C^[5] (Proposition prop:stabilization), by the paper's printed maps with inverses; NOT A not = C^[4], which carries the counterexample",
   "decider": "instruments/openaimath/finite/f103_affine_cancellation.py",
   "decides": "a finite component: the cylinder isomorphism A[w] = C^[5] (Proposition prop:stabilization), by the paper's printed maps with inverses; NOT A not = C^[4], which carries the counterexample",
   "value": {
    "H_terms": 12,
    "H_degree": 7,
    "Delta_nilpotency_orders": [
     1,
     4,
     2,
     7,
     4,
     1
    ],
    "e0_terms": 187,
    "W_terms": 381,
    "W_degree": 55,
    "cofactor_q_terms": 6641,
    "Phi_inv_M_terms": 669,
    "Phi_inv_M_degree": 58
   },
   "checks": [
    {
     "check": "0. printed: eq:example",
     "pass": true,
     "detail": "01-introduction.tex"
    },
    {
     "check": "0. printed: con:setup",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:setup y, z",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:identity",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:derivation",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: Delta H",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:cylinder-automorphism",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:linear-coordinates",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: determinant",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:linear-inverse",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:h-expansion",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:root C",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:root Q_1, L_*",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:root-divisibility",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:e-polynomial",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: H(L) + p^3 w = L - ph",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: eq:e-certificate",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: W(Z)",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: the five coordinates",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "1. Delta x = Delta y = Delta z = 0 and Delta u = -p^2 (Delta is -p^2 d/du in the coordinates p,x,y,z,u)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. xy - z(z+1) = p^2 (H + pu)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Delta H = p^3 and Delta^2 H = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Delta is locally nilpotent on the generators (first vanishing iterate on p,s,u,F,J,w)",
     "pass": true,
     "detail": "[1, 4, 2, 7, 4, 1]"
    },
    {
     "check": "1. Phi^-1(Phi(g)) = g for the six generators (computed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Phi(Phi^-1(g)) = g for the six generators (computed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Phi(H) = H + p^3 w (direct substitution)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. determinant 4s^2x0^2 + (1+2sx0)(1-2sx0) = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. iota(kappa(g)) = g on p,s,u,F,J,w and kappa(iota(g)) = g on p,s,u,M,L,w (computed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. eq:h-expansion H(L) = L - pu + p^2 Q(L) + p^4 F(L)^3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. C = Q(0) = M^2(2x0 + 6s x0^2 + 4s^2 x0^3 - x0^4) as printed",
     "pass": true,
     "detail": "9 terms, degree 14"
    },
    {
     "check": "3. L divides Q(L) - C, so Q_1 = (Q(L) - C)/L is in B[L]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. eq:root-divisibility H(L_*) = p^3((u - pC) Q_1(L_*) + p F(L_*)^3)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. H(L) + p^3 w = L - p h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. eq:e-certificate p^3 e_0 - (L - L_*) = (p^2 Q_1(L) - 1)(H(L) + p^3 w)",
     "pass": true,
     "detail": "e_0: 187 terms, degree 29"
    },
    {
     "check": "3. p^3 divides H(L_* + p^3 E), so W(E) = -H(L_* + p^3 E)/p^3 is in B[E]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. beta(G) = H(L_* + p^3 E) + p^3 W(E) = 0: beta is well defined on B[L,w]/(G)",
     "pass": true,
     "detail": "W: 381 terms, degree 55, degree 3 in E"
    },
    {
     "check": "3. beta(alpha(E)) = beta(e_0) = E (computed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. alpha(beta(L)) - L = L_* + p^3 e_0 - L = (p^2 Q_1 - 1) G (explicit cofactor)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. H(L) is cubic in L over B",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. alpha(beta(w)) - w = W(e_0) - w = q G (explicit cofactor q)",
     "pass": true,
     "detail": "W(e_0): 19860 terms, degree 103; q: 6641 terms, degree 84"
    },
    {
     "check": "spot (evidence, not proof): t in Q^5 -> chi(t) lies on H = 0 and the five coordinates return t, at 3 points (one with p = 0)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/build/sections/01-introduction.tex": "735089761fe623a36c91e57b98ea6d798a4b74933257fdc2d2b19fbc51269540",
    "preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/build/sections/02-construction.tex": "fd10c0e2eb35a5572f43263ba88ea893551415c1807f6f6c8ad6158e64caa882"
   },
   "forges": [
    {
     "forge": "Delta J printed as 3x u^2 instead of 3x^2 u^2",
     "verdict": "REFUTED"
    },
    {
     "forge": "the root C with 5 s x0^2 in place of 6 s x0^2",
     "verdict": "REFUTED"
    },
    {
     "forge": "M = (1 - 2s x0) F + 3s^2 J (the change is no longer unimodular)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the relation with -2pu in place of -pu",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1.7
  },
  {
   "id": "F-105",
   "index": 105,
   "family": "049",
   "title": "A stable coordinate that is not a coordinate in four variables",
   "tier": "A",
   "dir": "A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026",
   "leanChallenges": [
    "AbhyankarSathaye",
    "CommutingDerivations",
    "StableCoordinateFour"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: f is a one-stable coordinate (an explicit automorphism of C[x1..x4,w] with exhibited inverse sends f to x1), deg f = 5, and every fibre f = lambda is C^[3]; NOT that f is not a coordinate in four variables, which carries the counterexample",
   "decider": "instruments/openaimath/finite/f105_stable_coordinate.py",
   "decides": "a finite component: f is a one-stable coordinate (an explicit automorphism of C[x1..x4,w] with exhibited inverse sends f to x1), deg f = 5, and every fibre f = lambda is C^[3]; NOT that f is not a coordinate in four variables, which carries the counterexample",
   "value": {
    "f_terms": 16,
    "f_degree": 5,
    "H_terms": 12,
    "Delta_nilpotency_orders": [
     1,
     3,
     2,
     5,
     3,
     1
    ],
    "cofactor_terms_A": [
     960,
     760,
     760,
     55,
     0,
     0
    ],
    "cofactor_terms_B": [
     0,
     0,
     0,
     1,
     3,
     138
    ]
   },
   "checks": [
    {
     "check": "0. printed: abstract: degree five",
     "pass": true,
     "detail": "main.tex"
    },
    {
     "check": "0. printed: eq:explicit-counterexample",
     "pass": true,
     "detail": "01-introduction.tex"
    },
    {
     "check": "0. printed: thm:main, the stable half",
     "pass": true,
     "detail": "01-introduction.tex"
    },
    {
     "check": "0. printed: con:presentation",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:candidate",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: M, e, n",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: x = -det M, e^T n = 0",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: H = e^T M e - J - u",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:matrix-change",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: (ne^T)^2 = 0, det = 1",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: -det M' = x",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: the inverse substitution",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: u' and H = -J - u'",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: the lower-right entry",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: values 0 and 1",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:xyz",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:quadric-identity",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:localized-inverse",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: Delta = -p d/du",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:delta",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: Delta(H) = p",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: exp(w Delta)(H) = H + pw",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: x0",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:linear-coordinates",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: determinant 1",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:linear-inverse",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: con:H-expansion",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: w' = w + Q0",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: the stabilized quotient",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: fibre at 0",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "0. printed: fibre at lambda != 0",
     "pass": true,
     "detail": "02-construction.tex"
    },
    {
     "check": "A. x = -det M, e^T n = 0, H = e^T M e - J - u",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. (n e^T)^2 = 0 and det(I - 2 n e^T) = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. -det M' = x",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. u' = u - e^T M e and H = -J - u'",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. tau(sigma(g)) = g and sigma(tau(g)) = g on all generators: p',s',u',F',J is a coordinate system on P",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. the inverse's lower-right entry: p = p' - 2x(x(s' - u') + p'J)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pi^-1(H) = 0: pi^-1 is well defined on A = P/(H)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pi^-1(pi(x_i)) = x_i for x1..x4, w",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pi(pi^-1(g)) - g = c_g H with explicit cofactors, g = p,s,u,F,J,w",
     "pass": true,
     "detail": "cofactor terms [960, 760, 760, 55, 0, 0]"
    },
    {
     "check": "A. pi^-1(p) = f and pi^-1(x) = Q, as printed in thm:main (con:candidate)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. deg f = 5",
     "pass": true,
     "detail": "16 terms"
    },
    {
     "check": "A. f(0,0,0,0) = 0 and f(1,0,0,0) = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Delta x = Delta y = Delta z = 0 and Delta u = -p (Delta is -p d/du in p,x,y,z,u)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. xy - z(z+1) = p(H + u)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. con:localized-inverse: s = y - x(x+u^2), pF = x - s^2 + u^2, pJ = z - sx",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Delta H = p and Delta^2 H = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Delta is locally nilpotent on the generators (first vanishing iterate on p,s,u,F,J,w)",
     "pass": true,
     "detail": "[1, 3, 2, 5, 3, 1]"
    },
    {
     "check": "B. Phi^-(Phi(g)) = g and Phi(Phi^-(g)) = g on the six generators: Phi is an automorphism of P[w]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Phi(H) = H + p w (direct substitution)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. the (F,J) -> (L,N) change: determinant 1 and con:linear-inverse returns F and J",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. con:H-expansion H = L - u + p Q0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. eps(H + p w) = 0: eps is well defined on P[w]/(H + pw)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. eps(eps'(y)) = y for y = p,s,u,N,w'",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. eps'(eps(g)) - g = c_g (H + pw) with explicit cofactors, g = p,s,u,F,J,w",
     "pass": true,
     "detail": "cofactor terms [0, 0, 0, 1, 3, 138]"
    },
    {
     "check": "B. theta(f) = x1: pi(f) = p + c_p H (A), Phi(p) = p, eps(p) = p, eps(Phi(H)) = eps(H + pw) = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "spot (evidence, not proof): theta and its inverse are mutually inverse on 3 rational points (one with x1 = 0) and f(theta(t)) = t1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. lambda = 0: zeta(H|p=0) = 0, zeta(zeta'(N)) = N, zeta'(zeta(g)) - g = c_g H|p=0 for g = s,u,F,J: A/(p) = Q[s,u,N]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. lambda != 0 (symbolic): nu(H|p=lambda) = 0, nu(nu'(v)) = v for v = x,y,z, nu'(nu(g)) - g = c_g H|p=lambda for g = s,u,F,J in Q[lambda^+-1][...]: A/(p - lambda) = Q[x,y,z]",
     "pass": true,
     "detail": "well-defined True, one way True, cofactors [True, True, True, True]"
    }
   ],
   "sources": {
    "preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/main.tex": "d6916e604fb93f0258b1831dc144c245ce76b6641c18ba1b0de349d5944c3f07",
    "preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/01-introduction.tex": "18f064a0ac5200eae26ed5aec3d65981fc13956b83108115646efc4d153ec67d",
    "preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/build/source/sections/02-construction.tex": "8882699fbc553ef2194503b2c13bd633a737f0afac8499afb1b00945bc617fbf"
   },
   "forges": [
    {
     "forge": "f with Q = x2^2 + x4^2 + x1x3 (one sign flipped)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Delta(F) printed as -4sxu + 2u",
     "verdict": "REFUTED"
    },
    {
     "forge": "N = (1 - 2s x0) F + 3s^2 J (the change is no longer unimodular)",
     "verdict": "REFUTED"
    },
    {
     "forge": "M' = (I - n e^T) M (factor 1 in place of 2)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the relation with -2pJ^2 in place of -pJ^2",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 2.5
  },
  {
   "id": "F-106",
   "index": 106,
   "family": "049",
   "title": "An explicit noncoordinate polynomial with affine three-space zero fibre",
   "tier": "A",
   "dir": "An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026",
   "leanChallenges": [
    "AbhyankarSathaye",
    "CommutingDerivations",
    "StableCoordinateFour"
   ],
   "word": "CERTIFIED",
   "why": "the whole headline: R/(F) = C^[3] by the paper's printed mutually inverse maps, and grad F vanishes at a point, so F is not a coordinate",
   "decider": "instruments/openaimath/finite/f106_abhyankar_sathaye.py",
   "decides": "the whole headline: R/(F) = C^[3] by the paper's printed mutually inverse maps, and grad F vanishes at a point, so F is not a coordinate",
   "value": {
    "F_terms": 69,
    "F_degree": 17,
    "psi_w_terms": 244
   },
   "checks": [
    {
     "check": "the paper prints F, the critical point and the inverse maps",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. x^2 + y^3 = h s in R",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. grad F(2, 0, -1/2, 1/2) = 0",
     "pass": true,
     "detail": "['0', '0', '0', '0']"
    },
    {
     "check": "2. F(2, 0, -1/2, 1/2) = -1 (the critical point lies on the fibre F = -1)",
     "pass": true,
     "detail": "-1"
    },
    {
     "check": "3a. in C[X,Y]: x_B^2 + y_B^3 = h_B s_B",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3a. psi(x) = psi(u)^3 + psi(h) psi(v) = x_B",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3a. psi(y) = -psi(u)^2 + psi(h) psi(w) = y_B",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3a. psi(s) = s_B, computed directly from S(psi(h), psi(u), psi(v), psi(w))",
     "pass": true,
     "detail": "244 terms in psi(w), degree 51"
    },
    {
     "check": "3a. psi(F) = psi(h) - p(psi(x), psi(y), psi(s)) - 1 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3b. psi(phi(X)) = psi(x) + psi(s)^3 = X",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3b. psi(phi(Y)) = psi(y) - psi(s)^2 = Y",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3b. psi(phi(T)) = alpha(psi) psi(u) + beta(psi)(psi(v) + psi(u) psi(w)) = T",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. (x+s^3)^2 + (y-s^2)^3 - s = s F",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. h - 1 - p(x,y,s) = F (the definition, as polynomials)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. alpha h + beta y - 1 = (1 + 2s^2 x) F",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. h T_R - beta x - u = u (1 + 2s^2 x) F",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. y T_R + alpha x - (v + uw) = (v + uw)(1 + 2s^2 x) F",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3c. W_R - w = w (1 + 2s^2 x)(1 + beta y) F",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/build/paper.tex": "60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c"
   },
   "forges": [
    {
     "forge": "the critical point moved (w = 1/3)",
     "verdict": "REFUTED"
    },
    {
     "forge": "s perturbed by + h v^2 (x^2 + y^3 = hs breaks)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 4.4
  },
  {
   "id": "F-125",
   "index": 125,
   "family": "059",
   "title": "Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three",
   "tier": "A",
   "dir": "Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: Lemma seed:product (the integer relation with E > 0 and both products 1), by the shipped witness and by the printed F_1009 rank certificate, plus the chain arithmetic; not the spectra, the lattice/topology, or the homeomorphism",
   "decider": "instruments/openaimath/finite/f125_valuation_relation.py",
   "decides": "a finite component: Lemma seed:product (the integer relation with E > 0 and both products 1), by the shipped witness and by the printed F_1009 rank certificate, plus the chain arithmetic; not the spectra, the lattice/topology, or the homeomorphism",
   "value": {
    "E_digits": 140,
    "E_mod_10^12": 156269075320,
    "support": 849,
    "max_digits": 140,
    "cutoff_columns": 1118,
    "cutoff_rows": 1100,
    "core_columns": 896,
    "core_rows": 867,
    "rank_F1009": 868,
    "kernel_dim_M_over_Q": 29
   },
   "checks": [
    {
     "check": "the lemma and its certificate as printed (seeds.tex:7-56)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. every b of the witness lies in B (0 < b < d/2, 3 does not divide b), all distinct",
     "pass": true,
     "detail": "849 entries"
    },
    {
     "check": "1. every coefficient e_b is a nonzero integer",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. sum_b e_b v_p(A_b) = 0 for every prime p (complete factorizations)",
     "pass": true,
     "detail": "546 distinct primes divide the six integers of the support"
    },
    {
     "check": "1. sum_b e_b v_p(R_b) = 0 for every prime p (complete factorizations)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. E = sum e_b > 0",
     "pass": true,
     "detail": "140 digits"
    },
    {
     "check": "1. second route: prod A_b^e_b = prod R_b^e_b = 1 modulo q = [531457, 531481, 531497] (no factorization; necessary only)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. support size = 849 (printed in certificate.tex and the JSON)",
     "pass": true,
     "detail": "849"
    },
    {
     "check": "2. at most 140 decimal digits per coefficient (max_abs_e_digits)",
     "pass": true,
     "detail": "140"
    },
    {
     "check": "2. JSON sum_e, sum_abs_e and d equal the recomputed values",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. after the prime cutoff 4450: 1118 columns, 1100 nonzero valuation rows",
     "pass": true,
     "detail": "1118 columns, 1100 rows (|B| = 59049)"
    },
    {
     "check": "3. after all deletions: 896 columns, 867 nonzero rows",
     "pass": true,
     "detail": "896 columns, 867 rows"
    },
    {
     "check": "3. 1009 is prime",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. rank over F_1009 of M~ (868 x 896) = 868 = its row count, so M~ has full row rank over Q",
     "pass": true,
     "detail": "rank 868"
    },
    {
     "check": "3. corroboration: 1000003 is prime and rank over F_1000003 of M~ = 868",
     "pass": true,
     "detail": "rank 868"
    },
    {
     "check": "4. the witness is supported inside the cutoff columns and inside the 896-column core",
     "pass": true,
     "detail": "849 of 849 in the cutoff set, 849 in the core"
    },
    {
     "check": "5. every starting weight numerator (3b, b, d-b, d+b over D = 3^12) gives a chain with all a_i >= 2, terminal exponent >= 3, a_l c_l = D, weights in (0, 1/2)",
     "pass": true,
     "detail": "3396 chains"
    },
    {
     "check": "5. reduced denominators: b/d has denominator d; b/(3d), (d-b)/(3d), (d+b)/(3d) have denominator 3d",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/build/sections/seeds.tex": "f83599a45fbe0a60e6ee4dcbcb76dc94fed3b4efe30dd1469349231733da4e2e",
    "preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/build/sections/certificate.tex": "022082f8ba3a6807c5448358f17b23fce3f3222ec0c9d217e04887c2798231bd",
    "preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/build/sections/introduction.tex": "bc4e0fccbd95ac9c6b574707fe2e46c04513737fe55c10ef5df46560b263990e",
    "preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/verification/support/explicit-relation.json": "e5a3ad0b8901ac358374575dbaf7be9e03bb84063def63749f16618cf2ba1780"
   },
   "forges": [
    {
     "forge": "the witness: first coefficient + 1",
     "verdict": "REFUTED"
    },
    {
     "forge": "the witness: last b moved to the next admissible b",
     "verdict": "REFUTED"
    },
    {
     "forge": "the printed rank changed to 867",
     "verdict": "REFUTED"
    },
    {
     "forge": "the prime cutoff moved to 4440 (counts must change)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 2.1
  },
  {
   "id": "F-135",
   "index": 135,
   "family": "067",
   "title": "The Campana–Peternell conjecture in dimension six",
   "tier": "A",
   "dir": "The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: Lemma lem:finite-ratio (from the matrices A, C_1, N defined in Section 4, N(xi_i eta_j) = 0 with xi_0 = eta_0 = 1 forces 32 - 20 eta_1 + 3 eta_1^2 = 0); dim_Q K_3 <= 6 by a rank mod 10007 (a certified lower bound on the rational rank), everything else exact over Q; not the geometry producing these equations, nor the classification",
   "decider": "instruments/openaimath/finite/f135_campana_peternell.py",
   "decides": "a finite component: Lemma lem:finite-ratio (from the matrices A, C_1, N defined in Section 4, N(xi_i eta_j) = 0 with xi_0 = eta_0 = 1 forces 32 - 20 eta_1 + 3 eta_1^2 = 0); dim_Q K_3 <= 6 by a rank mod 10007 (a certified lower bound on the rational rank), everything else exact over Q; not the geometry producing these equations, nor the classification",
   "value": {
    "A": "1785 x 581, rank_Q 516",
    "N": "69 x 129",
    "dims_mod_10007": [
     60,
     163,
     6
    ],
    "minor": "-16/5",
    "t_exact_s": 11.7,
    "t_mod_s": {
     "10007": 5.4,
     "1000003": 5.5
    }
   },
   "checks": [
    {
     "check": "the theorem, the lemma and the certificate as printed (01:23-26, 05:10-22, 05:81-196, 08:23-29)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. D_1 = 5 (04:234)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. the printed beta_j equal -B_j / j! (Bernoulli numbers re-derived; beta_3 = beta_5 = beta_7 = 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. theta_i (eq:todd-recursion) equals the degree-i part of prod Q(y_k), Q(x) = x/(1-e^-x), at a rational test point, i <= 8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. every row polynomial is weight-homogeneous of weight 9",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. A: 581 columns (T_9) and 1785 nonzero rows",
     "pass": true,
     "detail": "1785 x 581 (1893 rows before dropping zero rows)"
    },
    {
     "check": "2. rank_Q A = 516 exactly (exact rational RREF)",
     "pass": true,
     "detail": "516"
    },
    {
     "check": "2. K (581 x 65, canonical from the RREF) satisfies AK = 0 over Q: a rational basis of ker A",
     "pass": true,
     "detail": "65 columns"
    },
    {
     "check": "2. M = B_6 and G = (I_3, I_4) in the recursive order equal the printed lists (04:335-339)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. seven M_i contain h, six indices of J have weight three, |C_1| = 129, the 129 selected monomials distinct",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. P is 129 x 65 with exact rank 60",
     "pass": true,
     "detail": "60"
    },
    {
     "check": "3. N (RREF of the left kernel of P) is 69 x 129 and NP = 0 over Q",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. ranks mod 10007 of A, K, P, N = 516, 65, 60, 69 (printed table, 05:93-97)",
     "pass": true,
     "detail": "(516, 65, 60, 69)"
    },
    {
     "check": "3. every denominator in A, K, P, N is prime to 10007",
     "pass": true,
     "detail": "largest denominator in N has 57 digits"
    },
    {
     "check": "4. |C_n| = 129, 1140, 6820 by enumeration and by the printed formula",
     "pass": true,
     "detail": "[129, 1140, 6820]"
    },
    {
     "check": "4. the double-slice matrix is 69 x 120 = 8280 rows by 6820 columns",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. 6820 * 10006^2 = 682818645520 < 2^63 - 1 (08:29)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. dim K_1, K_2, K_3 over F_10007 = 60, 163, 6 (our own slice merging), K_2 covers all of C_2 and the 15 slices cover all of C_3",
     "pass": true,
     "detail": "(60, 163, 6, 1140, 6820) in 5s"
    },
    {
     "check": "4. hence rank_F10007 (M mod 10007) = 6820 - 6 = 6814 and dim_Q K_3 <= 6 (modular rank is a lower bound on the rational rank)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. corroboration mod 1000003: dims 60, 163, 6",
     "pass": true,
     "detail": "(60, 163, 6, 1140, 6820) in 5s"
    },
    {
     "check": "5. every printed coordinate of the sparse columns, the minor rows and the functional lies in C_3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. N (B_i H_j)_{C_1} = 0 exactly for (B(0),H^a), (B(1),H^a), (B',H^b)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. N annihilates all 120 double slices of each of the six columns (exact integers)",
     "pass": true,
     "detail": "[True, True, True, True, True, True]"
    },
    {
     "check": "5. the 6 x 6 minor on the printed rows: upper-right block 0, blocks -8/15 and 6, full minor -16/5",
     "pass": true,
     "detail": "blocks -8/15, 6; minor -16/5"
    },
    {
     "check": "5. the target functional vanishes on all six columns",
     "pass": true,
     "detail": "['0', '0', '0', '0', '0', '0']"
    },
    {
     "check": "5. on the product columns it reads 32 - 20 H_1 + 3 H_1^2 with H_1 = 4, 4, 8/3",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/01-introduction.tex": "f903e8ff33d32f6a27a2eb7729cbeba232ad7063a1cc466fd4ae9921f72915f6",
    "preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-p1-cohomology.tex": "6ab56e8c89241c3d479a6cd781ed6dcf45313c6bc49d8dc55ae4b96e01506055",
    "preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/04-characteristic-classes.tex": "d97f37d7242268423525fb6212a4d6ee7660144b43e5c569514d660a959250b3",
    "preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/05-arithmetic.tex": "2926fa84a4d08fbbf8c30cc43781fa7f0ba558946b0e33bd6879aa4d8311b048",
    "preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/08-certificate.tex": "cad18e6f196c882f1abc10bb9dc2a60b453f9a8a46ef51d0f08bb07e4d9fccfb"
   },
   "forges": [
    {
     "forge": "s2 entry at (9,(10,10,10)) changed 25 -> 24",
     "verdict": "REFUTED"
    },
    {
     "forge": "target functional 32 - 21 eta_1 + 3 eta_1^2",
     "verdict": "REFUTED"
    },
    {
     "forge": "beta_4 = 1/721 instead of 1/720 (A, N, K_3 recomputed; mod 10007 only)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 23.1
  },
  {
   "id": "F-171",
   "index": 171,
   "family": "087",
   "title": "The Mahler Conjecture for General Convex Bodies",
   "tier": "A",
   "dir": "The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026",
   "leanChallenges": [
    "GeneralMahler",
    "MahlerConjecture",
    "SymmetricMahlerEquality",
    "SymmetricPolar"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the finite certificates of Appendices 07-08 (the 152-node profile enclosures and every printed node bound and finite sum; the stable-formula identities; the layer rectangles; the chord-error, tail-vector, interpolation and strip arithmetic; the segment threshold sums and sample tables; the two Bernstein certificates; the cubic and larger-width tables; the tail and short-width budgets) that support Proposition prop:scalar-input. Not the analytic interpolation/tail/segment-reduction arguments, and not sections 01-06 and 09, where the convex-geometric proof of the Mahler inequality lives",
   "decider": "instruments/openaimath/finite/f171_mahler_general.py",
   "decides": "a finite component: the finite certificates of Appendices 07-08 (the 152-node profile enclosures and every printed node bound and finite sum; the stable-formula identities; the layer rectangles; the chord-error, tail-vector, interpolation and strip arithmetic; the segment threshold sums and sample tables; the two Bernstein certificates; the cubic and larger-width tables; the tail and short-width budgets) that support Proposition prop:scalar-input. Not the analytic interpolation/tail/segment-reduction arguments, and not sections 01-06 and 09, where the convex-geometric proof of the Mahler inequality lives",
   "value": {
    "checks": 179,
    "nodes": 152,
    "node_precision_bits": 340,
    "seconds": 12.0
   },
   "checks": [
    {
     "check": "the headline as stated (abstract.tex)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the finite appendices feed one interface, Proposition prop:scalar-input (01-cones.tex line 138), proved in Appendices 07 and 08",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the parameters as printed: b = .602, c = .365, r = .22, s = 3.6, w = 4.6, eta = .022, kappa = 1.16, a_R = 7.5, lambda = .65; t_- = -.022, t_+ = .064, t_c = .021, t_r = .043, m_* = .352",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: X' = 1 + xX and Y' = xY - 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: f' = -phi Y / X and j' = phi X / Y (from f = -(1-p) - log p, j = -p - log(1-p))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: d = (1+xX)^2 f + B^2 j = B^2 j + (1 - pB)^2 h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: K = d - 2g = d + Y^2 (j + p^2 h) - 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: g' = Y (Bj - (1 - pB) p h)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: g'' = x g' - K",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: R_v = v' - x = phi/p - B/Y",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: V_2 = v'' = 2 - (phi/p)(x + phi/p) - B/Y^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: D_1 = d' = x(2d - 1) - R_v - 2g'",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: D_2 = d'' = 2d + 2x D_1 - 2g'' - V_2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: G_3 = g''' = x g'' + 3g' - D_1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F3. identity: G_4 = g'''' = x G_3 + 4g'' - D_2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. the certificate table lists the seventeen rows d .. n_q in order",
     "pass": true,
     "detail": "['d', '\\\\dot d', '\\\\ddot d', '\\\\mathsf K', '\\\\dot{\\\\mathsf K}', '\\\\ddot{\\\\mathsf K}', 'n_{\\\\mathsf K}', '\\\\mathsf C', '\\\\dot{\\\\mathsf C}', '\\\\ddot{\\\\mathsf C}', 'n_{\\\\mathsf C}', 'R_v', 'V_2', 'q', '\\\\dot q', '\\\\ddot q', 'n_q']"
    },
    {
     "check": "F1. d: every node value (i = 0..151) lies in [1931/5000, 1/2]",
     "pass": true,
     "detail": "node hull [0.38629436, 0.49991885]"
    },
    {
     "check": "F1. d: |F_{i+1} - 2F_i + F_{i-1}| <= 13/50000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00025907"
    },
    {
     "check": "F1. d_t: every node value (i = 0..151) lies in [-669/5000, 669/5000]",
     "pass": true,
     "detail": "node hull [-0.00000000, 0.13365422]"
    },
    {
     "check": "F1. d_t: |F_{i+1} - 2F_i + F_{i-1}| <= 23/25000 for i = 0..150 (F_{-1} by odd reflection)",
     "pass": true,
     "detail": "max 0.00091537"
    },
    {
     "check": "F1. d_t: |F_{i+1} - F_i| <= 1037/100000 for i = 0..150",
     "pass": true,
     "detail": "max 0.01035855"
    },
    {
     "check": "F1. d_tt: every node value (i = 0..151) lies in [-83/200, 83/200]",
     "pass": true,
     "detail": "node hull [-0.21210364, 0.41467485]"
    },
    {
     "check": "F1. d_tt: |F_{i+1} - 2F_i + F_{i-1}| <= 421/100000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00420671"
    },
    {
     "check": "F1. d_tt: |F_{i+1} - F_i| <= 367/10000 for i = 0..150",
     "pass": true,
     "detail": "max 0.03660804"
    },
    {
     "check": "F1. K: every node value (i = 0..151) lies in [-173/25000, 2737/100000]",
     "pass": true,
     "detail": "node hull [-0.00691588, 0.02735845]"
    },
    {
     "check": "F1. K: |F_{i+1} - 2F_i + F_{i-1}| <= 37/250000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00014705"
    },
    {
     "check": "F1. K_t: every node value (i = 0..151) lies in [-3271/50000, 3271/50000]",
     "pass": true,
     "detail": "node hull [-0.02101105, 0.06535105]"
    },
    {
     "check": "F1. K_t: |F_{i+1} - 2F_i + F_{i-1}| <= 673/1000000 for i = 0..150 (F_{-1} by odd reflection)",
     "pass": true,
     "detail": "max 0.00067147"
    },
    {
     "check": "F1. K_tt: every node value (i = 0..151) lies in [-471/2000, 471/2000]",
     "pass": true,
     "detail": "node hull [-0.19441728, 0.23541536]"
    },
    {
     "check": "F1. K_tt: |F_{i+1} - 2F_i + F_{i-1}| <= 373/100000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00371915"
    },
    {
     "check": "F1. n_K: every node value (i = 0..151) lies in [-471/2000, 471/2000]",
     "pass": true,
     "detail": "node hull [-0.21376905, 0.23541536]"
    },
    {
     "check": "F1. n_K: |F_{i+1} - 2F_i + F_{i-1}| <= 201/50000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00400372"
    },
    {
     "check": "F1. C: every node value (i = 0..151) lies in [-1/2, -3413/10000]",
     "pass": true,
     "detail": "node hull [-0.49992037, -0.34134117]"
    },
    {
     "check": "F1. C: |F_{i+1} - 2F_i + F_{i-1}| <= 49/100000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00048715"
    },
    {
     "check": "F1. C_t: every node value (i = 0..151) lies in [-307/1250, 307/1250]",
     "pass": true,
     "detail": "node hull [-0.24557168, 0.00000000]"
    },
    {
     "check": "F1. C_t: |F_{i+1} - 2F_i + F_{i-1}| <= 111/50000 for i = 0..150 (F_{-1} by odd reflection)",
     "pass": true,
     "detail": "max 0.00220937"
    },
    {
     "check": "F1. C_tt: every node value (i = 0..151) lies in [-39/50, 39/50]",
     "pass": true,
     "detail": "node hull [-0.77968801, 0.56107834]"
    },
    {
     "check": "F1. C_tt: |F_{i+1} - 2F_i + F_{i-1}| <= 129/10000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.01280724"
    },
    {
     "check": "F1. n_C: every node value (i = 0..151) lies in [-39/50, 39/50]",
     "pass": true,
     "detail": "node hull [-0.77968801, 0.65647924]"
    },
    {
     "check": "F1. n_C: |F_{i+1} - 2F_i + F_{i-1}| <= 69/5000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.01374075"
    },
    {
     "check": "F1. Rv: every node value (i = 0..151) lies in [-159/500, 159/500]",
     "pass": true,
     "detail": "node hull [-0.31795328, 0.00000000]"
    },
    {
     "check": "F1. Rv: |F_{i+1} - 2F_i + F_{i-1}| <= 167/100000 for i = 0..150 (F_{-1} by odd reflection)",
     "pass": true,
     "detail": "max 0.00166125"
    },
    {
     "check": "F1. V2: every node value (i = 0..151) lies in [-53/50, 53/50]",
     "pass": true,
     "detail": "node hull [0.72676046, 1.05737522]"
    },
    {
     "check": "F1. V2: |F_{i+1} - 2F_i + F_{i-1}| <= 93/50000 for i = 0..150 (F_{-1} by even reflection)",
     "pass": true,
     "detail": "max 0.00185800"
    },
    {
     "check": "F1. q: every node value (i = 0..151) lies in [777/10000, 1021/4000]",
     "pass": true,
     "detail": "node hull [0.07779634, 0.25513453]"
    },
    {
     "check": "F1. q_t: every node value (i = 0..151) lies in [-477/2500, 477/2500]",
     "pass": true,
     "detail": "node hull [-0.00000000, 0.19069252]"
    },
    {
     "check": "F1. q_t: |F_{i+1} - F_i| <= 127/10000 for i = 0..150",
     "pass": true,
     "detail": "max 0.01266723"
    },
    {
     "check": "F1. q_tt: every node value (i = 0..151) lies in [-127/250, 127/250]",
     "pass": true,
     "detail": "node hull [-0.27531716, 0.50690014]"
    },
    {
     "check": "F1. q_tt: |F_{i+1} - F_i| <= 47/1000 for i = 0..150",
     "pass": true,
     "detail": "max 0.04677021"
    },
    {
     "check": "F1. n_q: every node value (i = 0..151) lies in [-127/250, 127/250]",
     "pass": true,
     "detail": "node hull [-0.37097375, 0.50690014]"
    },
    {
     "check": "F1. b - r^2 - d_i >= .0536 at every node (d_i <= .5), before any q evaluation",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. simultaneous table printed as 0{:}151 & .537 & .00090 & -.00074 & .132",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. sup_i U_* <= .537",
     "pass": true,
     "detail": "0.53634700"
    },
    {
     "check": "F1. inf_i (lambda t_-^2 + C t_- + K) >= .00090",
     "pass": true,
     "detail": "0.00090823"
    },
    {
     "check": "F1. sup_i (lambda t_+^2 + C t_+ + K) <= -.00074",
     "pass": true,
     "detail": "-0.00075342"
    },
    {
     "check": "F1. inf_i (2(-.37) C - .37^2 - 4 lambda K) >= .132",
     "pass": true,
     "detail": "0.13293901"
    },
    {
     "check": "F1. for 56 <= i <= 144: 0 <= K <= .019",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 56 <= i <= 144: |K_t| <= .0207",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 56 <= i <= 144: n_K >= .00315",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 56 <= i <= 144: -.5 <= C <= -.49512",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 56 <= i <= 144: |C_t| <= .0071",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 56 <= i <= 144: q >= .2353",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F1. for 45 <= i <= 144: q >= .2204",
     "pass": true,
     "detail": "min 0.220421"
    },
    {
     "check": "F1. the finite integral sum at i = 56 is <= .126 and lies in the printed [.1250504493, .1250504494]",
     "pass": true,
     "detail": "0.125050449311"
    },
    {
     "check": "F1. the finite integral sum at i = 144 is <= .335 and lies in the printed [.3344848649, .3344848650]",
     "pass": true,
     "detail": "0.334484864946"
    },
    {
     "check": "F2. the V_2 chord error R(.00186) = .536(.00186) + .00001 <= .00101",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. the nine-row table of q, S_q, V_2 (sample extremes on each row of |t| through index 144, enlarged by .000042, .003, .00101; the last row joined with the tail bounds) and rho_max (rho at the row end; r sqrt(4+w^2) for 68+) bounds every printed entry",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "F2. rho increases in t: 1 - 6/s^2 - 2(w^2+1)/s^4 > .27",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. the table gives q < k and S_q < 2k (monotonicity of the scalar part of H)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. the 81 rectangle pairs x 4 vertices: the printed lower bounds [11, 14, 33, 64, 63, 39, 54, 45, 40] (min over rows of x of the vertex minimum of D, times 1000, in the order of the rows of z) hold, and every minimum is positive",
     "pass": true,
     "detail": "recomputed minima x 1000: 12.007, 14.700, 33.503, 64.232, 63.128, 39.297, 54.718, 46.085, 40.255 (floors [12, 14, 33, 64, 63, 39, 54, 46, 40]; printed [11, 14, 33, 64, 63, 39, 54, 45, 40])"
    },
    {
     "check": "F2. 4c + 4k(.113) - .113^2 - 1.104^2 > .25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. b + eta + kappa t_- + .125 > 0, 1.053 kappa^2 / [8(...)] < .256, .256 + (log 2 + .25)(2.26)/4 < .80",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. d <= .5 + R(.000260) <= .50015 and b - r^2 - .50015 >= .05345",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. l = 1/(2 ell) <= 2.164 (ell^2 >= .05345)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. |l_t| <= 2 l^3 (.1338 + R(.00092)) <= 2.8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. |l_tt| <= 2 l^3 (.415 + R(.00421)) + 12 l^5 (.1338 + R(.00092))^2 <= 19.2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord error of q_t: 2.164 R(.00092) + .1338 e + (2.8 delta)(.01037)/4 < .0016",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord error of q_t^2: (2(.1908) + .0016)(.0016) + .0127^2/4 < .000654",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord error of q_tt < .0095",
     "pass": true,
     "detail": "0.009279"
    },
    {
     "check": "F2. chord error of q: (.508 + .0095) delta^2 / 8 < .000042",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord error of n_q: .0095 + .0016 + .1908/12800 + delta(.0127)/4 < .012",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord error of S_q < .003",
     "pass": true,
     "detail": "0.002685"
    },
    {
     "check": "F2. U_* between nodes: the addition to .537 is < .010, and .547 < .54 r^2 w^2",
     "pass": true,
     "detail": "0.009113"
    },
    {
     "check": "F2. .238/12800 < .000019 and .79/12800 < .000062",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. chord errors of the three sampled inequalities < .000024, .000024, .0001 and the table margins suffice",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. n_K >= .00315 > R(.00402) on t >= 1.4, and s sinh(45/40) < 5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: K >= -.007, K <= .0275",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: |K_t| <= .066, |K_tt| <= .238",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: C in [-.501, -.341]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: |C_t| <= .247, |C_tt| <= .79, |n_C| <= .79",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: q in [.0775, .2559], |q_t| <= .193, |q_tt|, |n_q| <= .53",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: |v' - x| <= .319",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: on |t| >= 1.4: K in [-.0001, .0191], |K_t| <= .0211, |C_t| <= .0084, C <= -.495, q >= .235",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F2. Lemma scalar-ranges on the finite region: for |x| >= 5: q >= .2203",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: sin(.45) < .435 and tan(.45) < .5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: Y(0) = sqrt(pi/2) <= 1.254",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: e^{1.25} u(0) = e^{1.25}/2 <= 1.75",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: A <= .9: A |y| <= .9 sqrt(2.5 + .25(.81)) < pi/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: A >= .9: u(.9) e^{1.25 + .125(.81)} < .80, with u(.9) <= 1/2 - (.9 - .9^3/6)/sqrt(2 pi)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: Re Y >= sqrt(pi/2) e^{-2}/4 > .02",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: A >= 2: |y|/A <= sqrt(7/8) < .936, and e^{-1-1/8} > .324",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: |d| < 48 from |Bj| <= 12, |1 - pB| <= 3.75, |h| <= 2.5: 12 + 3.75^2 (2.5) < 48",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: |K| < 99: 48 + 19 + 1.254^2 (2.75)^2 (2.5) + 1 < 99",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: |C| <= 48 + 6.5(76 + 99) < 1300 and |V_2| <= 2 + 20 + 49 + 10000 < 11000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F4. analytic strip arithmetic: A e^{-.375 A^2} <= 1 (its maximum, at A^2 = 4/3, is e^{-1/2} sqrt(4/3))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F5. sum_{l=2}^8 2^(2l-4)/(2l-2)! prod_{j<=l-2} (j+1/2)^2 = 4387/8192 < .536",
     "pass": true,
     "detail": "4387/8192"
    },
    {
     "check": "F5. 4(10^5) prod_{j=0}^7 ((j+1/2)/(40(.36)))^2 < .000007336661 < .00001",
     "pass": true,
     "detail": "0.000007336660"
    },
    {
     "check": "F5. Cauchy constants: 1300/.09 + 2(1300)/.09^2 < 4(10^5)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F6. the four chord errors eps_j: R(.000673) + .021 R(.00222) = .00039592632 < .00040, R(.00222) = .00119992 < .00120, R(.00402) = .00216472 < .00217, R(.0138) = .0074068 < .00741",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F6. the threshold table: all 32 sums sum_i (delta_i/80)(A_j(t_i) + eps_j - v)_+ are below the printed 10^-5 I_j(v), and at both ends of each of the eight H-intervals I_j(v) <= 2H(alpha_j - beta_j H - v)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "F6. D(0) agrees with (1 - 6.5 t_c)((2 + pi) c_* - 1) - 2 t_c c_*, c_* = log 2 - 1/2, and D(0) < -.0140",
     "pass": true,
     "detail": "-0.0140840400"
    },
    {
     "check": "F6. sample table: 0:144  D - D(0) in [-10^-8, .0314]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F6. sample table: 0:144  P <= .01767",
     "pass": true,
     "detail": "extreme 0.017659805"
    },
    {
     "check": "F6. sample table: 0:144  B_0 <= .01773",
     "pass": true,
     "detail": "extreme 0.017708814"
    },
    {
     "check": "F6. sample table: 0:144  B_0 - .0146 t_i^2 <= .00001",
     "pass": true,
     "detail": "extreme 0.000000000"
    },
    {
     "check": "F6. sample table: 0:144  D_t >= -.0213",
     "pass": true,
     "detail": "extreme -0.021203244"
    },
    {
     "check": "F6. sample table: 0:56   B_1 + (P - .0049)_+ / 2 <= .000001",
     "pass": true,
     "detail": "extreme 0.000000000"
    },
    {
     "check": "F6. sample table: 0:56   Z_1 in [0, .00742]",
     "pass": true,
     "detail": "extreme 0.007414249"
    },
    {
     "check": "F6. sample table: 0:56   B_1 + 78 Z_1^2 <= .000001",
     "pass": true,
     "detail": "extreme 0.000000000"
    },
    {
     "check": "F6. sample table: 0:56   P_{q^2} - (q + q_M) P_q + (q^2 + q_M^2)/2 <= .01587",
     "pass": true,
     "detail": "extreme 0.015860457"
    },
    {
     "check": "F6. monotonicity line (0:144): -C and q increase by at least .000005 and .000012 between successive nodes; min(D, -.01065) and min(P, .0049) are nondecreasing (clipping applied to the endpoints)",
     "pass": true,
     "detail": "least increments 0.0000056, 0.0000127"
    },
    {
     "check": "F7. T_(4) has degree 32, and its Bernstein coefficients on [.16, .54], grouped 0:7, 8:15, 16:23, 24:32, are all positive with minima x 10^5 at least the printed [173, 576, 770, 412]",
     "pass": true,
     "detail": "degree 32; minima x 10^5: 173.690, 576.646, 770.739, 412.674"
    },
    {
     "check": "F7. T_(5) has degree 30, and its Bernstein coefficients on [.16, .54], grouped 0:7, 8:15, 16:23, 24:30, are all positive with minima x 10^5 at least the printed [1586, 701, 408, 407]",
     "pass": true,
     "detail": "degree 30; minima x 10^5: 1586.270, 701.700, 408.468, 407.602"
    },
    {
     "check": "F7. the omitted tail of P_u: 1 + 4w^2 < 9.26^2, the i = 8 terms at h = .54 sum to < .000153, the i = 9 ratio is < .10, and .000153/(1 - .10) < .00025",
     "pass": true,
     "detail": "i=8: 0.0001507"
    },
    {
     "check": "F7. the F_* tail ratio 8 w^2 h^2/(18 . 17) < 1 at h = .54",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F7. on [.16, .54]: L_N(h) >= L_N(.54) = .08521 > 0 and L_C(h) >= L_C(.54) > .26609",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F7. S_u, H_u remainder constants: 875/104271 < .0084 and 3125/74271 < .0421, with first terms 1/120, 1/24 and ratio .54^2/42",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F8. the seven rows of (h^2 a_s, Q, A, B, N_0): each printed entry bounds its defining formula in the stated direction (A_0 with the angular square root enclosed), and the cubic Q/25 + A Z^2 + B Z - N_0 Z^3 formed from the rounded entries has maximum strictly below the printed last column, every one below .027",
     "pass": true,
     "detail": "maxima 0.02028, 0.02448, 0.02526, 0.02307, 0.02242, 0.02217, 0.02195; []"
    },
    {
     "check": "F8. the fourth-row B = .4(.281 - .254(.4))/(5(.65)) = .02208 exactly",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F8. h^2 a_s <= .16025 gives 1 - (.16025 + .0163)/1.3 > .86419 > .855, and .193^2 + .54^2/25 = .048913 < .05",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F8. (h b_s)' = .281 - .508 h >= .00668 through .54",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F9. w(.54) > 3 pi/4 (so wh in (3pi/4, pi] on [.54, pi/w]) and the last endpoint .684 > pi/w > .68",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F9. the three rows (L = .54, .58, .63): p(L) rounded up, f(L) rounded down, U_P = .01815 + 1/(312 sinh^2 2L), R_P, R_O bound their formulas; the budgets (1+chi)f - chi(p + b_M) - [(R_j + p + b_M)^2 + (.08 + b_M)^2]/(4 lambda) from the printed entries reach the printed P, O; and P > U_P, O > .01815 in every row",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "F10. h >= pi/w: coth(.68) <= 1/.68 + .68/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: p > .077 for h >= pi/w (angular range with coth h <= coth(.68))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: |d_h| <= r^2 (sqrt(1 + w^2) - 1)/(1 + w^2) < .00810 when d_h < 0; p <= 2r^2 = .0968",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: pi <= wh <= 4.14 < 3pi/2 on [pi/w, .9]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: sinh(.78) > .85, sinh(.9) > 1.024, coth(.9) < 1.3965, sinh(1.36) > 1.819, sinh(1.8) > 2.94",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: f >= .0431 on [pi/w, .9] (|sin wh| <= w(h - pi/w) <= .46 with sinh h >= sinh .68 up to .78; |sin| <= 1, sinh >= .85 after)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: on [pi/w, .9]: budget >= .02005 > .01912 >= .01815 + 1/(312(1.819)^2) (2h >= 2pi/w > 1.36)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: h >= .9: f >= .04413, p <= .10882, cost <= .01815 + 1/(312(2.94)^2) <= .01853",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h >= pi/w: h >= .9 budgets: d_h < 0 (p <= .0968, |c_d - p_d| <= .08810) >= .02115 > .01853; d_h >= 0 (p <= .10882, .08) >= .01916 > .01853",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 3 + 7(30) xi_0 + 6(630) xi_0^2 + 22680 xi_0^3 < 3.1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: PN <= 1.02 P",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 1.02 sum_k k^3 (1.02 xi_0)^(k-1) = 1.02 (1 + 4z + z^2)/(1 - z)^4 <= 1.04",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 1.55 P(N + I) + 6.2 P + .5 P N^2 + 2(10^-6) xi_0 P N <= 10 P",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 1.02 P(M + I) + P N^2 <= 3.2 P",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: the tail table row K - xi(L - 3/2): (.0064, .027, .11) bounds the vector (0.00628, 0.02592, 0.10680)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: the tail table row d - 1/2: (.51, 1.02, 2.1) bounds the vector (0.50356, 1.01450, 2.05901)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: the tail table row C + 1/2: (.55, 1.16, 2.61) bounds the vector (0.53420, 1.14194, 2.58856)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 4.5 < L(64) = log 64 + log sqrt(2 pi) - 1/2 < 4.59",
     "pass": true,
     "detail": "4.577822"
    },
    {
     "check": "F10. tail lemma: (D^2 - D)K >= xi(6(4.5 - 1.5) - 5 - .11 - .027) > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: a_t/x = sqrt(1 + s^2/x^2) <= 1.002 on x >= 64",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: 0 <= K <= xi_0(4.59 - 1.5 + .0064) <= .0008, |K_t| <= 1.002 xi_0(2(4.59) - 4 + .027) < .002, |K_tt| <= |n_K| + |DK| < .02",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: |C + .5| <= .55 xi_0 <= .0002, |C_t| <= 1.002(1.16) xi_0 < .001, |n_C|, |C_tt| < .002",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: ell^2 >= .0536 - .51 xi_0 > .231^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: q in [.255, .2556], |q_t| < .00055, |n_q| < .0017",
     "pass": true,
     "detail": "q in [0.255041, 0.255579], q_t 0.000540, n_q 0.001657"
    },
    {
     "check": "F10. tail lemma: S_q = 2q + tanh(t) q_t - a^-2 n_q in [.509, .512]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: x^20 e^{-x^2/2} < 10^-200 at x = 64 (20 log 64 - 2048 + 200 log 10 < 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: V_2 = 2 - (phi/p)(x + phi/p) - n/m^2 in [.999, 1.001] (Gaussian term < .0001, n in [1-3 xi, 1], m in [1 - xi, 1])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: |R_v| <= .319 on the tail: |n/(xm)| <= 1/(64(1 - xi_0)) plus the Gaussian term",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: tail covariance bounds: lambda t_-^2 + t_- C + K >= .010, lambda t_+^2 + t_+ C + K <= -.027, .74(.499) - .37^2 - 2.6(.0008) > m_*^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: U_* < .001 on the tail",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. tail lemma: log sqrt(2 pi) - 2 + .0064 < 0, (log 64 + 1)/64 < .086, .335 + .086 < .422",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. h <= .16: W <= (3/2) S <= 1.507 and int W <= S(1 + cosh h)/2 <= 1.012",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. mean of beta^2: (1/3 + h^2/10 + (7/5) h^4/168) <= (1/3 + .05 h^2)(1 + h^2/6) for h <= .54, with cosh(.54) <= (1 - .54^2/2)^-1 < 7/5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. short widths: the displayed rational lower bound is > .55510 > .55 > .54",
     "pass": true,
     "detail": "0.555109"
    },
    {
     "check": "F10. sinh(1.1) > 1.335, coth(1.1) < 1.25, sinh(1.4) > 1.904, coth(1.4) < 1.13",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. central/distant (h >= 1.1): f_0^* >= .0449, .0818 <= p_a <= .1074, |p_d| <= .0132",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. opposite tails (h >= 1.4): f_0^* >= .0456, .0830 <= p_a <= .1062, |p_d| <= .0129",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. x_+ > 65.8 and |x_-| < 6.86; mean K <= max{(.422 + .126)/65.8, .422/(65.8 - 6.86)} < .0084",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. b_+ <= (10/58.94)(.1781)^2 + (1 - 10/58.94)(.0353)^2 < .0068, with .1781 = .2556 - .0775, .0353 = .2556 - .2203; b_- <= .1784^2 < .032",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. C ranges: central/distant c_a in [-.081, .160], |c_d| <= .080; opposite tails c_a in [-.006, .160], |c_d| <= .003",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. the e_K + Gamma table: vertex maxima give <= .0144 (central/distant) and <= .0277 (opposite tails)",
     "pass": true,
     "detail": "0.014294, 0.027602"
    },
    {
     "check": "F10. the tail budgets: 1.064(.0449) - (.2078^2 + .1092^2)/2.6 - .064(.1074 + .0194) > .018 > .0144 and 1.064(.0456) - (.1442^2 + .0319^2)/2.6 - .064(.1062 + .032) > .031 > .0277, with the cost components as stated",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. positive tail, h in [.16, .4]: f/h^2 > .18 (short-width bound at h = .4) and Gamma/f <= .10; h >= .4: f >= .0332 (sinh .4 > .410)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. positive tail: p_a + |p_d| <= (4 + .15h^2 + 2h)u_0 <= .150 and <= 6.65 f up to h = .3; coth(.3) < 3.434 gives <= .150 after; f >= .02256 (sinh .3 > .304), .150 < 6.65(.02256); f(.16) >= .008",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F10. positive tail: .10 + (sqrt(.150(6.65)) + .00645/sqrt(.008))^2/(4 lambda) + t_+(6.65 + .00045/.008) < 1 + t_+, with |C - C_pm| <= .006, b_pm <= (.2559 - .235)^2 <= .00045",
     "pass": true,
     "detail": "0.97026"
    },
    {
     "check": "F10. middle widths: 2r^2/T_0^2 > .33, c_a <= .79 g_u(.54)/(1.5 . 3) h^2 < .33 h^2, .53 g_u(.54)/(1.5 . 3) < 1/5, 2(.193)/5 <= .078",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/build/abstract.tex": "42cff64862e210ca6a51de1a659c3e52305cd8b2ddc4e89516fcca734b90d62d",
    "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/build/sections/01-cones.tex": "3336db65cf53a2a38d1ac37fccedd1acf5f9a246ba87c104dba9b166451a3344",
    "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/build/sections/07-scalar.tex": "e665398d6b0f73581f7cfc186197f4612e01f72841b007c6df15c37559ab2969",
    "preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/build/sections/08-segments.tex": "e8c784e04ffae0d9d5a0feff21ad5479ed7f2e6b8f0ebf15367b30849e6cd59b"
   },
   "forges": [
    {
     "forge": "the |Delta^2 K| bound printed one unit lower (.000148 -> .000147; the true maximum is .00014705)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the first Bernstein group bound of T_(4) printed one unit higher (173 -> 174; the true minimum is 173.69e-5)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the layer table q_min of row 0:4 printed one unit higher (77 -> 78; q(0) - .000042 = .077754)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 12.0
  },
  {
   "id": "F-174",
   "index": 174,
   "family": "088",
   "title": "A product counterexample to the simplex maximum for projection-body volume",
   "tier": "A",
   "dir": "A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026",
   "leanChallenges": [
    "PettyProjectionVolume",
    "ProjectionCounterexample",
    "ProjectionVolume"
   ],
   "word": "CERTIFIED",
   "why": "the whole headline: K = T10 x T10 violates the proposed simplex bound in dimension 20, with the exact ratio printed",
   "decider": "instruments/openaimath/finite/f174_projection_body.py",
   "decides": "the whole headline: K = T10 x T10 violates the proposed simplex bound in dimension 20, with the exact ratio printed",
   "value": {
    "ratio": "5588869/5505024",
    "decimal": "1.015230632964",
    "R_K": "295410156250000/321489",
    "c_20": "640000000000000000/707107401"
   },
   "checks": [
    {
     "check": "the theorem as stated in the paper",
     "pass": true,
     "detail": "main.tex Theorem thm:counterexample"
    },
    {
     "check": "R_d(T_d) = (d+1)d^d/d! re-derived for d = 2..7 by the zonotope sum",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R_20(T_20) = c_20 re-derived (21 determinants)",
     "pass": true,
     "detail": "640000000000000000/707107401"
    },
    {
     "check": "K = T10 x T10 has 22 facets",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R_20(K) / c_20 equals the printed value",
     "pass": true,
     "detail": "5588869/5505024 vs printed 5588869/5505024"
    },
    {
     "check": "R_20(K) / c_20 > 1 (the counterexample)",
     "pass": true,
     "detail": "ratio - 1 = 83845/5505024"
    },
    {
     "check": "the printed integer certificate 121 C(20,10) - 21 2^20 > 0",
     "pass": true,
     "detail": "335380"
    }
   ],
   "sources": {
    "preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/main.tex": "5302b6298402cf1665e5747766b53055ecad84a5027b1657811731389879f0e0"
   },
   "forges": [
    {
     "forge": "printed value off by one in the numerator",
     "verdict": "REFUTED"
    },
    {
     "forge": "T2 x T2 against c_4 (ratio 27/40, must not exceed 1)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.3
  },
  {
   "id": "F-177",
   "index": 177,
   "family": "090",
   "title": "An atomic certificate for triangular-lattice universal optimality",
   "tier": "A",
   "dir": "An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026",
   "leanChallenges": [
    "AtomicGaussian",
    "PlanarPacking",
    "TriangularEnergy",
    "TriangularGaussian"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every inequality of Lemma lem:finite-certificate (the two 20x20 interval blocks and their inverses, the parameter polytope, the jet / curvature / Taylor envelopes, all 659 Bernstein row minima) and the printed rational consequences; not the analytic interpolation, sign propagation or energy-transfer arguments, hence not the universal-optimality theorem",
   "decider": "instruments/openaimath/finite/f177_triangular_atomic.py",
   "decides": "a finite component: every inequality of Lemma lem:finite-certificate (the two 20x20 interval blocks and their inverses, the parameter polytope, the jet / curvature / Taylor envelopes, all 659 Bernstein row minima) and the printed rational consequences; not the analytic interpolation, sign propagation or energy-transfer arguments, hence not the universal-optimality theorem",
   "value": {
    "Q_min": "0.157468",
    "inv_norm_I-R": "16.152943",
    "inv_norm_I+R": "3.882191",
    "pivot_abs_range": "0.12346..1.72711",
    "bernstein_rows": 659,
    "seconds": 5.5
   },
   "checks": [
    {
     "check": "the paper prints the finite lemma, the polytope and the 659 row minima",
     "pass": true,
     "detail": "main.tex l.1129-1385, 2260-2265"
    },
    {
     "check": "CERTIFICATE.json names the scope as finite constants, the two 20x20 blocks, the polytope, envelopes, Bernstein",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. P_j exactly in Z[w]: P_{-j} = conj(P_j)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. P(n) = P'(n) = 0 exactly at every residue of I (double zeros)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. folded zero-frequency atomic masses are real (l.821-826), exactly",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. folded atomic masses reproduce kappa^2 P csc^2 and kappa P cot exactly at s = 2, 5, 6, 8, 33",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. Q_n, D_n (appendix recipe) agree with the closed product / cotangent forms at all 15 residues",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 2: 157/1000 < Q_n < 40 at every residue",
     "pass": true,
     "detail": "min 0.157468, max 39.8795"
    },
    {
     "check": "item 2: |D_n| < 283/100 at every residue",
     "pass": true,
     "detail": "max 2.821466"
    },
    {
     "check": "item 2: min(Q_16, Q_19, Q_21) > 16",
     "pass": true,
     "detail": "16.5038, 20.7972, 30.5480"
    },
    {
     "check": "item 2: the 15 gap-midpoint values of P(s)/(s-m)^2 exceed the printed bounds",
     "pass": true,
     "detail": "0.1657, 0.1206, 0.0548, 0.4821, 0.6568, 1.1190, 0.3376, 3.1305, 9.2615, 8.1943, 19.0883, 2.3473, 0.7069, 6.8080, 14.1155"
    },
    {
     "check": "item 1: every pivot of diagonal Gauss-Jordan on I-R and I+R excludes 0 (40 pivots)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 1: 3/25 < |pivot| < 87/50 for all 40 pivots",
     "pass": true,
     "detail": "min 0.12346, max 1.72711"
    },
    {
     "check": "item 1: ||(I-R)^-1|| < 404/25",
     "pass": true,
     "detail": "16.152943"
    },
    {
     "check": "item 1: ||(I+R)^-1|| < 389/100",
     "pass": true,
     "detail": "3.882191"
    },
    {
     "check": "|Zw - Delta| <= max(Z W*, Delta*) < 0.0024 (the coordinate bound used for e_3)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 1: sup_E ||A_1 e|| < 129/50 (columns 1-2 at z = 0, Z, the rest by coordinate bounds)",
     "pass": true,
     "detail": "2.576529"
    },
    {
     "check": "item 1: sup_E ||A_2 e|| < 14/5 (columns 1-2 at z = 0, Z, the rest by coordinate bounds)",
     "pass": true,
     "detail": "2.794585"
    },
    {
     "check": "polytope: 1/(pi(1/b - h)) < Z",
     "pass": true,
     "detail": "0.390708"
    },
    {
     "check": "polytope: e^{-2/Z} < W*",
     "pass": true,
     "detail": "0.006005"
    },
    {
     "check": "polytope: e^{-3/Z} < U*",
     "pass": true,
     "detail": "0.0004654"
    },
    {
     "check": "polytope: e^{-6/Z} < P*",
     "pass": true,
     "detail": "2.166e-07"
    },
    {
     "check": "polytope: Delta = (Z - z)e^{-2/z} has its maximum at z = sqrt(1+2Z) - 1, value < 0.00015 < Delta*",
     "pass": true,
     "detail": "0.000143016"
    },
    {
     "check": "eq. polytope-row-minimum equals the vertex minimum of E (40 rational rows, exact enumeration)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 3: ||W|| < 119 (tail variation matrix, eq. analytic:tail-variation)",
     "pass": true,
     "detail": "118.9065"
    },
    {
     "check": "item 3: Psi_{f,h,H}(W) < 335",
     "pass": true,
     "detail": "334.7"
    },
    {
     "check": "item 3: Psi_{T,H,h}(M) < 7/1000000000000000",
     "pass": true,
     "detail": "6.893e-15"
    },
    {
     "check": "item 3: Psi_{T,h,h}(W) < 23/1000000000",
     "pass": true,
     "detail": "2.209e-08"
    },
    {
     "check": "item 3: Psi_{T,h,h}(V) < 127/1000000000",
     "pass": true,
     "detail": "1.262e-07"
    },
    {
     "check": "item 3: (1 + pi eta)/(e^{36 pi eta} - 1) < 2.5e-10",
     "pass": true,
     "detail": "2.4447e-10"
    },
    {
     "check": "item 3: omitted target envelope over f cap [9,21] < 8.1e-8",
     "pass": true,
     "detail": "8.074e-08"
    },
    {
     "check": "item 3: omitted target envelope over T < 9.2e-29",
     "pass": true,
     "detail": "9.1067e-29"
    },
    {
     "check": "item 4: L curvature envelope below 1498, 45.1, 13.8, .00458, .000022",
     "pass": true,
     "detail": "s=0: L 1497.9, V 13522, T 26209; s=1: L 45.02, V 150.37, T 1146.6; s=2: L 13.726, V 30.877, T 780.05; s=29/2: L 0.0045732, V 0.0012152, T 411.3; s=23: L 2.1917e-05, V 3.179e-06, T 411.23"
    },
    {
     "check": "item 4: transformed-constant curvature envelope below 13530, 151, 31, .00122, .000004",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 4: T curvature envelope below 26500, 1160, 800, 800, 800",
     "pass": true,
     "detail": ""
    },
    {
     "check": "item 4: divided Taylor remainder envelope < .00161 at m = 1 and < .0000075 elsewhere (14 half-cells)",
     "pass": true,
     "detail": "(1,-1) 0.0016051, (1,1) 0.0016051, (3,-1) 1.72402e-08, (3,1/2) 6.9906e-15, (4,-1/2) 5.26082e-16, (4,3/2) 7.46215e-06, (7,-3/2) 7.68695e-08, (7,1) 1.35012e-11, (9,-1) 1.89734e-12, (9,3/2) 1.07266e-08, (12,-3/2) 8.98729e-10, (12,1/2) 6.8417e-20, (13,-1/2) 3.21924e-20, (13,3/2) 4.21356e-10"
    },
    {
     "check": "item 5: all 27 degree-20 sign tests, 567 Bernstein row minima over E, exceed the printed bounds",
     "pass": true,
     "detail": "T2(1,-1)>87/500: 0.1748405; Q1/2Z T1(1,-1)>57/1000: 0.0578388; T2(1,1)>11/1000: 0.0118632; 0.8Q1w T1(3,-1)>11/20000: 0.0037733; 0.8Q1w T1(3,1/2)>11/20000: 0.0005534; T2(3,-1)>97/100000: 0.0049426; T2(3,1/2)>97/100000: 0.0009708; -T T1(4,-1/2)>47/100000: 0.0004751; -T T1(4,3/2)>47/100000: 0.0022294; T2(4,-1/2)>97/100000: 0.0009708; T2(4,3/2)>97/100000: 0.0024462; -T T1(7,-3/2)>7/2500: 0.0028307; -T T1(7,1)>7/2500: 0.0039540; T2(7,-3/2)>7/2500: 0.0028861; T2(7,1)>7/2500: 0.0039603; -T T1(9,-1)>39/10000: 0.0039540; -T T1(9,3/2)>39/10000: 0.0067274; T2(9,-1)>39/10000: 0.0039603; T2(9,3/2)>39/10000: 0.0067298; -T T1(12,-3/2)>1/500: 0.0065626; -T T1(12,1/2)>1/500: 0.0020199; T2(12,-3/2)>1/500: 0.0065637; T2(12,1/2)>1/500: 0.0020202; -T T1(13,-1/2)>1/500: 0.0020199; -T T1(13,3/2)>1/500: 0.0084036; T2(13,-1/2)>1/500: 0.0020202; T2(13,3/2)>1/500: 0.0084045"
    },
    {
     "check": "item 5: the four exceptional degree-22 tests at m = y = 1 (92 Bernstein row minima) exceed .046, .056, .024, .027",
     "pass": true,
     "detail": ">23/500: 0.0464909; >7/125: 0.0562081; >3/125: 0.0243396; >27/1000: 0.0273774"
    },
    {
     "check": "Section 4 table: e'_+ < 1.800004e-9, e_+ < 1.179803e-5, e'_- < 3.360002e-9, e_- < 6.122004e-6 (exact rationals)",
     "pass": true,
     "detail": "1.800004e-09 1.179802e-05 3.360002e-09 6.122003e-06"
    },
    {
     "check": "Section 4: individual errors (e_+ + e_-)/2 < 1e-5 and (e'_+ + e'_-)/2 < 3e-9; 1 - delta - 340 beta N > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 4: the finite bounds feed ||B|| < 340, ||D|| < 3e-8, ||C||+||A|| < 3e-10, ||K_TV|| < 1.3e-7, d = 8.2e-8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma deleted-jets: .0171795 < .018, .00246348 < .003, .0001524 < .00016 (exact sums)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 5: r(Z, .5) = 836500/974643 > .8 and 1/(2Z) > .8; B_2(1) = 1 + 4Z + 6Z^2 = 3.481286",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 5 table: K = .0140109 / .00008363, .006*3 = .018 / .006*.054 = .000324, margins .0039891 / .00024037",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 3: the four endpoint differences kX_k(5/6) - v are 100079/455650, 8949/455650, 216419/554650, 105489/554650",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 4: 1/X <= (k^2 + 25/36)/B < 3k for k = h, H",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/build/main.tex": "bfa619a454d49596b65f310f0bfdce3ae3f5ed6545bd9058d0c6316843ded2a6",
    "preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/verification/CERTIFICATE.json": "76e1ad0b480f84dc00308c76498125426090918df1ecd1630b2ab1e6da28040d"
   },
   "forges": [
    {
     "forge": "the polytope constant W* = 0.0061 moved to 0.0060 (below e^{-2/Z} = 0.0060050)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the fixed coefficient c_{i,0} = 0.44 changed to 0.30",
     "verdict": "REFUTED"
    },
    {
     "forge": "the printed bound ||(I-R)^-1|| < 16.16 tightened to 16.10",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 5.5
  },
  {
   "id": "F-178",
   "index": 178,
   "family": "090",
   "title": "Universal optimality of the triangular lattice",
   "tier": "A",
   "dir": "Universal-optimality-of-the-triangular-lattice-September-23-2026",
   "leanChallenges": [
    "AtomicGaussian",
    "PlanarPacking",
    "TriangularEnergy",
    "TriangularGaussian"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Proposition prop:finite-certificate through its three stronger tables (27 power, 30 column, 37310 Bernstein and 10 tail comparisons) on enclosures of the exact quadrature arrays, the same tables on an exact execution of the floor-rounded procedure of Appendix C, the exact node constants, and the scalar budgets; not the analytic interpolation, sign-propagation, energy-transfer or mixture arguments, hence not the theorem",
   "decider": "instruments/openaimath/finite/f178_triangular_universal.py",
   "decides": "a finite component: Proposition prop:finite-certificate through its three stronger tables (27 power, 30 column, 37310 Bernstein and 10 tail comparisons) on enclosures of the exact quadrature arrays, the same tables on an exact execution of the floor-rounded procedure of Appendix C, the exact node constants, and the scalar budgets; not the analytic interpolation, sign-propagation, energy-transfer or mixture arguments, hence not the theorem",
   "value": {
    "A_seconds": 16.6,
    "A_bernstein_min_by_group": {
     "(0, '59/25')": 0.3378864,
     "(0, '53/20')": 0.3487801,
     "(0, '16/5')": 0.3629395,
     "(0, '4')": 0.3768646,
     "(0, '6')": 0.3955472,
     "(1, '59/25')": 0.0204609,
     "(1, '53/20')": 0.0181503,
     "(1, '16/5')": 0.0340869,
     "(1, '4')": 0.0375484,
     "(1, '6')": 0.0120037,
     "(2, '59/25')": 0.0034722,
     "(2, '53/20')": 0.0038829,
     "(2, '16/5')": 0.0063695,
     "(2, '4')": 0.0087667,
     "(2, '6')": 0.0107646,
     "(3, '59/25')": 0.0012061,
     "(3, '53/20')": 0.0019328,
     "(3, '16/5')": 0.0027156,
     "(3, '4')": 0.0032953,
     "(3, '6')": 0.0039561
    },
    "B_seconds": 74.7,
    "seconds": 105.4
   },
   "checks": [
    {
     "check": "CERTIFICATE-INPUTS.json: the 13 manuscript sources hash to the recorded sha256 (the arrays are defined there)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the paper and README state the floor-rounded matrix construction is not executed by the supplied programs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the paper prints the counts: 84 nodes, 171 coordinates, 2311 mass points, 46 centres, 91 half-gaps, 37310 comparisons",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the counts recomputed: |F| = 84, |N_0 cap [0,88]| = 46, half-gaps 91, 91*2*5*41 = 37310; the half-gaps cover [0, 89.5]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exact: P_l from prod_{a in L} (w - zeta^a)^2 equals the printed (5, -1+2ib, 1+2ib, -2, -1/2+ib, -1-2ib, 1), P_-l = conj P_l",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exact: P(n) = P'(n) = 0 at the six residues (in Q(sqrt3)(i))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exact: Q_a = (pi/6)^2 prod (2 sin(pi(a-d)/12))^2 and D_a = (pi/6) sum cot(pi(a-d)/12) equal the printed table",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exact: the same Q_a, D_a from the Fourier coefficients, P''(a) = 2Q_a and P'''(a) = 6 Q_a D_a",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma rational-pi-b: p_* < pi < p_* + 1e-80 (Machin brackets, exact rationals)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma rational-pi-b: b_*^2 < 3/4 < (b_* + 1e-80)^2 (exact)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "pi > 16(1/5 - 1/(3 5^3)) - 4/239 = 281476/89625 > 157/50; sqrt3 < 1733/1000; pi(2/sqrt3 - 2/5) > 2.36 chain",
     "pass": true,
     "detail": ""
    },
    {
     "check": "quadrature lemma: e < 95901/35280 < 2719/1000; 220(2.719)^182 < 1e83; 1100(2.719)^160 < 1e83; 4e114 < 2^384",
     "pass": true,
     "detail": ""
    },
    {
     "check": "quadrature lemma: pi M/3 + 1.65 pi < 182 and pi M/6 + .081 pi M + 9.3 pi < 160 (pi < 22/7, M = 168); |z| <= B/(h-1/6) + h = 214/35; Im z >= -1402/17505 at u = 11/12; 37 pi^2/3 < 220",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma kernel-bounds: |lambda(0)| = 5/sqrt3 < 3, |z(0)| = 44/15, q(1) = 26pi/435 > .187, q(5/6) = 862pi/3845 > .70, |lambda(5/6)| = 60/sqrt2307 < 5/4, |z(5/6)|^2 = 33476/19225 < (33/25)^2, -q'(1) = 2000pi/2523 > 2.48",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma kernel-bounds: 65pi^2/18 < 36, 32pi/3 < 36, 3(1+2.12+3pi)/1.75 < 22, 1.25(1+2.12+1.32pi)/1.75 < 5.3, 22*36/(e^.187 - 1) < 4000 via e^.187 - 1 > .198, 22*36 <= 800, 2pi/2.48 < 2.54",
     "pass": true,
     "detail": ""
    },
    {
     "check": "node-constant bounds (2:151-157): Q_n > 7/4 via (2/3)(157/50)^2(267/1000) > 7/4 and 2 - sqrt3 > .267; |D_n| < 53/25 via (22/18)(1733/1000) < 53/25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq. scalar-error-budgets: all six quantities lie below their printed outward bounds, which lie below the allowances",
     "pass": true,
     "detail": "1.06693e-14, 8.15742e-14, 2.27107e-07, 7.79146e-12, 4.71848, 9.59326e-05"
    },
    {
     "check": "Section 4 constants: c_* = 548/413 < 4/3; 1.1e-14 + margin 1.425e-15 > 1e-170",
     "pass": true,
     "detail": ""
    },
    {
     "check": "half-gap remainders R(m,Y,nu) for (0,1/2,0), (1,1,2), (3,1,2), (4,3/2,2) below 1.8e-19, 1e-11, 3e-16, 2e-11; plus 41*4*5*1e-31 = 8.2e-29, each sum < 1e-8",
     "pass": true,
     "detail": "3.016e-20, 2.435e-12, 5.61e-17, 1.909e-11"
    },
    {
     "check": "target-support tails: (e^6/6) 3^41/41!/(1-3/42) < 8e-29 and 6 6^41/43!/(1-6/44) < 1e-20",
     "pass": true,
     "detail": "7.895e-29, 9.223e-21"
    },
    {
     "check": "tail second-derivative majorant at s0 = 89.5 < 8.5485e-5 < .0002; s0 - 1 > 1/2.36",
     "pass": true,
     "detail": "8.54841e-05"
    },
    {
     "check": "12 - 8 sqrt2 > .6862 > .68 (exact: 8 sqrt2 < 11.3138); final margins .000888 > 0, .0021 - 2e-8 > .002, .68*.002 > .0001",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma sine-product-barrier: the six midpoint values of P(s)/(s-m)^2 match 12-8sqrt2, 1, (8/9)(3+sqrt2+sqrt3+sqrt6), 9 and exceed .68 (from the sine product)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Section 4 arithmetic: 36/(1-.004^8) < 37, 37/(1-74e-27) < 40, 1.1e-14(1+160000) = 1.760011e-9, 20000(.004)^8 = 1.31072e-15, 8e-12+1e-24+1.4e-15 < 1e-11, u_F + u_T < 1.3681e-8 < 2e-8, 4.8 + 2e-8 < 5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "local target tails: P_7(59/25) > 10.549 > 21/2 termwise; (548/413)(21/19) < 3/2; (21/20)^10 > 3/2; 3^16 > 150 2^16; first-five jets (548/413)(1+(2/21)^2+...) < 1.35; 1.35 + (4000 + 2e-27)4.8 < 20000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "roundoff lemma majorants: e_* Taylor error < 1.263e-78 (with e^.416 < 2719/1000); (4e-78 + 24 eps)3^12 e^7 < 2.472e-69 and 41*4*230*3e^15*1e-29 < 3.700e-18 (e^7, e^15 enclosed; both FAIL with e < 2719/1000); matrix recurrence max e_r < 5.765e-38, v_9 < 6.529e-36; 40000 v_9 + 72e-43 + 336 eps < 2.612e-31; E column max 3.3505, 2 + (4000+2e-27)3.3505 < 13405",
     "pass": true,
     "detail": "Taylor 1.26274e-78, e^7 term 2.47105e-69, e^15 term 3.69922e-18, v_9 6.52808e-36"
    },
    {
     "check": "route A (exact-array enclosures): Table power-bounds, all 27 strict upper comparisons (||U|| before, ||V+||, ||V-|| after each step)",
     "pass": true,
     "detail": "0: 1.72578 2.72578 2.05368; 1: 2.07688 6.92516 2.164; 2: 2.25363 15.3035 2.114; 3: 1.73882 25.2799 1.95551; 4: 0.739031 31.0187 1.86133; 5: 0.117763 32.0742 1.84422; 6: 0.00295639 32.1014 1.84378; 7: 1.86308e-06 32.1014 1.84378; 8: 7.39905e-13 32.1014 1.84378"
    },
    {
     "check": "route A (exact-array enclosures): Table column-bounds, every column of X_1, X_2 and every same-column sum (30 strict comparisons)",
     "pass": true,
     "detail": "max cols 1-5 / 6-10: X1 1.28379 / 2.64247, X2 1.00224 / 3.02131, sum 2.16550 / 5.34258"
    },
    {
     "check": "route A (exact-array enclosures): Table sign-bounds, all 37310 half-gap Bernstein lower comparisons strictly above their cutoffs",
     "pass": true,
     "detail": "minima per (K; m=0, m=1, m=3,4, 7<=m<=88): K=59/25: 0.337886 0.020461 0.003472 0.001206; K=53/20: 0.348780 0.018150 0.003883 0.001933; K=16/5: 0.362939 0.034087 0.006370 0.002716; K=4: 0.376865 0.037548 0.008767 0.003295; K=6: 0.395547 0.012004 0.010765 0.003956"
    },
    {
     "check": "route A (exact-array enclosures): Table sign-bounds, the 10 tail comparisons strictly above .0022, .0029, .0036, .0043, .0052",
     "pass": true,
     "detail": "0.002455 0.002268 0.002980 0.003114 0.003721 0.004256 0.004432 0.005317 0.005372 0.006682"
    },
    {
     "check": "route A: the Fejer weights are positive (o_v >= 1/(N(N-1))) and their enclosed sum contains 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "route B (floor-rounded listing, exact): Table power-bounds, all 27 strict upper comparisons (||U|| before, ||V+||, ||V-|| after each step)",
     "pass": true,
     "detail": "0: 1.72578 2.72578 2.05368; 1: 2.07688 6.92516 2.164; 2: 2.25363 15.3035 2.114; 3: 1.73882 25.2799 1.95551; 4: 0.739031 31.0187 1.86133; 5: 0.117763 32.0742 1.84422; 6: 0.00295639 32.1014 1.84378; 7: 1.86308e-06 32.1014 1.84378; 8: 7.39905e-13 32.1014 1.84378"
    },
    {
     "check": "route B (floor-rounded listing, exact): Table column-bounds, every column of X_1, X_2 and every same-column sum (30 strict comparisons)",
     "pass": true,
     "detail": "max cols 1-5 / 6-10: X1 1.28379 / 2.64247, X2 1.00224 / 3.02131, sum 2.16550 / 5.34258"
    },
    {
     "check": "route B (floor-rounded listing, exact): Table sign-bounds, all 37310 half-gap Bernstein lower comparisons strictly above their cutoffs",
     "pass": true,
     "detail": "minima per (K; m=0, m=1, m=3,4, 7<=m<=88): K=59/25: 0.337886 0.020461 0.003472 0.001206; K=53/20: 0.348780 0.018150 0.003883 0.001933; K=16/5: 0.362939 0.034087 0.006370 0.002716; K=4: 0.376865 0.037548 0.008767 0.003295; K=6: 0.395547 0.012004 0.010765 0.003956"
    },
    {
     "check": "route B (floor-rounded listing, exact): Table sign-bounds, the 10 tail comparisons strictly above .0022, .0029, .0036, .0043, .0052",
     "pass": true,
     "detail": "0.002455 0.002268 0.002980 0.003114 0.003721 0.004256 0.004432 0.005317 0.005372 0.006682"
    },
    {
     "check": "route B: every e_* argument satisfies |w| < 1700 and Re w < 7 (Lemma rational-roundoff, item 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Proposition finite-certificate follows from the tables: .0031 < .004, 32.12 + 1.85 < 36, 2.20 < 2.3, 5.40 < 5.6, .00116 > .001, .0022 > .0021, and (floor route) .00116 - 1e-6 > .001, .0022 - 1e-6 > .0021",
     "pass": true,
     "detail": ""
    },
    {
     "check": "cross-check: every entry of S# lies within 1e-44/168 of the exact S~ (roundoff lemma item 2: operator error < 1e-44)",
     "pass": true,
     "detail": "max entry deviation 1.28e-52"
    },
    {
     "check": "cross-check: combined column error |X#_1 - X_1| + |X#_2 - X_2| < 1e-29 in every column (roundoff lemma item 3)",
     "pass": true,
     "detail": "max 5.21e-45"
    },
    {
     "check": "cross-check: every Bernstein row entry L#_{i,d} lies within 1e-16 of L_{i,d} (roundoff lemma item 3), all 7462 rows",
     "pass": true,
     "detail": "max 3.75e-37"
    }
   ],
   "sources": {
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/main.tex": "0659cc480c9fe569812056d9f19f7dc757f3eb073487c2983c694033fa7b374f",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/02-cardinal-fourier.tex": "6c056e68cefadd5f5d5ef20ba4fd5c3397de52369a16ecd56b4b8446e720dc8c",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/03-quadrature-finite-data.tex": "58ae00cd73a9de0d85fc31d3e399c1e1afb25a752ae9088c1424b9161408ae8d",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/04-exact-interpolation.tex": "a0890d277c3b58dbe2dfa5bdf8c3e18c107cb16e71f9588ba20eed071cceaede",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/05-global-signs.tex": "dfb6a62f0bda56c56e3f82d02e2d4eebbd111766ac5e0a94b8f00d753f9606d4",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/08-arithmetic-verification.tex": "ce6acf5421bea6ba06cca6b411eca1245a1642d13a710b2334ae78ce67067dd5",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/build/sections/09-conditional-floor-verification.tex": "6c1ecac1f36c1ce6ac9dac8225d514b4114ab6aef5707d6836e77b9eca12f646",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/verification/README.md": "45c7e0543446d36cbefb64bc6f95cc62c618846b12b0b6977259986ca9410e89",
    "preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/verification/CERTIFICATE-INPUTS.json": "7f5ff0bafbe83b4c06d83daa3c953f6130dadfca65f1e06df79034567478d539"
   },
   "forges": [
    {
     "forge": "Table sign-bounds, K = 2.36, 7 <= m <= 88: printed .00116 raised to .00121 (the enclosed minimum is .0012061)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table power-bounds, step 8: ||V+|| < 32.12 tightened to 32.0",
     "verdict": "REFUTED"
    },
    {
     "forge": "extra data E(k): the constant coordinate C_1 = -.0074 + .0115/k flipped to +.0074 + .0115/k",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 105.4
  },
  {
   "id": "F-179",
   "index": 179,
   "family": "090",
   "title": "A sharp Fourier certificate for planar circle packing",
   "tier": "A",
   "dir": "A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026",
   "leanChallenges": [
    "AtomicGaussian",
    "PlanarPacking",
    "TriangularEnergy",
    "TriangularGaussian"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every finite certificate the proof cites -- Lemma interpolation:finite (the two inverse certificates via W_+-, the exterior block, the 204 residuals, the table norms) and Lemma signs:bernstein (2436 Bernstein coefficients), recomputed with an independent quadrature and error bound, the exact node data and envelope constants, and every scalar substitution of certificates.tex:487-572; NOT the theorem, whose infinite inversion, sign argument and normalisation are analytic",
   "decider": "instruments/openaimath/finite/f179_circle_packing.py",
   "decides": "a finite component: every finite certificate the proof cites -- Lemma interpolation:finite (the two inverse certificates via W_+-, the exterior block, the 204 residuals, the table norms) and Lemma signs:bernstein (2436 Bernstein coefficients), recomputed with an independent quadrature and error bound, the exact node data and envelope constants, and every scalar substitution of certificates.tex:487-572; NOT the theorem, whose infinite inversion, sign argument and normalisation are analytic",
   "value": {
    "quadrature_error_bound_per_moment": "9.398e-29",
    "D_block": [
     [
      -0.2459266632296453,
      -0.023655100336718873,
      -0.001097737034296297,
      0.424700589287547,
      0.14567104797035005,
      -0.049222269312866614
     ],
     [
      0.08491908392062632,
      -0.05868197098898521,
      0.011552892703545774,
      -0.4018186176423354,
      -0.013628559633735632,
      -0.08551572048310671
     ],
     [
      0.03851356741224271,
      -0.050367220418701314,
      0.016906689819273464,
      0.22714513313159101,
      -0.14050143849687882,
      -0.07839077610943805
     ],
     [
      0.03045255506692664,
      0.08052883858673528,
      -0.056406703537031366,
      0.931968827424316,
      0.5217563428109545,
      0.027834593699646377
     ],
     [
      -0.10645038355983477,
      -0.009335195132489238,
      -0.046599334773430375,
      0.05116063584659245,
      0.6966037598782777,
      -0.11396537120443254
     ],
     [
      -0.035236824314615944,
      -0.06556376796860185,
      -0.03213831529258536,
      0.013947539579869522,
      0.6176419715067059,
      -0.19777206623826185
     ]
    ],
    "defect_+1": [
     "0.000254226249106525452079",
     "0.000254226249106525452080"
    ],
    "defect_-1": [
     "0.000422881651025533561534",
     "0.000422881651025533561535"
    ],
    "exterior": [
     "0.086887481822896415914050",
     "0.086887481822896415914051"
    ],
    "max_residual": "6.3418e-11",
    "bernstein_minima": {
     "0,2": "0.763187",
     "1,1": "0.315557",
     "1,2": "0.192424",
     "3,1": "0.028830",
     "3,2": "0.024805",
     "4,1": "0.028830",
     "4,2": "0.024805",
     "7,1": "0.115318",
     "7,2": "0.112836",
     "9,1": "0.168569",
     "9,2": "0.161067",
     "12,1": "0.012161",
     "12,2": "0.011197",
     "13,1": "0.012161",
     "13,2": "0.011197",
     "15,1": "0.010575",
     "15,2": "0.009519",
     "16,1": "0.010575",
     "16,2": "0.009519",
     "19,1": "0.124084",
     "19,2": "0.114093",
     "21,1": "0.133061",
     "21,2": "0.130158",
     "24,1": "0.011632",
     "24,2": "0.011580",
     "25,1": "0.011632",
     "25,2": "0.011580",
     "27,1": "0.011214",
     "27,2": "0.011395",
     "28,1": "0.011214",
     "28,2": "0.011395",
     "31,1": "0.123661",
     "31,2": "0.127051",
     "33,1": "0.122469",
     "33,2": "0.128933",
     "36,1": "0.010913",
     "36,2": "0.011595",
     "37,1": "0.010913",
     "37,2": "0.011595",
     "39,1": "0.010770",
     "39,2": "0.011587",
     "40,1": "0.010770",
     "40,2": "0.011587"
    },
    "refuting": [],
    "seconds": 6.6
   },
   "checks": [
    {
     "check": "A. the theorem as stated: fhat(0)=1, f(0)=2/sqrt3, fhat>=0, f<=0 for |x|>=1 (introduction.tex:34-41)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. coefficients.tsv = data/coefficients.tex = certificate-inputs.json (51 rows, scale 1e10)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. the table nodes are exactly I_f = N cap [1,100] (fourier.tex:32-33, interpolation.tex:206)",
     "pass": true,
     "detail": "51 nodes"
    },
    {
     "check": "A. W_+ and W_- agree in .tsv, .tex and .json (scale 1e4, order c1,c3,c4,d1,d3,d4)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. fixed node-zero data and constants: json = fourier.tex:348-351 = certificates.tex:27-31",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. P_j from w^-6 prod_a (w - rho^a)^2 equal the printed (P_0..P_6) (fourier.tex:66-68), P_-j = conj P_j",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. P(n) = P'(n) = 0 exactly at every residue n mod 12 in A (double zeros)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Q_n = P''(n)/2, D_n = P'''(n)/(6Q_n) (Fourier route) = product/cot formula (fourier.tex:88-96) = printed table (fourier.tex:98-107), exactly in Q(sqrt3)",
     "pass": true,
     "detail": "a=0: Q/pi^2=1/3+0 r3, D/pi=0+-7/18 r3; a=1: Q/pi^2=4/3+-2/3 r3, D/pi=1/6+1/18 r3; a=3: Q/pi^2=4/3+-2/3 r3, D/pi=-1/6+-1/18 r3; a=4: Q/pi^2=1/3+0 r3, D/pi=0+7/18 r3; a=7: Q/pi^2=4/3+2/3 r3, D/pi=-1/6+1/18 r3; a=9: Q/pi^2=4/3+2/3 r3, D/pi=1/6+-1/18 r3"
    },
    {
     "check": "B. rational-check.tex:138-145: Q_1=Q_3=(4pi^2/3)(1-b), Q_4=pi^2/3, D_1=-D_3=pi(3+2b)/18, D_4=7pi b/9",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Q_n > 197192/112500 > 7/4 and |D_n| < 286/135 < 53/25 by the printed chain (pi in (157/50, 22/7), sqrt3 < 26/15)",
     "pass": true,
     "detail": "min Q_n = 1.763035, max |D_n| = 2.116099"
    },
    {
     "check": "B. j^2+jk+k^2 mod 12 takes exactly the residues A = {0,1,3,4,7,9}; 15 is in N but is not a value (fourier.tex:35-45)",
     "pass": true,
     "detail": "[0, 1, 3, 4, 7, 9]"
    },
    {
     "check": "B. |P_j| = 5,2,2,2,1,2,1: atom masses (5,4,4,4,2,4,2), sum_j |P_j| = 25, pi^2 sum |P_l| t_l^2 = 65pi^2/18, 2pi sum |P_l| t_l = 32pi/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Im z(t) = h(B/(t^2+h^2) - 1) has minimum 26/435 on [-1,1], at |t| = 1 (fourier.tex:258-260, 303-304)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. |lambda(t_j)| <= b*_j, |z(t_j)| <= a*_j, Im z(t_j) >= g_j at all seven endpoints (exact rational)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. beta_j <= both endpoint values of -d Im z/dt = 2hBt/(t^2+h^2)^2 (unimodal, max at t = h/sqrt3), and B^2 - 2Bh^2 = 304/225 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the density bound 2 max_a max{pi|L_{a,j}|, pi^2 max_u |H_{a,j} - u L_{a,j}|} < M_j for j = 1..6 (interpolation.tex:95-102)",
     "pass": true,
     "detail": "55.6916<56, 47.5523<48, 34.7273<35, 24.7753<25, 18.2798<19, 6.2832<1571/250"
    },
    {
     "check": "E. row envelope at cut 0 < 27618723/1000000 (cert:row-substitutions), hence < 277/10 and < 28",
     "pass": true,
     "detail": "[27.6187223397337, 27.6187223397337] < 27618723/1000000"
    },
    {
     "check": "E. row envelope at cut 4 < 134547/500000 (cert:row-substitutions), hence < 34/125 and < 7/25",
     "pass": true,
     "detail": "[0.269093906605057, 0.269093906605057] < 134547/500000"
    },
    {
     "check": "E. row envelope at cut 16 < 7061/500000 (cert:row-substitutions), hence < 29/2000 and < 3/200",
     "pass": true,
     "detail": "[0.014121791598236, 0.014121791598236] < 7061/500000"
    },
    {
     "check": "E. row envelope at cut 100 < 461701/1000000000000000 (cert:row-substitutions), hence < 47/100000000000 and < 1/2000000000",
     "pass": true,
     "detail": "[4.61700965256316e-10, 4.61700965256316e-10] < 461701/1000000000000000"
    },
    {
     "check": "E. the atom-column envelope beyond 100 < 3.131186e-8 (certificates.tex:511-512), hence < 3.2e-8",
     "pass": true,
     "detail": "[3.13118538656315e-08, 3.13118538656315e-08] < 1565593/50000000000000"
    },
    {
     "check": "E. rational lower sum (signs:rational-lower-sum), index 1 = 0.0120683484 > 603417/50000000 > 3/250, and - delta/(s0-40) = -.00002 leaves > .011 (exact)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. rational lower sum (signs:rational-lower-sum), index 2 = 0.0165646804 > 414117/25000000 > 2/125, and - delta/(s0-40) = -.00002 leaves > .011 (exact)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. sum_j M_j (pi t_j)^2 / 6 < 76.043 < 77",
     "pass": true,
     "detail": "[76.0427431240673, 76.0427431240673] < 76043/1000"
    },
    {
     "check": "E. the second-derivative tail (signs:tail-second-derivative) < .005303 < .006",
     "pass": true,
     "detail": "[0.0053021040742676, 0.0053021040742676] < 5303/1000000"
    },
    {
     "check": "E. perturbation cap (signs:perturbation-envelope)/nu! at (s*,nu) = (0,0) < 3189/1000000",
     "pass": true,
     "detail": "[0.003188475, 0.003188475] < 3189/1000000"
    },
    {
     "check": "E. perturbation cap (signs:perturbation-envelope)/nu! at (s*,nu) = (1/2,1) < 467/125000",
     "pass": true,
     "detail": "[0.00373531089838195, 0.00373531089838195] < 467/125000"
    },
    {
     "check": "E. perturbation cap (signs:perturbation-envelope)/nu! at (s*,nu) = (1/2,2) < 4573/500000",
     "pass": true,
     "detail": "[0.00914529310566468, 0.00914529310566468] < 4573/500000"
    },
    {
     "check": "E. perturbation cap (signs:perturbation-envelope)/nu! at (s*,nu) = (2,2) < 449/250000",
     "pass": true,
     "detail": "[0.00179530475939406, 0.00179530475939406] < 449/250000"
    },
    {
     "check": "E. hence the caps of signs:perturbation-caps: < .004 (m=0), < .011 (m=1, both orders), < .002 (m>=3)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 2.29 sum M_j/6 + .017 sum m*_j < 72.669 < 80 (direct-measure variation)",
     "pass": true,
     "detail": "[72.6683933333333, 72.6683933333333] < 72669/1000"
    },
    {
     "check": "E. Taylor representative E(m,Y,nu) = (0,1/2,0) < 789/5000000000000 < .001 (cert:taylor-substitutions)",
     "pass": true,
     "detail": "[1.57702579927588e-10, 1.57702579927588e-10] < 789/5000000000000"
    },
    {
     "check": "E. Taylor representative E(m,Y,nu) = (1,1,1) < 2007/100000000 < .001 (cert:taylor-substitutions)",
     "pass": true,
     "detail": "[2.00678760960183e-05, 2.00678760960183e-05] < 2007/100000000"
    },
    {
     "check": "E. Taylor representative E(m,Y,nu) = (1,1,2) < 1147/200000000 < .001 (cert:taylor-substitutions)",
     "pass": true,
     "detail": "[5.73458543674384e-06, 5.73458543674398e-06] < 1147/200000000"
    },
    {
     "check": "E. Taylor representative E(m,Y,nu) = (3,1,2) < 9581/100000000000000 < .001 (cert:taylor-substitutions)",
     "pass": true,
     "detail": "[9.58092657130689e-11, 9.58092657130712e-11] < 9581/100000000000000"
    },
    {
     "check": "E. Taylor representative E(m,Y,nu) = (4,3/2,2) < 6133/10000000000 < .001 (cert:taylor-substitutions)",
     "pass": true,
     "detail": "[6.13255109716883e-07, 6.13255109716909e-07] < 6133/10000000000"
    },
    {
     "check": "E. P(s)/(s-m)^2 > .68 at the six gap midpoints of a period (Lemma signs:P-lower-bound); (283/200)^2 > 2 and 12 - 8(283/200) = .68",
     "pass": true,
     "detail": "1/2:0.686292, 7/2:0.686292, 2:1.000000, 8:9.000000, 11/2:7.640670, 21/2:7.640670"
    },
    {
     "check": "E. the printed midpoint values 12-8sqrt2, 1, 9, (8/9)(3+sqrt2+sqrt3+sqrt6) and P = (4c^2-3)^2(1-2c)^2 agree with enclosures of the product (consistency, not equality)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. arithmetic: the quotient margins .735, .168, .017, .006 (signs:finite-quotient) and .011(.68) - .006/2 = .00448 (signs:tail-margin)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. arithmetic of Lemma interpolation:inverse and Prop. interpolation:solution: .09+.015<.11; .28+.11(3.7)=.687<1; 84.8562<90; 2521 delta<1; coefficients 90.000113<91, 2521.0032<2540; 102e-9<1.1e-7; 2(91R_f+2540R_t) = .00002860012 < .00003; 2.29+.00003<2.3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the paper's quadrature-lemma arithmetic: Delta_1 = 167/1200, Delta_6 = 389/400, Im z >= -1402/17505 > -.081, the 4.774/5.912 squarings, integrand bounds < 5.38e50 and < 1.51e50, 2^-254 1e53 < 3.455e-24 (their bound; this decider uses its own)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. Fejer rule N=256: all weights > 0 and their sum encloses 2; error bound per moment (derived here) < 1e-24",
     "pass": true,
     "detail": "min weight 6.571e-05, bound 9.398e-29"
    },
    {
     "check": "D. corroboration of exactness: the rule integrates x^k on [-1,1] to within 1e-40 for k = 0,5,...,255",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_+1|| = 308353/10000 < 771/25 (exact column sums; cert:matrix-table)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||I - W_+1(I - D)|| in [0.000254226249106525452079, 0.000254226249106525452080] < 1/1000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the printed enclosure [0.00025422624910652544, 0.00025422624910652546] of that defect norm contains the one computed here (certificates.tex:405-410)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_+1 U_{7:9}|| <= 3.527816 < 177/50",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_+1 U_{12:16}|| <= 0.424592 < 11/25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_+1 U_{19:100}|| <= 0.233376 < 1/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the W argument: (I - D) invertible, ||(I - D)^-1|| <= ||W||/(1-defect) = 30.8431 < 32 and ||(I - D)^-1 U|| <= 3.5287 < 3.7",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_-1|| = 18277/10000 < 183/100 (exact column sums; cert:matrix-table)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||I - W_-1(I + D)|| in [0.000422881651025533561534, 0.000422881651025533561535] < 1/1000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the printed enclosure [0.00042288165102553355, 0.00042288165102553357] of that defect norm contains the one computed here (certificates.tex:405-410)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_-1 U_{7:9}|| <= 0.721065 < 73/100",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_-1 U_{12:16}|| <= 0.307873 < 8/25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||W_-1 U_{19:100}|| <= 0.156935 < 17/100",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the W argument: (I + D) invertible, ||(I + D)^-1|| <= ||W||/(1-defect) = 1.8285 < 32 and ||(I + D)^-1 U|| <= 0.7214 < 3.7",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. ||S_{E cap [7,16],J}|| in [0.086887481822896415914050, 0.086887481822896415914051] < 11/125 (cert:exterior-block) < .09",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the printed enclosure [0.08688748182289641591, 0.08688748182289641592] of the exterior norm contains the one computed here (certificates.tex:412-413)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "SECOND WAY: the 36 entries of D = S_{J,J} recomputed with a Fejer rule of 192 nodes per piece (own error bound) overlap the N=256 enclosures",
     "pass": true,
     "detail": "max radius 4.71e-27"
    },
    {
     "check": "SECOND WAY: all 1364 direct moments p^(l)(m)/l! (both tables, 22 centres, l <= 30) by quadrature of the folded measure overlap the exact node-Taylor values",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. all 204 residuals of cert:residual-bounds are < 3/10000000000 (largest enclosure upper end 6.3418e-11), hence < 1e-9 (interpolation:finite-residuals)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the four table norms (cert:table-norms) by integer addition: 2.2516666247, 0.8927477314, 1.97471e-05, 1.61623e-05",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the half-gaps: 22 centres in [0,40], 43 for index 2 and 41 for index 1, 84 in all (signs.tex:19-37)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. all 2436 Bernstein coefficients B_k exceed their grouped bounds of cert:bernstein-table",
     "pass": true,
     "detail": "group minima: m=0,i=2:0.76319, m=1,i=1:0.31556, m=1,i=2:0.19242, m=3,i=1:0.02883, m=3,i=2:0.02481, m=4,i=1:0.02883, m=4,i=2:0.02481, m=7,i=1:0.11532, m=7,i=2:0.11284, m=9,i=1:0.16857, m=9,i=2:0.16107, m=12,i=1:0.01216, m=12,i=2:0.01120, m=13,i=1:0.01216, m=13,i=2:0.01120, m=15,i=1:0.01057, m=15,i=2:0.00952, m=16,i=1:0.01057, m=16,i=2:0.00952, m=19,i=1:0.12408, m=19,i=2:0.11409, m=21,i=1:0.13306, m=21,i=2:0.13016, m=24,i=1:0.01163, m=24,i=2:0.01158, m=25,i=1:0.01163, m=25,i=2:0.01158, m=27,i=1:0.01121, m=27,i=2:0.01139, m=28,i=1:0.01121, m=28,i=2:0.01139, m=31,i=1:0.12366, m=31,i=2:0.12705, m=33,i=1:0.12247, m=33,i=2:0.12893, m=36,i=1:0.01091, m=36,i=2:0.01159, m=37,i=1:0.01091, m=37,i=2:0.01159, m=39,i=1:0.01077, m=39,i=2:0.01159, m=40,i=1:0.01077, m=40,i=2:0.01159"
    },
    {
     "check": "D. hence all exceed the thresholds of Lemma signs:bernstein (.74 at m=0, .18 at m=1, .02 at m=3,4, .009 beyond), and every table bound exceeds its threshold",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. B at (i,m,y,k) = (2,16,-1/2,28) in [0.0095187595568082337321, 0.0095187595568082337322], inside the printed [0.00951875955680, 0.00951875955684] (certificates.tex:477-479)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/introduction.tex": "dd16705680392b3c9113d18843a7615559f842dd35de3b9aacc02400aed2cfa9",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/fourier.tex": "d77e937bd06de5e444e0a31b8133fc54c75b1bb2d155a77de67cc04d9d9ab818",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/interpolation.tex": "afd0494a9b1790e2df3ed77f7fb3f2cab9764c396ac9c155e5feea6a43ce5e5d",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/signs.tex": "bde028b69c1cb68358d2920b2dbd6001c51ab0adb8c8484bd261bfef9d57b438",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/certificates.tex": "2e3d6227f744e3eab60ce0c8dc66bb55f4efef9d18189d5b90cc2ee890101168",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/sections/rational-check.tex": "efa68deb925c58992ead4ba5e1b87eb6fa9e84f3a97689f9c134e4fa17625cdc",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/coefficients.tsv": "b62c6b3ea62ab8d7906214f009b23b232c16602cb6ae0487ae8fa362c0ad4ced",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/data/coefficients.tex": "2dc6a2a19f0abdb0e0ca3f57261d44938a7c9f1ae8697dc816d7911875038577",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/certificate-inputs.json": "f967c6559e568142f1a9c174f54e7087bfa7934dc7f38792084db6351c5676c4",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/W_plus.tsv": "c8588c3c6c8bd0278980bda25dbfd1018683e3df5bf38c8fc3c5b5dc1c2d9398",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/data/W_plus.tex": "b73b2488a0884682d17a00a28467bc58a2f07fcee905997c844ce9662d8b59d5",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/verification/data/W_minus.tsv": "ccab423b5a6575b4c539da031f575d9504bc14770373948b09f9c342c83e6141",
    "preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/build/data/W_minus.tex": "0b8e7055844b8dc23d27e54e0526a1a62b8b69f4386f1518638c1f62cfa3c77d"
   },
   "forges": [
    {
     "forge": "table entry c~_{1,7} moved by 1e-9 (10 integer units)",
     "verdict": "REFUTED"
    },
    {
     "forge": "W_+ entry (c1,c1) moved by 2e-3 (the defect bound .001 must fail)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Bernstein group bound for m in {12..16}, i=2 printed .0096 instead of .0094",
     "verdict": "REFUTED"
    },
    {
     "forge": "constant C_1 = -0.012 instead of -0.013",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed exterior-norm enclosure shifted by one unit in its last digit",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 6.6
  },
  {
   "id": "F-180",
   "index": 180,
   "family": "090",
   "title": "Triangular minimality for planar Coulomb renormalized energy",
   "tier": "A",
   "dir": "Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026",
   "leanChallenges": [
    "AtomicGaussian",
    "PlanarPacking",
    "TriangularEnergy",
    "TriangularGaussian"
   ],
   "word": "CERTIFIED",
   "why": "a finite component -- the whole finite certificate of Section sec:computation (Proposition comp:finite-soundness), by an independent route: the 38 lattice coefficients (two ways) and |c_1| < 1; lambda, mu, K, B(l), f_0(l), f(l), -pi B(l) - lambda/2 (closed-form/Fejer route) each inside its saved pair; the exterior signs comp:tail-signs (K < 0 included); the finite-core premise; the global derivative bounds L_1..L_4; the outer primitive assertion (2 g_d(x) > K off int D_0) and the negative-part assertion on the paper's 2186 x 3612 grid; the Hessian assertions on the 601 x 401 cells of D_0; the printed arithmetic of the truncation sections. Not the headline: analytic, not finite: the reduction (sec:reduction), the geometry and dual identity (sec:geometry), Lemma cal:stationarity (used by the Hessian step), the truncation estimates of sec:tails (their printed arithmetic is checked in part 7; the sweeps add their 10^-11 query error, which is conditional on the finite-core premise decided here), Corollary cor:bhs. The paper's own radial table (midpoint propagation) and its K-integral G_ss mesh are not reproduced: the table is built here by another route (closed forms on [b, R_0], Taylor integration on [R_0, l]) and K by Fejer quadrature.",
   "decider": "instruments/openaimath/finite/f180_coulomb_triangular.py",
   "decides": "a finite component -- the whole finite certificate of Section sec:computation (Proposition comp:finite-soundness), by an independent route: the 38 lattice coefficients (two ways) and |c_1| < 1; lambda, mu, K, B(l), f_0(l), f(l), -pi B(l) - lambda/2 (closed-form/Fejer route) each inside its saved pair; the exterior signs comp:tail-signs (K < 0 included); the finite-core premise; the global derivative bounds L_1..L_4; the outer primitive assertion (2 g_d(x) > K off int D_0) and the negative-part assertion on the paper's 2186 x 3612 grid; the Hessian assertions on the 601 x 401 cells of D_0; the printed arithmetic of the truncation sections. Not the headline: analytic, not finite: the reduction (sec:reduction), the geometry and dual identity (sec:geometry), Lemma cal:stationarity (used by the Hessian step), the truncation estimates of sec:tails (their printed arithmetic is checked in part 7; the sweeps add their 10^-11 query error, which is conditional on the finite-core premise decided here), Corollary cor:bhs. The paper's own radial table (midpoint propagation) and its K-integral G_ss mesh are not reproduced: the table is built here by another route (closed forms on [b, R_0], Taylor integration on [R_0, l]) and K by Fejer quadrature.",
   "value": {
    "c_1": [
     "0.634691798595",
     "0.634691798835"
    ],
    "c_1_agm": "0.634691798712461814605305045792",
    "c_2..c_5": [
     0.21127640525499658,
     0.09133432931546531,
     0.04449471528203999,
     0.023120085789109925
    ],
    "c_1_used": [
     "0.634691798712461814605305045792",
     "0.634691798712461814605305045793"
    ],
    "truncation": {
     "eps_f": "2.93e-24",
     "eps_e": "7.03e-23",
     "dS1R": "7.83e-61",
     "dS2R": "2.32e-57",
     "dS1l": "1.31e-46",
     "dS2l": "6.80e-42",
     "dBl": "4.76e-58",
     "B1max": "2.94e+01"
    },
    "quadrature_error_total": "2.51e-17",
    "lam": [
     "-6.730029288563031",
     "-6.730029288563030"
    ],
    "mu": [
     "0.413883384578748",
     "0.413883384578749"
    ],
    "K": [
     "-0.003481586664481",
     "-0.003481586664480"
    ],
    "Bl": [
     "1.046464356261010",
     "1.046464356261011"
    ],
    "f0l": [
     "-1.213596565652968",
     "-1.213596565652967"
    ],
    "fl": [
     "0.004476344685998",
     "0.004476344685999"
    ],
    "slope": [
     "0.077449910408352",
     "0.077449910408353"
    ],
    "W_from_constants": [
     "-4.150412807679",
     "-4.150412807678"
    ],
    "W_kronecker": [
     "-4.150412807678964",
     "-4.150412807678963"
    ],
    "seconds_table": 62.5,
    "table_cells": 6825,
    "f0(l)_table": [
     "-1.213596565713360",
     "-1.213596565592567"
    ],
    "premise": [
     "5424.5742",
     "1765949.50"
    ],
    "L1..L4": [
     "7.4479113",
     "2.4935595",
     "142.7578546",
     "313.5095805"
    ],
    "G_hull": [
     "-0.0416168",
     "0.5279370"
    ],
    "seconds_bounds": 8.5,
    "seconds_outer": 9.7,
    "outer": {
     "m_star": "0.0000147963",
     "min_node_upper": "0.0000148001",
     "at": [
      "680797/1270000",
      "32621/100000"
     ],
     "E0": "0.0000005502",
     "EI": "0.0000008576",
     "m_star_minus_E0_EI": "0.0000133886",
     "grid": "2186 x 3612",
     "Dh": "24319/128000000",
     "Sh": "32379/179900000",
     "slow_queries": 0
    },
    "negative": {
     "N_star": "0.0024880427",
     "N_max_lower": "0.0024880223",
     "at_d": "28209/100000",
     "er_drop": "0.0005328195",
     "lhs": "-0.0004607246"
    },
    "seconds_hessian": 49.5,
    "hessian": {
     "g_dd_min": "1.1001680",
     "g_xx_min": "0.0235243",
     "det_min": "0.0230750",
     "hulls": {
      "g_dd": [
       "1.1001680",
       "1.7339734"
      ],
      "g_dx": [
       "0.0508063",
       "0.1482686"
      ],
      "g_xx": [
       "0.0235243",
       "0.0845289"
      ],
      "det": [
       "0.0230750",
       "0.1240894"
      ]
     },
     "cells": [
      601,
      4079
     ],
     "tested": 241001
    },
    "not_decided": "analytic, not finite: the reduction (sec:reduction), the geometry and dual identity (sec:geometry), Lemma cal:stationarity (used by the Hessian step), the truncation estimates of sec:tails (their printed arithmetic is checked in part 7; the sweeps add their 10^-11 query error, which is conditional on the finite-core premise decided here), Corollary cor:bhs. The paper's own radial table (midpoint propagation) and its K-integral G_ss mesh are not reproduced: the table is built here by another route (closed forms on [b, R_0], Taylor integration on [R_0, l]) and K by Fejer quadrature.",
    "refuting": [],
    "seconds": 154.2
   },
   "checks": [
    {
     "check": "0. the definitions read off the paper: a, p, R_0, x_0 (cal:geometry), k_m = 6m, h_s = 3/40, c = 3/50, the cutoffs, l = R_0 + 19/250, b = 28209/100000, the scalar bounds (cal:scalar-bounds)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. the paper's finite procedure as summarised (06-computation.tex:653-679): denominator 2^144, modes 0..38, D = 256/48, densities 8000/24000, 45000 midpoint steps, meshes .0008/.00019/.00018/.00004/.00008",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0. the swept objects read off the paper: D = [b,.697] x [0,.65] (comp:domain), D_0 = [.52528,.54930] x [.29420,.32621] (comp:inner-domain), E_0, E_I (comp:outer-errors), the negative-part assertion, the Hessian entries and assertions, the premise |B_1|_{2,t} < 6000, sum (k_i+1)^3 |D_i|_{3,t} < 2e6 (tail:finite-core-premise), the query error 10^-11 (comp:lookup)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. the lattice coefficients c_1..c_38 by the paper's recipe (integer numerators, shells divided once, square tail added): c_1 in [0.634691798595, 0.634691798835]",
     "pass": true,
     "detail": "24192 norm shells for m=1 (D=256), 1029 for m>=2 (D=48)"
    },
    {
     "check": "1. |c_1| < 1 (06-computation.tex:132)",
     "pass": true,
     "detail": "0.634691798835"
    },
    {
     "check": "1. SECOND WAY: Weierstrass recurrence with g_2 = 0 (only n = 0 mod 3 survive, exactly), g_3 from AGM(1, cos 15deg) = 0.982889082896578...: all 38 coefficients overlap the lattice enclosures",
     "pass": true,
     "detail": "rho_1 = 1/28, rho_2 = 1/10192; c_1 = 0.634691798712461814605305045792 (AGM route)"
    },
    {
     "check": "1. the coefficients USED below: X = (6m(6m-1)c_m/rho_m)^(1/m) from the lattice enclosure of c_10 (radius 8.7e-60), then c_m = rho_m X^m/(6m(6m-1)) by the recurrence: c_1 in [0.634691798712461814605305045792, 0.634691798712461814605305045793]; every one inside the lattice enclosure and overlapping the AGM value",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. omitted modes m >= 39 bounded here: |Delta f_0| <= 2.9e-24, |Delta e| <= 7.0e-23 on |s| <= x_0, |Delta S_1(l)|, |Delta S_2(l)| <= 6.8e-42, |Delta B(l)| <= 4.8e-58; quadrature total 2.5e-17",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. lambda in [-6.730029288563031, -6.730029288563030], inside the saved pair [-6.7300312, -6.7300275] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. mu in [0.413883384578748, 0.413883384578749], inside the saved pair [0.4138830, 0.4138838] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. K in [-0.003481586664481, -0.003481586664480], inside the saved pair [-0.0034823, -0.0034809] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. B(l) in [1.046464356261010, 1.046464356261011], inside the saved pair [1.0464643, 1.0464644] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. f_0(l) in [-1.213596565652968, -1.213596565652967], inside the saved pair [-1.2135966, -1.2135965] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. f(l) in [0.004476344685998, 0.004476344685999], inside the saved pair [0.0044755, 0.0044772] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2-4. -pi B(l) - lambda/2 in [0.077449910408352, 0.077449910408353], inside the saved pair [0.0774490, 0.0774509] (certificate-record.txt; 06-computation.tex:734-740)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. comp:tail-signs: -6.731 < lambda < -6.729 (decided here, by the independent route)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. comp:tail-signs: K < 0 (decided here, by the independent route)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. comp:tail-signs: f(l) = f_0(l) - lambda l^2/2 - mu > 0 (decided here, by the independent route)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. comp:tail-signs: -pi B(l) - lambda/2 > 0 (decided here, by the independent route)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. SECOND WAY (theory-dependent): 6K + lambda + 2 pi mu in [-4.150412807679, -4.150412807678] overlaps the Kronecker-limit value -pi log(2 pi sqrt(Im tau)|eta(tau)|^2) = -4.150412807678964 (Lemma cal:energy-identity)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the domain edge b = 0.28209 < 1/(2 sqrt pi) (the geometric threshold d_*): 4 b^2 pi < 1 by an enclosure of pi",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. |c_m| <= 7/(d a^d): the bracket 6 + 2 3^{-6} + 24 (4/3)^6 2^{-11} = 6 + 50/729 < 7 at d = 12, both nonconstant terms x1/27 per step (05-tails.tex:50-58)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the cutoff S: |S'| <= 15/8 <= 2, |S''| <= 10/sqrt3 < 6 ((10/sqrt3)^2 = 100/3 < 36), |S'''| <= 60; tau: 2/.075 <= 27, 6/.075^2 <= 1067, 60/.075^3 <= 142223; 1+3(27)+3(1067)+142223 = 145506 < 10^6 (05-tails.tex:83-87, 196-203)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. .537<p<.538, .620<R_0<.621, .696<l<.697, l/a<13/20, R_0/a<289/500 (05-tails.tex:89-93); the ratios used here (5774/10000, 789/1000, 13/20) bound R_0/a, (p+x_0)/a, l/a",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the U_m derivative coefficients at l > .69, k >= 234: orders 0..3 below .000001, .000197, .070572, 25.445 < 30 (05-tails.tex:97-110; the order 0-2 expressions re-derived by Leibniz)",
     "pass": true,
     "detail": "order 3: 25.444618"
    },
    {
     "check": "7. the inner continuation: |b_m'''| <= (60/c^3)|q| + (18/c^2)|q'| + (6/c)|q''| < 8500 k^3 2^{-k} at k >= 234 (05-tails.tex:112-133)",
     "pass": true,
     "detail": "8405.54 k^3"
    },
    {
     "check": "7. on [p, p+.01]: .02p + .0001 < .04 p^2 (p > .537), and 7(8(.55)^3 2^6 + 12(.55)2^4/234^2) < 597; on [p+.01,R_0]: 7(8(343) + 12(7)(300)/234 + 2(60000)/234^2) < 20000 (05-tails.tex:137-181)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the Hermite continuation: P, Q and their first three derivatives bounded by 2e5 and 4000 on [0, h_s] (coefficient sums), (1+200000+4000) 3e6 < 7e11, P(h_s) = Q(h_s) = 0 (05-tails.tex:183-194, 04-calibration.tex:130-133)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the omitted-mode sum (tail:weighted-sum, supp:tail-bounds): the three integer comparisons, rho_i(234) < 1/10, T_0 = sum t_i(234) and 10^29 235^4 C_234 < 156 500^234, P_600 < P_600 + R_606 <= A_component <= A_common < A_coarse < 2e-21 (exact)",
     "pass": true,
     "detail": "A_component = 1.6504e-21, P_600 = 1.6503e-21"
    },
    {
     "check": "7. F(A,4A) = A(56,264,180 + 104A) (three points of a quadratic), 56,264,180 + 208e-21 < 56,264,181 < 6e7, F(A_0,4A_0) < 12e-14, 2(16(2e6)A_0 + 32A_0^2) < 13e-14 (supp:rational-error-coefficient; tail:f-estimate, tail:e-estimate)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. |Delta B(l)| <= (7/234)(289/500)^234/(1-(289/500)^6) < 6.075e-58; sum_{k>=12} 7k^2(2/7)^{k-3} < .014 (first 40 terms + geometric rest); l/(k+1)+5/2+k/b < 6(k+1) for k = 6..228",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. log and arctan remainders below 2^-144: 2(3/5)^401/(401(1-(3/5)^2)), 5^-301/301, 239^-301/301 (comp:logarithm, comp:pi)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. b = 28209/100000 < 1/(2 sqrt pi) from pi < 3.1416 (4 b^2 (3.1416) < 1); p > 1/2 > b; x_0 > .01; sqrt(b^2+.65^2) > l; t(x_0) = R_0 (p^2 + R_0^2/4 = R_0^2) (06-computation.tex:393-398, 466-467; 07-conclusion.tex:18)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "8. the record's 22 saved pairs: the first label equals the article's projection table; L_1..L_4 = 7.4636170, 2.5091315, 143.7890461, 315.1489689 are the largest endpoints of the G_d, G_s, G_dd, G_ss hulls; m_*-E_0-E_I >= .0000120, <= -.0004585, g_dd >= 1.0529222, g_xx >= .0128335, det >= .0086252 match; gmin-factor and drneg+er_drop+K are consistent with their parts; the slope pair overlaps -pi B(l) - lambda/2 of the lambda and B(l) pairs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. the radial table (an independent construction, not the paper's midpoint propagation): breakpoints b, p-.06, p, R_0, R_0+.001, R_0+.075, l (comp:breakpoints), floor(8000 L)+1 / floor(24000 L)+1 cells per piece = [1562, 1441, 1995, 25, 1777, 25], 6825 cells; node balls strictly increasing (comp:node-separation)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. the sector formula cal:f0-e with the closed-form integrals comp:initial-integrals reproduces the exact identity comp:inner-exact f_0 = (pi t^2 - 1/2) log t - 3 pi^2 t^4/8, e = 0 at t = p - c, through the second derivative",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. continuity of the table: at every one of the 6824 interior nodes, f_0, f_0', e, e' from the left cell's Taylor model (order 2 at its node + third-derivative range on the cell) overlap the right cell's node values (C^1), and f_0'', e'' too inside each piece",
     "pass": true,
     "detail": "0 disagreements"
    },
    {
     "check": "9. two routes agree: f_0(p), f_0(R_0), e(R_0) at the table nodes overlap the sector-edge complex-ball values of part 3, and f_0(l) = -2 pi S_1(l) - (log l + S_2(l))/2 from the table's Taylor-integrated S_1, S_2 on [R_0, l] (-1.213596565713360..-1.213596565592567) overlaps the Fejer-route f_0(l) of part 4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. the finite-core premise tail:finite-core-premise on every one of the 6825 cells covering [b, l] (direct interval jets of the truncated B_1 and D_i, orders 2 and 3, mesh <= 1/8000 and 1/24000 finer than the paper's .00022): max |B_1|_{2,t} <= 5424.5742 < 6000 and max sum (k_i+1)^3 |D_i|_{3,t} <= 1765949.50 < 2000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. global derivative bounds (comp:global-derivative-bounds) on 519 x 813 closed rectangles of side <= .0008 covering D = [b,.697] x [0,.65], by interval substitution in comp:G-derivatives with the radial query of comp:lookup: L_1 = 7.4479113, L_2 = 2.4935595, L_3 = 142.7578546, L_4 = 313.5095805; each at most the printed L_1 <= 7.4636170, L_2 <= 2.5091315, L_3 <= 143.7890461, L_4 <= 315.1489689, which are therefore valid bounds",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. the G hull on D, [-0.0416168, 0.5279370], overlaps the record's G pair [-.0417514, .5279383] (both enclose the range of G on D)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. comp:outer-assertion: on the 2186 d-nodes x 3612 s-steps grid (D_h = 0.000189993 <= .00019, S_h = 0.000179984 <= .00018, cut at the sides of D_0 = [0.52528,0.54930] x [0.29420,0.32621]), every node outside int D_0 has 2Q(d,x) - K >= m_* = 0.0000147963 (at d = 0.536060, x = 0.326210); E_0 = 0.0000005502, E_I = 0.0000008576; m_* - E_0 - E_I = 0.0000133886 > 0, so 2 g_d(x) > K on D minus int D_0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. comp:negative-assertion: N_* = 0.0024880427 (the largest negative-part midpoint sum, at d = 0.282090), N_* + K + .65(S_h L_2/4 + D_h L_1/2) = -0.0004607246 < 0, so int_0^inf max(0, -G(d,s)) ds < -K for every d >= b",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. printed outer/negative numbers: m_* - E_0 - E_I = 0.0000133886 >= .0000120 and the left side -0.0004607246 <= -.0004585 (06-computation.tex:741-743); the min over eligible nodes of 2Q - K, [0.0000147963, 0.0000148001], overlaps the record's gmin pair [.0000134, .0000162]; max_d N(d), [0.0024880223, 0.0024880427], overlaps the record's drneg pair [.0024879, .0024882] (same node counts 2186 x 3612)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. comp:hessian-assertions on all 241001 cells of D_0 = [0.52528,0.54930] x [0.29420,0.32621] (601 d-steps <= .00004; s-steps <= .00008 from 0, 2451479 cells accumulated for A_j): inf A_j >= 1.1001680 > 0, inf C_j >= 0.0235243 > 0, inf (A_j C_j - B_j^2) >= 0.0230750 > 0 on every cell; (p, x_0) in the interior of D_0, so with the exact stationarity (Lemma cal:stationarity, analytic) 2 g_d(x) >= K on D_0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "9. printed Hessian numbers (06-computation.tex:744-746): g_dd >= 1.0529222, g_xx >= .0128335, det >= .0086252 hold (the enclosures here are tighter), and each of the four hulls overlaps the record's hess-hull pair",
     "pass": true,
     "detail": ""
    },
    {
     "check": "10. Theorem comp:scalar, its finite hypotheses: K < 0, the premise, the outer and negative-part assertions, the Hessian assertions and the exterior signs (with sqrt(b^2+.65^2) > l and l < .697 from part 7) are ALL decided here -- by an independent radial table and the paper's meshes; what remains is analytic (stationarity, the truncation estimate 10^-11, the reduction and the geometry)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/01-introduction.tex": "6326c2926659a7d3bdf3946e843a0ba9d921259c9432a4132ba8da9b3d2f0496",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/04-calibration.tex": "3a432cdeffb306087a7e24dbf0135b7fff36770e9a5c89126fc89a77913b88ce",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/05-tails.tex": "d9793ed1ece39363a2407a83aac00e908f19d2dfa087ff162596a164850a63ce",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/06-computation.tex": "fa15ff08f64cd539300d3bef2ea4a5c753cf971e37975f8f056755d4ec76e381",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/07-conclusion.tex": "8ec26728f5f2cb1fa5497c874dbeb051de182c1a57c8199fd8f565f185f42070",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/build/sections/08-scientific-support.tex": "adf0f7905596f53d7fc9b261c4be556a7212d05dfbdb21346d5b22dc4ebc19e0",
    "preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/verification/data/certificate-record.txt": "90e4f4055ce7f589303ad246dac23845db75f397aecbad71533230e34a86ee01"
   },
   "forges": [
    {
     "forge": "saved lambda pair with upper end -6.7300295 (excludes the true lambda)",
     "verdict": "REFUTED"
    },
    {
     "forge": "saved mu pair shifted up by 4e-7",
     "verdict": "REFUTED"
    },
    {
     "forge": "K printed with the opposite sign",
     "verdict": "REFUTED"
    },
    {
     "forge": "domain edge b = 0.2821 (above 1/(2 sqrt pi))",
     "verdict": "REFUTED"
    },
    {
     "forge": "mu raised by 2/1000 in G (K, lambda kept): the outer and negative-part sweeps re-run on the forged G",
     "verdict": "REFUSED"
    },
    {
     "forge": "the excluded rectangle D_0 shrunk to [.5368,.5378] x [.3098,.3107] around (p, x_0): the outer sweep re-run on the re-cut grid",
     "verdict": "REFUSED"
    },
    {
     "forge": "convexity asserted on [.52528,.54930] x [0,.32621] (D_0 extended down to s = 0, where G_s = 0 by evenness; (p, x_0) still inside): the Hessian sweep re-run on all 601 x 4079 cells",
     "verdict": "REFUSED"
    },
    {
     "forge": "the finite-core premise with 2e6 replaced by 1.7e6",
     "verdict": "REFUSED"
    }
   ],
   "seconds": 154.2
  },
  {
   "id": "F-189",
   "index": 189,
   "family": "096",
   "title": "The Gaussian propeller bound in every dimension",
   "tier": "A",
   "dir": "The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026",
   "leanChallenges": [
    "GaussianPropeller"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every finite numerical comparison and enclosure of the m >= 5 exclusion (Appendix cert:arithmetic, Tables sc:interval-table, cert:quantiles, elim:small-table, elim:four-table, elim:final-table and the inline bounds), plus the finite algebra of geometry.tex; NOT the case m <= 4 (Heilman-Jagannath-Naor, external, computer-assisted) and not the prose analysis",
   "decider": "instruments/openaimath/finite/f189_gaussian_propeller.py",
   "decides": "a finite component: every finite numerical comparison and enclosure of the m >= 5 exclusion (Appendix cert:arithmetic, Tables sc:interval-table, cert:quantiles, elim:small-table, elim:four-table, elim:final-table and the inline bounds), plus the finite algebra of geometry.tex; NOT the case m <= 4 (Heilman-Jagannath-Naor, external, computer-assisted) and not the prose analysis",
   "value": {
    "pi_machin_7_6": [
     "11950827742763724038336441397624078583300672396/3804066618602651140286948940075677032470703125",
     "1820381950672004239437026206052012466686191144368/579445571523205428758215494416167327391357421875"
    ],
    "paper_recipe_does_not_reproduce": [],
    "n_enclosure_claims": 47
   },
   "checks": [
    {
     "check": "the theorem as printed (main.tex:81-92): sum ||z(A_i)||^2 <= 9/(8 pi)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every inline printed bound read from the LaTeX (20 strings)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "tables parsed: cert:quantiles 9 rows, sc:interval-table 10, elim:small-table 4, elim:four-table 2, elim:final-table 2",
     "pass": true,
     "detail": "[9, 10, 4, 2, 2]"
    },
    {
     "check": "A. Machin through indices 7 and 6: 3.14159 < pi < 3.14160",
     "pass": true,
     "detail": "width 3.5e-11"
    },
    {
     "check": "A. the tight pi enclosure lies inside the paper's",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pi in (3.14159, 3.14160)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. -0.000131 < D_f(0.6119) < -0.000126",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. 0.000120 < D_f(0.6121) < 0.000124",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. -0.000430 < D_f(x) < 0.000424 on [0.6119, 0.6121] (endpoint enclosures)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. f(0.612) > 0.884383832",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. -0.0047010 < F(0.29) < -0.0046981",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. 0.0007683 < F(0.294) < 0.0007712",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. R(0.29) > 0.9312526",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. p(0.294) - 0.294/pi > 0.22198",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.05 q_a=-2.033: 10^6(P_0(a) - Phi(q_a)) > 98",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.05 q_a=-2.033: G(q_a) < 0.887728",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.14 q_a=-1.548: 10^6(P_0(a) - Phi(q_a)) > 228",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.14 q_a=-1.548: G(q_a) < 0.854397",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.24 q_a=-1.238: 10^6(P_0(a) - Phi(q_a)) > 381",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.24 q_a=-1.238: G(q_a) < 0.826606",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.33 q_a=-1.024: 10^6(P_0(a) - Phi(q_a)) > 366",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.33 q_a=-1.024: G(q_a) < 0.803691",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.41 q_a=-0.860: 10^6(P_0(a) - Phi(q_a)) > 470",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.41 q_a=-0.860: G(q_a) < 0.783737",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.48 q_a=-0.728: 10^6(P_0(a) - Phi(q_a)) > 452",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.48 q_a=-0.728: G(q_a) < 0.766006",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.54 q_a=-0.621: 10^6(P_0(a) - Phi(q_a)) > 539",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.54 q_a=-0.621: G(q_a) < 0.750456",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.59 q_a=-0.535: 10^6(P_0(a) - Phi(q_a)) > 739",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.59 q_a=-0.535: G(q_a) < 0.737150",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.64 q_a=-0.450: 10^6(P_0(a) - Phi(q_a)) > 684",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. a=0.64 q_a=-0.450: G(q_a) < 0.723262",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0, 0.05], U = 1: 10^6 Delta > 542",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.05, 0.14], U = 0.91: 10^6 Delta > 6173",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.14, 0.24], U = 0.87: 10^6 Delta > 2976",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.24, 0.33], U = 0.836: 10^6 Delta > 1328",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.33, 0.41], U = 0.809: 10^6 Delta > 523",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.41, 0.48], U = 0.788: 10^6 Delta > 392",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.48, 0.54], U = 0.771: 10^6 Delta > 1076",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.54, 0.59], U = 0.756: 10^6 Delta > 2787",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.59, 0.64], U = 0.743: 10^6 Delta > 4115",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. interval [0.64, 2/3], U = 0.73: 10^6 Delta > 8886",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. (.44^2 - q0^2)(.44^2 - 2q0^2) - 8(3/pi)^2 q0^2 > .00321132",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. [.23, .30]: (a^2 - q0^2) M(b) - 8(3/pi)^2 q0^2 > .00330237",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. [.30, .35]: (a^2 - q0^2) M(b) - 8(3/pi)^2 q0^2 > .02063552",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. [.35, .41]: (a^2 - q0^2) M(b) - 8(3/pi)^2 q0^2 > .01621418",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. [.41, .44]: (a^2 - q0^2) M(b) - 8(3/pi)^2 q0^2 > .00803616",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: e(1/sqrt5) > .02979364",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: e(1/sqrt5) > .03879364",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: F(.066) - (1-A) > .01522916",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: F(.066) - (1-A) > .00253590",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 1.5(2.6-.12)^2 - 10(3/pi)^2 > .10667806",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. |D_f| < 0.001 on [0.6119, 0.6121] (from the printed interval)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. 0.884383832 (1 - 10^-7) > 0.8843837 > 0.884",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. |F| < 0.004702 on [0.29, 0.294] (F increasing there; endpoint enclosures)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. (9/4) 0.004702 / (1 - 0.294^2)^4 < 0.015189 < 0.03",
     "pass": true,
     "detail": "0.015188302"
    },
    {
     "check": "C. 0.9312526 - 0.03(0.004) = 0.9311326 > 0.929",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. R' = (9/4) F/(1-x^2)^4, F' = 6H, H' = (4x^2-3)/(pi sqrt(1-x^2)) (exact differential algebra)",
     "pass": true,
     "detail": "(True, True, True)"
    },
    {
     "check": "D. each G(q_a) bound is strictly below the U of its interval row (row a = 0 uses G < 1, U = 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the interval rows tile [0, 2/3] in order",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. P_0(2/3) = 103/300 < 1/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: d = 779/1000, s = .415/d, and the common charge normalises to (.415/sqrt c - .734) <= .116",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: e(u)(s+u) = d u (s^2 - u^2) (degree 3, checked at 5 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: d = 367/500, s = .415/d, and the common charge normalises to (.415/sqrt c - .734) <= .116",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: e(u)(s+u) = d u (s^2 - u^2) (degree 3, checked at 5 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. L <= 2 (2/3)^2 = 8/9, and (1/9)(.415/.23 - .734) - .116 = 101/34500 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. both factors positive: .44^2 - q0^2 > 0 and .44^2 - 2 q0^2 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. elim:small-table: .415/b - .734 > 0, M(b) > listed > 0 (M(b) exact), a^2 - q0^2 > 0",
     "pass": true,
     "detail": "M(.30)=0.72264324, M(.35)=0.61198851, M(.41)=0.40621448, M(.44)=0.24314650"
    },
    {
     "check": "E. elim:small-table intervals tile [.23, .44]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. e(.066) = .023996676 (A=.929) and .024192696 (A=.884), exactly",
     "pass": true,
     "detail": "(Fraction(5999169, 250000000), Fraction(3024087, 125000000))"
    },
    {
     "check": "E. A=.929, m=5: K(4s^2 - 1 + (m-4)(.066)^2) + (m-4)e(.066) - (1-A) > .00092980 (exact: 0.0009298024)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: e(.066) < listed e(1/sqrt5) bound (so the min is e(.066))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884, m=6: K(4s^2 - 1 + (m-4)(.066)^2) + (m-4)e(.066) - (1-A) > .02229774 (exact: 0.0222977493)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: e(.066) < listed e(1/sqrt5) bound (so the min is e(.066))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: k e(.42) - (1-A) = .0027688 exactly, > 0",
     "pass": true,
     "detail": "3461/1250000"
    },
    {
     "check": "E. A=.884: k e(.42) - (1-A) = .0184672 exactly, > 0",
     "pass": true,
     "detail": "5771/312500"
    },
    {
     "check": "E. A=.929: .1070554 < t0 < .1070555 (by squaring, no root)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: .066 < t0 < .109",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: .1176565 < t0 < .1176566 (by squaring, no root)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: .066 < t0 < .12",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.929: e(.23) + k e(.066) - (1-A) = .031234252 exactly",
     "pass": true,
     "detail": "7808563/250000000"
    },
    {
     "check": "E. A=.929: s^2 - 2.6 t0 > .00546157 (by squaring)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. A=.884: e(.23) + k e(.066) - (1-A) = .013199488 exactly",
     "pass": true,
     "detail": "103121/7812500"
    },
    {
     "check": "E. A=.884: s^2 - 2.6 t0 > .01376450 (by squaring)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the minima feed the lemma: 0.8843837 > .884 = A (m >= 6), 0.9311326 > .929 = A (m = 5)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. 8 - (1+x)^2((2-x)^2+1) = (1-x)^2(3-x^2) (geometry.tex:187)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. sum_{j<k}(1+a_j)^2(1+a_k)^2 = 3+4s1+2s1^2+2s1s2+s2^2-(6+2s1)s3 (geometry.tex:219-220)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. 3+4+2+2/3+1/9 = 88/9 < 10",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. the pair-bound constant: (27/6)(1/(2pi))(2/sqrt(2pi)) / (3/(2 sqrt(2pi))) = 3/pi",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. the attaining propeller: 3 sin^2(pi/3)/(2pi) = 9/(8pi) (sin^2(pi/3) = 3/4)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/build/main.tex": "8d4ee009275cbd6866413ee2348f878077d9ec834320ac92eaf52aecb70127bc",
    "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/build/sections/scalars.tex": "49dbe014d994fd65e99c25610ac6c8328b335748aca94f17f433b3149fe5a075",
    "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/build/sections/geometry.tex": "d33c3b97205de25544c8b7b71cc11ce5642b0ccaf0495ba0b5efc9705083e745",
    "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/build/sections/elimination.tex": "88f7bf42e5af37cab8aaffa1c7b0bd386aeb1c6647afb94f267158868ce9d98c",
    "preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/build/sections/certificates.tex": "e4ecb33cc61e6d6d16f2962496e6ff8fd4c6571026aaf7db073c11774e2728d9"
   },
   "forges": [
    {
     "forge": "t0 enclosure for A=.929 moved up one unit: .1070555 < t0 < .1070556",
     "verdict": "REFUTED"
    },
    {
     "forge": "elim:small-table row [.35,.41]: M(b) bound .40621 -> .40622",
     "verdict": "REFUTED"
    },
    {
     "forge": "cert:quantiles row a=0.41: q_a moved -0.860 -> -0.850",
     "verdict": "REFUTED"
    },
    {
     "forge": "sc:interval-table row [0.41,0.48]: U lowered .788 -> .783 (below the G(q_a) bound)",
     "verdict": "REFUTED"
    },
    {
     "forge": "cap budget for A=.929 printed .0027689 instead of .0027688",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.1
  },
  {
   "id": "F-195",
   "index": 195,
   "family": "100",
   "title": "Finite angular cylinder covers below the half-area bound",
   "tier": "B",
   "dir": "Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026",
   "leanChallenges": [
    "CylinderCovering",
    "RuledCovering",
    "TriangularCovering"
   ],
   "word": "REFUSED",
   "why": "a finite component: A_min(K) = sqrt 2, the construction's parameters, every polynomial identity of the coverage proof, and the exact total base area at eps = 1/2000 and tau = 1/4000 (strictly below A_min/2, inside the printed remainder bounds); COVERAGE -- the decisive geometric part -- is REFUSED (a continuum covered by 1.6e7 cylinders; ~3.2e13 sector pairs), so the row is REFUSED",
   "decider": "instruments/openaimath/finite/f195_cylinder_covers.py",
   "decides": "a finite component: A_min(K) = sqrt 2, the construction's parameters, every polynomial identity of the coverage proof, and the exact total base area at eps = 1/2000 and tau = 1/4000 (strictly below A_min/2, inside the printed remainder bounds); COVERAGE -- the decisive geometric part -- is REFUSED (a continuum covered by 1.6e7 cylinders; ~3.2e13 sector pairs), so the row is REFUSED",
   "value": {
    "n": 8000000,
    "cylinders": 16000000,
    "S/H enclosure (eps=1/2000)": [
     "295147904859614271203163352691972088291862394476145975537886689599604670591896339273825288877405394410382579308099889003276890964167919748788110589838231839902603737752611/590295810358705651712000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
     "180143984899666913576205234342230142826612804368229700211870469104459726109975710618008248216312386605825816029275476352800219991816765351/360287970189639680000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
    ],
    "C/H enclosure (tau=1/4000)": [
     "365375409285150926194263891942582340247664384381808017138836703418526904824665069630110616090214119291007634243887944406540223349807177536402347171152130232338850887214014434195724384854069437752611/730750818665451459101842416358141509827966271488000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
     "850705917191577437786072627651162103066841579519840304426348846971446303074314067844913827395561947781263712007216638458759381062683070405893120054271136765351/1701411834604692317316873037158841057280000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
    ],
    "runtime_s": 0.8
   },
   "checks": [
    {
     "check": "the theorem, the construction and the cubic proposition as printed",
     "pass": true,
     "detail": "introduction.tex:32-43, construction.tex:52-128, cubic-buffer.tex:17-45"
    },
    {
     "check": "M. the face area vectors from the vertices are the printed (0,+-H,-1), (+-H,0,1), each of length sqrt 3, summing to zero",
     "pass": true,
     "detail": ""
    },
    {
     "check": "M. A_min(K) = sqrt 2: at every zonotope facet-normal candidate A(n)^2 >= 2|n|^2, with equality attained, and A(1,0,0) = H",
     "pass": true,
     "detail": ""
    },
    {
     "check": "M. the closed form A(u) = max(H|u_x|,|u_z|) + max(H|u_y|,|u_z|) agrees at those normals",
     "pass": true,
     "detail": ""
    },
    {
     "check": "P. eps = 1/2000: n = ceil(2/eps^2) = 8,000,000, 2n = 16,000,000 cylinders, Delta = 2/n <= eps^2",
     "pass": true,
     "detail": "Delta = 1/4000000"
    },
    {
     "check": "P. tau = 1/4000: N = ceil(8/tau^2) = 128,000,000, 2N = 256,000,000 cylinders, Delta <= tau^2/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. eq:shared-boundaries: alpha_j - q beta_j = (1-q^2)/4 at q = q_j and q = q_{j+1}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. the line through (0, q t0, H t0) along (1, eps alpha, H eps beta) lies in y = q t + eps x (alpha - q beta)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. eq:radial-budget, modulo q t0 = y0 and p s0 = x0 (cofactors -eps x, +eps y)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. eq:radial-budget remark: a + b = -eps x beta_j + eps y beta_k",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. d(p) + d(q) - (p^2+q^2+2p^2q^2)/16 = (p^2-q^2)^2/16",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. int_{-1}^1 d = 1/15 and int_{-1}^1 A = 17/30",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. 2 eta - 1/240 = -13/6000 (eta = 1/1000)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. 59/64 = 1/8 + 3/8 + 27/64 (the f'' bound)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. eq:uniform-coverage-gap at eps = eta/2 equals eta - 17 eta^2/16 > 0, decreasing in eps",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. 2 eps^2 <= 1/2,000,000 < 13/6000 at eps = 1/2000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. cubic budget: 2eta - (2+2eta)tau - tau^2/4 = tau(21/2 - 51 tau/4) at eta = 25tau/4; >= 33tau/8 for tau <= 1/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. T_j >= 1/2 + (25/4)(1/2)^3 = 41/32 > 1 for tau >= 1/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. 25 tau/2 + 2 tau^2 <= 1/320 + 1/8,000,000 < 1/240 at tau = 1/4000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. tau = 1/50 gives eta = 25 tau/4 = 1/8 (the variable-margin hypothesis eta <= 1/8)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. Remark rem:angular-cubic-scale: 4(1/2 - tau^2/240 + 25tau^3/2) at tau = 2eps = 2 - eps^2/15 + 400 eps^3; 4 * 2 (2eps)^4 = 128 eps^4; 1/50, 1/4000, 1 <-> 1/100, 1/8000, 1/2; D_I = 8 d; delta_I = eps^2 D_I + 100 eps^3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. g_lo <= (1+x)^(-1/2) <= g_hi on [0, eps^2] (g_hi^2(1+x) - 1 = 5x^3/8 - 15x^4/64 + 9x^5/64; 1 - g_lo^2(1+x) has lowest term 35/64 x^4)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. the closed-form sector sums equal the direct sector-by-sector sums at eps = 1/20 (n = 800)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. eps = 1/2000: total base area / H < 1/2 (strictly below A_min/2), exactly enclosed",
     "pass": true,
     "detail": "S/H in [0.499999999458341837, 0.499999999458341837]; 1/2 - S/H >= 5.416582e-10"
    },
    {
     "check": "A. eps = 1/2000: |S/H - 1/2 + (13/6000) eps^2| <= 2 eps^4 (the printed remainder bound)",
     "pass": true,
     "detail": "remainder / eps^4 in [0.135777, 0.135777]"
    },
    {
     "check": "A. eps = 1/2000: the Taylor form (eq:taylor-bound) |S/H - 1/2 - eps^2 sum Delta(M_j - A_j/8)| <= (59/64) eps^4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. tau = 1/4000: C/H < 1/2 (strict saving of Proposition prop:angular-cubic), exactly enclosed",
     "pass": true,
     "detail": "C/H in [0.499999999934896022, 0.499999999934896022]"
    },
    {
     "check": "A. tau = 1/4000: |C/H - 1/2 + tau^2/240 - (25/2) tau^3| <= 2 tau^4 (eq:angular-cubic-cost)",
     "pass": true,
     "detail": "remainder / tau^4 in [0.041946, 0.041946]"
    },
    {
     "check": "A. projected base area^2 = (intercept area H Delta T_j^2/2)^2 / |v_j|^2, |v_j|^2 = 1 + eps^2(alpha^2 + 2beta^2), both families, six sectors, exact in Q(sqrt 2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "S. sample: 300 exact points just past their first-family cutoff near t = 1/2 are covered by the second family (evidence, not a proof of coverage)",
     "pass": true,
     "detail": "uncovered: []"
    },
    {
     "check": "COVERAGE of the closed tetrahedron by the 1.6e7 (2.56e8) cylinders: NOT DECIDED (REFUSED: ~3.2e13 sector pairs at eps = 1/2000; see the docstring)",
     "pass": false,
     "detail": "the decisive part of the headline"
    }
   ],
   "sources": {
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/introduction.tex": "19c5b71d779403be3fd80bca70cfa2414d4a10286a0743e51fcb894fa309841d",
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/construction.tex": "028c3da94d3bda61e918277330ceac4df46b6224b9e90c5cae33c87bf76c0a5f",
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/coverage.tex": "e8be6425d41b8649156871d655f62a4d637164bac8a42328e7496a5972dcbc76",
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/area.tex": "6880eb73332cbebd6660819af6053c7231a2ae82593fa563dea57a6c580637c7",
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/cubic-buffer.tex": "e566122d6f65a338a54ce7dc7892fe247629ab531fb99d80431700d4e6a68719",
    "preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/build/sections/scaling-specializations.tex": "842f55e97e4fe9114e5627521387959eed62f08747b6ad3045f33730a1c5d8ae"
   },
   "forges": [
    {
     "forge": "printed coefficient 13/6000 changed to 14/6000",
     "verdict": "REFUTED"
    },
    {
     "forge": "margin eta = 1/100 instead of 1/1000 (2 eta - 1/240 > 0: no saving)",
     "verdict": "REFUTED"
    },
    {
     "forge": "no radial enlargement (T_j = 1/2): the coverage sample finds uncovered points",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.8
  },
  {
   "id": "F-197",
   "index": 197,
   "family": "100",
   "title": "Slope-field perturbations of the two-cylinder covering",
   "tier": "C",
   "dir": "Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026",
   "leanChallenges": [
    "CylinderCovering",
    "RuledCovering",
    "TriangularCovering"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED as a specific instance; construction in preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/build/sections/threshold.tex and dictionary.tex with non-explicit smallness thresholds.",
   "decider": null
  },
  {
   "id": "F-198",
   "index": 198,
   "family": "100",
   "title": "Finite triangular approximation of radial sweeps",
   "tier": "C",
   "dir": "Finite-triangular-approximation-of-radial-sweeps-September-27-2026",
   "leanChallenges": [
    "CylinderCovering",
    "RuledCovering",
    "TriangularCovering"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED as a specific instance; preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/build/beams.tex l.292 ('For every zeta > 0, the tetrahedron K has a finite cylinder cover ...'), coordinates.tex.",
   "decider": null
  },
  {
   "id": "F-199",
   "index": 199,
   "family": "101",
   "title": "A sharp entropy bound and the simplex inequality for isotropic constants",
   "tier": "A",
   "dir": "A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: the scalar certificate behind Lemmas sc:bounds and tr:scalar-basic — the central piecewise-polynomial certificate on [-6,10] rebuilt from its eleven seed pairs (residuals, joins, anchors, error budgets, all 44 group bounds), the rational inequalities of both tail arguments at their extreme points, and the finite comparisons of Section sc:scalar; NOT the isotropic-constant headline",
   "decider": "instruments/openaimath/finite/f199_entropy_simplex.py",
   "decides": "a finite component: the scalar certificate behind Lemmas sc:bounds and tr:scalar-basic — the central piecewise-polynomial certificate on [-6,10] rebuilt from its eleven seed pairs (residuals, joins, anchors, error budgets, all 44 group bounds), the rational inequalities of both tail arguments at their extreme points, and the finite comparisons of Section sc:scalar; NOT the isotropic-constant headline",
   "value": {
    "group_min_1e-3": {
     "-21/4": [
      "7.9429",
      "1.737",
      "1.6764",
      "1.9455"
     ],
     "-15/4": [
      "1.3544",
      "1.423",
      "3.3706",
      "3.3743"
     ],
     "-5/2": [
      "2.2665",
      "2.9488",
      "2.7234",
      "6.7547"
     ],
     "-3/2": [
      "2.2714",
      "3.2268",
      "3.1674",
      "9.8114"
     ],
     "-1/2": [
      "2.7459",
      "3.5966",
      "6.1815",
      "15.6621"
     ],
     "1/4": [
      "2.6471",
      "3.0578",
      "9.4579",
      "12.5861"
     ],
     "5/4": [
      "2.312",
      "1.0655",
      "8.9793",
      "21.9907"
     ],
     "3": [
      "3.1296",
      "1.0625",
      "9.3654",
      "8.0368"
     ],
     "5": [
      "1.385",
      "1.1154",
      "11.4475",
      "17.2885"
     ],
     "7": [
      "3.8941",
      "0.8248",
      "13.0253",
      "12.9544"
     ],
     "9": [
      "2.4181",
      "0.5083",
      "13.9701",
      "10.3046"
     ]
    },
    "residual_P_max": "1.8303e-09",
    "budget_q": "4.2090e-09",
    "budget_k": "3.6751e-07",
    "theta0": "[3.221523626198719, 3.221523626198719]",
    "polynomials_bounded": 196,
    "runtime_s": 1.9
   },
   "checks": [
    {
     "check": "tables parsed: 11 seed rows, 11 group-bound rows, 6 bins, 2 derivative rows, 6 substitution rows, m_0",
     "pass": true,
     "detail": "certificate.tex:378-388, 561-571; scalar.tex:46-51, 76-77, 329-334; transport.tex:93-98"
    },
    {
     "check": "constants alpha = 3/40, s = 1/(1-alpha), beta = 7/25, c = 3/2, gamma = c/2, epsilon = 3/20, kappa = 49/100 as printed",
     "pass": true,
     "detail": "matrix.tex:178, 233, 419"
    },
    {
     "check": "the eleven rows a = w + r x are consecutive and cover [-6,10]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every row lies inside one bin of Table tr:bins",
     "pass": true,
     "detail": "-21/4:O, -15/4:O, -5/2:P_1, -3/2:R_1, -1/2:U_1, 1/4:U_1, 5/4:V_1, 3:V_1, 5:T_1, 7:T_1, 9:T_1"
    },
    {
     "check": "pi enclosed two ways (Gauss three-arctangent and Machin, alternating bounds) and the enclosures intersect",
     "pass": true,
     "detail": "width 7.1e-75"
    },
    {
     "check": "sqrt(2 pi) > 12/5 (2 pi > 144/25) and (5/12) / sum_{i<60} 18^i/i! < 7e-9, so phi(6) < 7e-9",
     "pass": true,
     "detail": "bound 6.3458e-09"
    },
    {
     "check": "|q(10) - 10.098093234| < 4e-10 from 1/S_12(10) <= q(10) <= 1/S_11(10)",
     "pass": true,
     "detail": "[10.098093233962, 10.098093233965]"
    },
    {
     "check": "|k(-6) - .16237766| < 4e-8 from S_13(6) <= I_0(6) <= S_14(6) and |k(-6) - I_0(6)| <= 2 phi(6)/36",
     "pass": true,
     "detail": "[0.162377657256, 0.162377663827]"
    },
    {
     "check": "11 residual bounds r[(PD)_{deg>=14}] < 2e-9 (w = -3.75), 7e-11 (otherwise)",
     "pass": true,
     "detail": "max 1.830e-09"
    },
    {
     "check": "11 residual bounds r[(kD)_{deg>=14}] < 2e-10",
     "pass": true,
     "detail": "max 3.150e-11"
    },
    {
     "check": "20 joining bounds: P and k^ at each common endpoint differ by < 2e-10",
     "pass": true,
     "detail": "max 1.150e-10 (P), 1.131e-12 (k)"
    },
    {
     "check": "D > .09 on every row by the quartic Bernstein rule (discarded coefficients charged)",
     "pass": true,
     "detail": "min 0.09809"
    },
    {
     "check": "[k^] < 2 on rows w <= .25 and < 5.6 on rows w > .25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "first-row k^(-1) within 3e-9 of .16237766; last-row P(1) within 2e-10 of 10.098093234",
     "pass": true,
     "detail": "0.162377660817, 10.098093233958"
    },
    {
     "check": "e_q budget as displayed: 6e-10 + 2(2e-9 + 10*7e-11) + 10*2e-10 = 8e-9 < 1e-8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "e_k budget as displayed: 4.3e-8 + 22e-10*2 + 20e-10 + 1e-8(6.5*2 + 9.5*5.6) = 7.114e-7 < 8e-7",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the same budgets re-summed from our recomputed anchors, residuals, joins and row norms: < e_q and < e_k; lengths 6.5 and 9.5",
     "pass": true,
     "detail": "q: 4.209e-09, k: 3.675e-07"
    },
    {
     "check": "all 44 entries of Table cert:group-bounds: recomputed group minimum >= the printed lower bound, and > 0",
     "pass": true,
     "detail": "-5.25:7.94/1.74/1.68/1.95 -3.75:1.35/1.42/3.37/3.37 -2.5:2.27/2.95/2.72/6.75 -1.5:2.27/3.23/3.17/9.81 -.5:2.75/3.60/6.18/15.66 .25:2.65/3.06/9.46/12.59 1.25:2.31/1.07/8.98/21.99 3:3.13/1.06/9.37/8.04 5:1.39/1.12/11.45/17.29 7:3.89/0.82/13.03/12.95 9:2.42/0.51/13.97/10.30 (units 1e-3)"
    },
    {
     "check": "group membership as listed: G1 6, G2 6, G3 4 (3 in bin O), G4 2 per row",
     "pass": true,
     "detail": "196 polynomials bounded"
    },
    {
     "check": "q(-1) > .287, q(-2) < .056, q(-3) < .005, q(4) > 4 from P -+ e_q at the endpoint (both adjacent rows)",
     "pass": true,
     "detail": "q(-1)~0.287600, q(-2)~0.055248, q(-3)~0.004438, q(4)~4.225607"
    },
    {
     "check": "log(q(-1)/q(-2)) > 1.62, log(q(-1)/q(-3)) > 4, log(q(4)/q(-2)) > 4, log(q(4)/q(-3)) > 6.5 (exp upper bounds)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3.221523626198 < theta_0 = log(200 pi)/2 < 3.221523626199 (exp bounds against pi bounds; atanh series agrees)",
     "pass": true,
     "detail": "[3.221523626198719, 3.221523626198719]"
    },
    {
     "check": "q <= 1.01 phi(r): 1/(1 - phi(6)/6) <= 1.01",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the error constants 4, 6, 9 re-derived from identities (cert:identities) at r >= 6",
     "pass": true,
     "detail": "b: 2P + kq; l = k - q - rb; l' = 2b - q(q+r) - rl"
    },
    {
     "check": "(cert:left-errors) at r = 6: |g_j - I_j| <= .01 u j!/r^(j+1), j = 0..3; r^N phi(r) decreasing for N <= 7 as 36 > 7",
     "pass": true,
     "detail": ""
    },
    {
     "check": "u zeta < .028 at u = 1/36; l' >= 6r^-4(1 - 10.01u) > 0",
     "pass": true,
     "detail": "u zeta = 0.027785"
    },
    {
     "check": "p >= u(1-4.54u) >= 1/(7+r^2): 3.01 + 4 zeta^2 u/alpha < 4.54 and (1-4.54u)(1+7u) >= 1 at u = 1/36 (the latter = u(2.46 - 31.78u))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "h_0 summands at most .501/r and .43/r; .501^2 < .67^2 (1 - 4.54u)",
     "pass": true,
     "detail": "0.50014, 0.41493"
    },
    {
     "check": ".93 < b/p < 1.05 from (cert:left-ratios) at u = 1/36",
     "pass": true,
     "detail": "b/p in [0.94281, 1.04705]"
    },
    {
     "check": "D_* < .295; L <= 2 zeta/(alpha r^3) < .60; .37 < K_0 < .56 (corners of b, b/p, l^2/(alpha b)); J_0 < .72",
     "pass": true,
     "detail": "K_0 in [0.4777, 0.5257], J_0 <= 0.4632"
    },
    {
     "check": "z <= 3 zeta/(r(1-6.01u)) < .61; h_* < 1 + 3(.028) + zeta(.61)/6 < 1.77; chi <= 2 zeta u^2 < .070",
     "pass": true,
     "detail": "z <= 0.60037, h_* < 1.18569"
    },
    {
     "check": "E < .05072 (E <= b(1.75 + kl/b), b < .028)",
     "pass": true,
     "detail": "0.050698"
    },
    {
     "check": "N_11 < .224, N_22 < .802, .34 < N_12 < .386 (cert:N with b < .028, K_0 < .56, J_0 < .72)",
     "pass": true,
     "detail": "N11 0.22177 N22 0.79912 N12 [0.36767, 0.38444]"
    },
    {
     "check": "left-tail determinant (.43-.224-.13)(2.72-.802-.75) - (.13-.386)^2 = .023232 > 0, leading minor > 0",
     "pass": true,
     "detail": "363/15625"
    },
    {
     "check": "the left tail lands inside row O and row a <= -1: b < .028 <= .09, L < .60, K_0 in [.37,.56], J_0 <= .72, l_1 = .13, z in (0,.61), h_* in (1,1.77), chi < .070, m_0 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|F^(j)| <= (2j)!/2^j gives 1, 6, 90, 2520",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|G''| <= 15 + 6 + 1/.99 < 23 and |G'''| <= 315 + 90 + 18/.99 + 1/.99^2 < 435 (15 = 90/(2*3), 315 = 2520/(2*4))",
     "pass": true,
     "detail": "22.0101, 424.2021"
    },
    {
     "check": "Taylor data F(0) = 1, F'(0) = -1, G(0) = -1/2, G'(0) = 5/2 (formal power series of F and G = (F-1)/(2u) - F log F)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "linear-in-u parts: (1-D)C, (2-D)B, (3-D)J give theta-1.5+(8.5-3theta)u, 2theta-4+(37-12theta)u, 6theta-14+(197-60theta)u",
     "pass": true,
     "detail": ""
    },
    {
     "check": "remainder constants (5g+3), (30g+33), (210g+321+8c_0u) re-derived by expanding (k-D)...(1-D), D = d_theta - 2U",
     "pass": true,
     "detail": "[(5, 3, 0), (30, 33, 0), (210, 321, 8)]"
    },
    {
     "check": "uniformity table: B <= .024, J <= .102, R <= .624, hence < .04, .13, .75 (cert:right-errors); g = (23 + 6 theta)/2, c_0 = 435 + 90 theta",
     "pass": true,
     "detail": "B 0.023151 J 0.101628 R 0.623717"
    },
    {
     "check": "J > 2theta - 4.13 > 0; R - 1.7J > 2.6theta - 8.171 > 0; 3J - R > .86 (theta >= 3.221)",
     "pass": true,
     "detail": ""
    },
    {
     "check": ".482 < b; l < .0026; k/r < .528 (at r = 10, theta < 3.222); B > 0, J > 0 give b < .5, l > 0",
     "pass": true,
     "detail": "b > 0.48238, l < 0.002574, k/r < 0.52722"
    },
    {
     "check": "C >= .99theta - .5 > 0 and C - (theta - .5) <= u{2.5 - theta + .01(11.5 + 3theta)} < 0 for theta >= 3.221",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0 < uCJ < .075 and the majorant decreases: 1/(theta-.475) + 2/(2theta-3.87) < 2 at theta = 3.221",
     "pass": true,
     "detail": "0.07071"
    },
    {
     "check": "-.605 < -B + .5J < -.395 (theta cancels) and .493 < p < .5",
     "pass": true,
     "detail": "p in (0.49386, 0.49680)"
    },
    {
     "check": "B/r < .18; |h_0| <= .528(.18) + 2(.22)(.0026)/alpha < .14 < .67 sqrt(.493)",
     "pass": true,
     "detail": "0.11029"
    },
    {
     "check": ".367 < K_0 < .4; D < D_*/.493 < .640; J_0 < D_* c (.5)/.493 < .48; L < .0026/alpha < .20",
     "pass": true,
     "detail": "K_0 in [0.36759, 0.37118]"
    },
    {
     "check": "-.3 < z < 0 (z = -R/(rJ), 1.7 < R/J < 3, r >= 10) and .8 < 1 + 3(.482) - 3(.528) <= h_* <= 1 + 3(.5) - 1.7(.5) < 1.77",
     "pass": true,
     "detail": ""
    },
    {
     "check": "d >= .97/r, Var V >= .88 (u <= .01); chi <= J/(.97 r^2) < .027 < .097; slope >= 2(.88) - .027/(1-.5^2) > .68",
     "pass": true,
     "detail": "chi < 0.02654"
    },
    {
     "check": "N_11 < .403 (first part increasing in b: 2A_*(.482) > eta_*), N_22 < 1.12, .080 < N_12 < .13",
     "pass": true,
     "detail": "N11 0.40220 N22 1.10446 N12 [0.08266, 0.11739]"
    },
    {
     "check": "right-tail determinant (.43-.403-.022)(2.72-1.12-.75) - .05^2 = .00175 > 0, |.13 - N_12| < .05",
     "pass": true,
     "detail": "7/4000"
    },
    {
     "check": "the right tail lands inside row T_1 and row a >= -1: b in (.482,.5), L < .0347 <= .20, D < .640, K_0 in (.367,.4), J_0 < .48, l_1 = .022, z in (-.3,0), h_* in (.8,1.77), chi < .027, slope > .68 = m_0, p > .46",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D_* = 5/17, A_* = s + 1 + s/8, eta_*, nu_* computed; rho <= epsilon + kappa = .64; 1 - epsilon - kappa = .36",
     "pass": true,
     "detail": "A_* = 82/37, eta_* = 427/370, nu_* = 56897/74000"
    },
    {
     "check": "(sc:y-cost) s(.28^2 + .075 * .67^2/(4 * .36)) < .111",
     "pass": true,
     "detail": "0.110033"
    },
    {
     "check": "p-minorant: 1 - (1-x/2)(x^2+7u) >= x(x-1)^2/2 + .045(2-x) on [0,2] x [0,.13] (difference affine, corners >= 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "p-minorant on [0,4]: u + 2u^(3/2) < .25 at u <= .13 (u^3 < .0036); on [4,inf): u + n_u^2/(4d_u) = .46 identically",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(sc:p-mean) 1 - 1.28^2/[16(.46-.13)] > .689",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(sc:rho-loss) .64 * 1.28^2 * .13/[16(.46-.13)] < .02582 and alpha s * .64 * .13^3/[16(.46-.13)] < .027",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tangent of F_0 at S = .28: intercept < .16, slope < .52 (log .28 enclosed by the atanh series with z = -9/16, checked by exp)",
     "pass": true,
     "detail": "intercept 0.15659 slope 0.50743"
    },
    {
     "check": ".075(.16 + .52S) < .089(.15 + .45325S) coefficientwise; kappa/s = .45325",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(sc:F0-sqrt): x^2 J'(x) = P(x) exactly; Descartes: 3 sign changes; P(.24) < 0 < P(.26), P(.6) < 0, P(1) = 0 < P'(1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "J(1) = 0; J(.25) > .14 from log 2 < .694; |P(.25)| < .01, |P'| < 4 on [.24,.26]; |J'| < 2 there; J at the first minimum > .14 - .02 > 0",
     "pass": true,
     "detail": "J(.25) > 0.14959"
    },
    {
     "check": "sqrt(.075) * .478/.15 < .88 ((.88*.15/.478)^2 > .075); the ratio decreases (1.4*.15 < .478*.45325); .089 < .688*.13, .88 < .688*1.28",
     "pass": true,
     "detail": ""
    },
    {
     "check": ".027 + .111 = 69/500",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the 36 entries of (sc:substitution-table): the maximum over every corner of the box is <= the printed bound",
     "pass": true,
     "detail": "Ole:1.866/0.519/1.312 Oge:2.340/0.547/1.154 P_1le:1.766/0.488/1.812 P_1ge:2.563/0.432/1.154 R_1le:1.680/0.662/2.056 R_1ge:2.620/0.419/1.154 U_1le:1.550/0.846/2.100 U_1ge:2.461/0.613/1.154 V_1le:1.262/0.911/1.785 V_1ge:1.885/0.768/1.204 T_1le:0.912/0.867/1.384 T_1ge:1.070/0.839/1.250"
    },
    {
     "check": "(sc:F-simple): F_j <= .295, .23, .23, .39 (R_1, U_1, V_1, T_1, all a); R_1 <= .20 for a <= -1; P_1 <= .42^2 / .61^2",
     "pass": true,
     "detail": "P_1ge:0.3454 P_1le:0.1391 R_1ge:0.2726 R_1le:0.1287 T_1ge:0.3775 T_1le:0.2262 U_1ge:0.1928 U_1le:0.1205 V_1ge:0.1894 V_1le:0.1385"
    },
    {
     "check": "(sc:F-negative): F_j/chi^2 <= 20 + 1.04(7+t_j^2) (a <= -1), 12 + 1.63(7+t_j^2) (a >= -1)",
     "pass": true,
     "detail": "le: 13.1317 + 1.0283(7+t^2); ge: 9.3302 + 1.6153(7+t^2)"
    },
    {
     "check": "Theta_m increases: d/dy numerator = 1/180 + 2y/1680 + y^2/(1680*36); r_ij delta^2 = delta^2(1+delta^2/36)Theta/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "threshold comparisons (sc:threshold) in all 14 endpoint cases",
     "pass": true,
     "detail": "Theta_.19(4) = 0.347192, Theta_.19(5) = 0.387203, Theta_0(6.5) = 0.399206"
    }
   ],
   "sources": {
    "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/certificate.tex": "0434e78f918a386cc37af50b16305460d5d2af14cb8d7cced98254189ac61ed1",
    "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/scalar.tex": "daea10d96fedaaee1d69c6e46ddc69490add7e1d32176a36b209870540539a85",
    "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/transport.tex": "dfb3651f6b821051d8b17f5dca2a921074fde1d2a4bd9278591659007946bc23",
    "preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/build/manuscript/matrix.tex": "4e57766d2d6324f0ab04ad3118f506f79db6511bdc0571ac9f0576aa70ac76a2"
   },
   "forges": [
    {
     "forge": "seed P_0 of row w = 3 moved by 1e-9 (a join exceeds 2e-10)",
     "verdict": "REFUTED"
    },
    {
     "forge": "seed k^_0 of row w = -5.25 moved by 1e-8 (k^(-1) leaves the anchor window, a join exceeds 2e-10)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table cert:group-bounds G2 at w = 9 printed .6 instead of .35",
     "verdict": "REFUTED"
    },
    {
     "forge": "substitution table T_1, a >= -1 middle entry .840 -> .80",
     "verdict": "REFUTED"
    },
    {
     "forge": "left-tail determinant printed .023233",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1.9
  },
  {
   "id": "F-209",
   "index": 209,
   "family": "104",
   "title": "Randomized quasipolynomial-time mean-payoff games",
   "tier": "A",
   "dir": "Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026",
   "leanChallenges": [
    "RandomizedMeanPayoff",
    "TruffetCounterexample"
   ],
   "word": "CERTIFIED",
   "why": "a finite component (the appendix side result, not the main theorem): on the printed two-variable instance the true minimum is exactly 0, while the substitution rule as formalised in the challenge is forced through x2 = h, x1 = h + 1 to value 1; all 13 conjuncts of finite_execution_not_optimal. Faithfulness of the formalisation to Truffet v4 is a reading, not decided.",
   "decider": "instruments/openaimath/finite/f209_truffet.py",
   "decides": "a finite component (the appendix side result, not the main theorem): on the printed two-variable instance the true minimum is exactly 0, while the substitution rule as formalised in the challenge is forced through x2 = h, x1 = h + 1 to value 1; all 13 conjuncts of finite_execution_not_optimal. Faithfulness of the formalisation to Truffet v4 is a reading, not decided.",
   "value": {
    "true_min": "0",
    "min_with_x2_eq_h": "1",
    "rule_output_value": "1"
   },
   "checks": [
    {
     "check": "the appendix prints the instance (09-comparison.tex:12-19) and is an appendix of the paper",
     "pass": true,
     "detail": "side result; NOT the paper's main theorem"
    },
    {
     "check": "the transcription matches the challenge statement text (21 fragments)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the challenge statement is a statement (one sorry, theorem finite_execution_not_optimal)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every piece of the feasible set decided two ways (Fourier-Motzkin; Farkas recombination)",
     "pass": true,
     "detail": "min 0 ['row1: x1 >= h +0', 'row2: x1 >= x2 +1', 'row2: x1 >= h -2']; min 1 ['row1: x2 >= h +0', 'row2: x1 >= x2 +1', 'row2: x1 >= h -2']"
    },
    {
     "check": "(0,-1) is feasible (09-comparison.tex:17)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the true minimum is exactly 0 (no feasible x1 < 0; 09-comparison.tex:17-19)",
     "pass": true,
     "detail": "exact min 0"
    },
    {
     "check": "the first saturation removes every optimum: with x2 = h = 0 adjoined the minimum is 1 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the optimal set is {x1 = 0, x2 <= -1}: of the pieces cut by x1 <= 0 one is infeasible, the other projects to x2 in (-inf, -1]",
     "pass": true,
     "detail": "[('ok', None, Fraction(-1, 1)), ('infeasible', Fraction(0, 1), Fraction(-1, 1))]"
    },
    {
     "check": "crossRowRestriction initial (Truffet (5b), both ordered pairs; checked argument + instances)",
     "pass": true,
     "detail": "[((0, 1), True), ((1, 0), True)]"
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: wellFormed s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: wellFormed s'",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: bounded s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: not stop s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: not switch s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: not failure s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: eligible set = {(0,1)} exactly (found [(0, 1)])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: substituted cost is not null",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep initial middle 0 1: s' = advance s 0 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: wellFormed s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: wellFormed s'",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: bounded s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: not stop s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: not switch s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: not failure s",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: eligible set = {(1,0)} exactly (found [(1, 0)])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: substituted cost is not null",
     "pass": true,
     "detail": ""
    },
    {
     "check": "ForcedSingletonStep middle finalState 1 0: s' = advance s 1 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "no variable remains in finalState",
     "pass": true,
     "detail": ""
    },
    {
     "check": "stop finalState",
     "pass": true,
     "detail": ""
    },
    {
     "check": "feasible initial output = ('1', '0', '0')",
     "pass": true,
     "detail": ""
    },
    {
     "check": "feasible initial better = ('0', '-1', '0')",
     "pass": true,
     "detail": ""
    },
    {
     "check": "output 2 = 0 and better 2 = 0 (h normalised to 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "output 1 = eval (lowerForm initial 0 1) output (x2 = h)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "output 0 = eval (lowerForm middle 1 0) output (x1 = h + 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eval initial.cost output = eval finalState.cost output",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eval initial.cost better < eval finalState.cost output",
     "pass": true,
     "detail": "0 < 1"
    },
    {
     "check": "occurrence table: x1 lower-bound rows = [1, 2]",
     "pass": true,
     "detail": "found [1, 2]"
    },
    {
     "check": "occurrence table: x1 upper-bound rows = none",
     "pass": true,
     "detail": "found []"
    },
    {
     "check": "occurrence table: x2 lower-bound rows = [1]",
     "pass": true,
     "detail": "found [1]"
    },
    {
     "check": "occurrence table: x2 upper-bound rows = [2]",
     "pass": true,
     "detail": "found [2]"
    },
    {
     "check": "(53a) the dominating set is exactly {x2}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(53b) the eligible pairs are exactly {(row 1, x2)}: no tie-breaking",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the saturation is x2 = h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "after it, row 2 reads x1 >= h + 1 and row 1 is trivially satisfied (setrowtozero)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the next saturation is x1 = h + 1, forced",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the stop test (39) fails before both substitutions and holds after",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the objective depends on x1 at both stages, so the switch rule does not apply",
     "pass": true,
     "detail": ""
    },
    {
     "check": "both variables have lower or upper occurrences (Def. 4.2, eq. 34)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "each left side lacks an h term; the other row's right side has one",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(23b) for max(x2+1, h-2): 1 + lambda = -3 < -2 + mu = -2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the witness (0,-1,0) lies in R = [lambda,inf]^2 x [mu,inf] with lambda = -4, mu = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "normalised to h = 0 the rule's objective value is 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "simulated from the initial state: 2 forced steps, output value 1, exact minimum 0; output > minimum",
     "pass": true,
     "detail": "ok"
    },
    {
     "check": "the simulation reproduces the printed middle and final states",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/build/source/sections/09-comparison.tex": "1bc76c1832a2eeddaa4a927b321922137e3c63dfde0a2c2e79ec3b3bf9c6a830",
    "preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/build/source/main.tex": "e20659ded023e02f3b21df300526078750b99428713546fb3f47c3c5de9a73ba",
    "lean/ComparatorChallenges/TruffetCounterexample.lean": "03fe8de8feab6eeb5144931898f079a0f900e695bf3f8c5d3de3dbc177e21169",
    "lean/ComparatorChallenges/TruffetCounterexample.json": "5b6a0aaf8f27c927a9c0b6a07fb82f43b282c3265058616b0a58fed5eb212f3b"
   },
   "forges": [
    {
     "forge": "row 2 offset +1 -> +0 (rule output 0, minimum 0)",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed middle state x1 >= h + 2",
     "verdict": "REFUTED"
    },
    {
     "forge": "better point (0,0,0)",
     "verdict": "REFUTED"
    },
    {
     "forge": "claimed true minimum -1",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.0
  },
  {
   "id": "F-213",
   "index": 213,
   "family": "107",
   "title": "Complex Matrix Multiplication Below 2.258 and Rectangular Bounds",
   "tier": "B",
   "dir": "Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026",
   "leanChallenges": [
    "MatrixFields",
    "MatrixMultiplication"
   ],
   "word": "REFUSED",
   "why": "finite components only: the three rectangular witnesses (alpha > 0.465, w(0.709) < 2.092, family omega < 2.267) as finite computations; for omega < 2.258 the barrier, both charts (the 18 MB chart verified as the recursion's output), the residual eps, the chart comparisons and the W-seed polynomial. NOT the interior lemma (cost), NOT the boundary lemma / W-seed library (records not published), hence NOT Theorem thm:certificate; no theory",
   "decider": "instruments/openaimath/finite/f213_complex_omega.py",
   "decides": "finite components only: the three rectangular witnesses (alpha > 0.465, w(0.709) < 2.092, family omega < 2.267) as finite computations; for omega < 2.258 the barrier, both charts (the 18 MB chart verified as the recursion's output), the residual eps, the chart comparisons and the W-seed polynomial. NOT the interior lemma (cost), NOT the boundary lemma / W-seed library (records not published), hence NOT Theorem thm:certificate; no theory",
   "value": {
    "eps_upper": "3.328666321e-22",
    "timing": {
     "rectangular": 0.0,
     "barrier_and_W": 0.5,
     "P420": 20.1,
     "P780": 568.9,
     "residual_and_table": 1.2
    },
    "seconds": 590.6,
    "workers": 6,
    "full": true,
    "charts": true,
    "decided": 36,
    "refused_because": [
     "boundary lemma and W-seed finite check: the three-state record library is not published (inputs are only newc22, qP420_gmp, qP780_gmp); the paper specifies it by its program menu.hpp (\"The supplied finite program fixes all these selection choices, including ties\", certificate-w.tex:272-273) and its recorded gaps are that program's output",
     "interior lemma: branch-and-bound over 16,384 initial polar boxes, five fields of degree < 320 and < 52,000 monomials, degree-14 Taylor arrays, bisection to depth 28; ~6e9 big-integer operations before subdivision (hours of pure-Python CPU per pass), the subdivision count unpublished",
     "geometry tests: 230,400 centers of the degree-40 chart with derivatives (~1e9 operations) — not attempted"
    ]
   },
   "checks": [
    {
     "check": "the headline as printed: omega < 1129/500 = 2.258, alpha > 0.465, w(0.709) < 2.092",
     "pass": true,
     "detail": ""
    },
    {
     "check": "#212 (omega <= 9/4) as read: Theorem thm:main states omega <= 9/4; its TeX has no verbatim / tabular / listing block",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R1 dual witness: the script -2,+2,+2,-3,+2,-2,+3 and centers B,A,C,B,C,B,A (07-certificates.tex:145-154)",
     "pass": true,
     "detail": "['B', 'A', 'C', 'B', 'C', 'B', 'A']"
    },
    {
     "check": "R1 at b = 8: rho_8 = 1-(3/4)^4 exactly; delta_8, D_8 = (0.5625/4) log 3 and the two z (log 3, log 7/9) agree to 1e-60",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R1 the printed rational: (2/0.6932)[0.5625*1.098/4 + 0.463*0.635/48 + 0.360*0.628/288] = 1548637/3327360 > 0.465",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R1 THE DUAL WITNESS: alpha >= min(1, 2 D_288/log 2) > 1548637/3327360 > 0.465",
     "pass": true,
     "detail": "[0.466101554898471, 0.466101554898472]"
    },
    {
     "check": "R1 the reference value 2 D_288/log 2 = 0.466101554898 (to 5e-13)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R1 the printed table at b = 24, 48, 96, 288 (07-certificates.tex:184-198)",
     "pass": true,
     "detail": "all eight rows"
    },
    {
     "check": "R1 D_288 from the h-recursion agrees with the delta/z shortcut of eq. (delta-D) to 1e-60",
     "pass": true,
     "detail": "[0.16153848931624116501, 0.16153848931624116502]"
    },
    {
     "check": "R2 the fiber histogram re-enumerated from d(x) = 2^#zeros, d(0) = 3^3 - 2^3: {1: 8, 2: 12, 4: 6, 19: 1}, 75 leaves",
     "pass": true,
     "detail": "{19: 1, 4: 6, 2: 12, 1: 8}"
    },
    {
     "check": "R2 cube-root brackets: 1.58739^3 < 4 < 1.58742^3 and 7.1203^3 < 361 < 7.1205^3; u = 2^(2/3), v = 19^(2/3) inside them",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R2 49.2878 < Z(2/3) < 49.2890, 1.11844 < g < 1.11856, 3.1519 < h < 3.1522",
     "pass": true,
     "detail": "Z [49.288232, 49.288233] g [1.1184936, 1.1184937] h [3.152022, 3.152023]"
    },
    {
     "check": "R2 THE CURVE WITNESS: 2g/h > 0.709 and 6 log 3/h < 2.092 (and the printed chains 2*1.11844/3.1522 > 0.709, 6*1.0987/3.1519 < 2.092)",
     "pass": true,
     "detail": "2g/h [0.709698932519, 0.709698932520], 6log3/h [2.091251835621, 2.091251835622]"
    },
    {
     "check": "R2 reference values 2g/h = 0.709698932519 and 6 log 3/h = 2.091251835621 (to 5e-13)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R3 completion counts: after three steps (20691584, 13993344, 10144225), after four (428141648429056, 774902581936128, 522705468255425)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R3 N = 2995461856366867388223311649574852466276480577 >= 10^36 (1202^3 + 1296^3 - 2*774^3) > 2.95e45; b = 72",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R3 THE FAMILY SQUARE WITNESS: 216 log 3 / log N < 2.267 (and the printed chain 216*1.0987/(1.0817 + 45*2.3025) < 2.267)",
     "pass": true,
     "detail": "[2.266187438998, 2.266187438999]"
    },
    {
     "check": "R3 reference value 216 log 3 / log N = 2.266187438999 (to 5e-13)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R4 the log/exp table of subsec:log-certificate (14 enclosures)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "S1 barrier: Table cert:barrier-coefficients equals inputs/newc22 entry for entry (52 exact decimals) on the index set I, in order",
     "pass": true,
     "detail": "52 table rows, 52 file values"
    },
    {
     "check": "S1 phi(0) = a_00 = 1.098619383270680538 as printed (certificate.tex:23, 378)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "S1 phi(0) - log 3 > 7.094e-6 (certificate.tex:977) and > 7.0946e-6 (reproducibility.tex:109)",
     "pass": true,
     "detail": "[0.000007094602570, 0.000007094602571]"
    },
    {
     "check": "S1 the five fourth-derivative sums (certificate.tex:706-715) are < 435e6, the largest < 434439204",
     "pass": true,
     "detail": "[42009382, 61488577, 97201581, 175179092, 434439203]"
    },
    {
     "check": "S1 radial sums sum |a_ij| R^(i+j) (i+j)^k, k = 0..4, below 35000 / 750000 / 1.5e7 / 3e8 / 6e9 (certificate.tex:686-690, read so)",
     "pass": true,
     "detail": "[10509, 198597, 3795405, 73250174, 1426050319]"
    },
    {
     "check": "S5 H_W: 6441 monomials; L_W H_W has b_ij = 0 exactly for i+j <= 110 and sum |b_ij| R_W^(i+j) < 6.739223e-13 (certificate-w.tex:123-141)",
     "pass": true,
     "detail": "residual 6.739223e-13"
    },
    {
     "check": "S5 S_0 < .058178, S_1 < 1.490164, S_2 < 53.686778 (certificate-w.tex:111-121)",
     "pass": true,
     "detail": "0.0581771, 1.4901640, 53.6867779"
    },
    {
     "check": "S5 shell: r_* = .35670524, r_1 = .33437745, r_2 = 2.13289155, rho >= .20947544 > .2, |z'| <= .69108269, |z''| <= 3.15835169 < 3.16, |(H_W o z)''| < 30.348",
     "pass": true,
     "detail": "30.3470"
    },
    {
     "check": "S6 library error recurrences: f_12^init <= 1159.991/B, e_14 = 3637206/B < 8.5e-10, f_14 < 3.635e-9 < 1e-8 (certificate-w.tex:313-362)",
     "pass": true,
     "detail": "f12 1159.990/B, e14 3637206/B, f14 3.634e-09"
    },
    {
     "check": "S2 qP420_gmp is the fixed point of the chart recursion (certificate.tex:111-149) at N = 420, b = 224",
     "pass": true,
     "detail": "pure True, symmetric True, 44100 recursion equations checked, 0 fail []"
    },
    {
     "check": "S2 21 < C_00/Q < 22 at b = 224",
     "pass": true,
     "detail": ""
    },
    {
     "check": "S2 qP780_gmp (18 MB) is the fixed point of the chart recursion at N = 780, b = 500 (so it IS the defined chart P)",
     "pass": true,
     "detail": "pure True, symmetric True, 152100 recursion equations checked, 0 fail []"
    },
    {
     "check": "S2 21 < C_00/Q < 22 at b = 500; Theorem thm:certificate's p = 2^-500 rnd(2^500 18623/100000) = A_00/2^500; E_1(0) = 1/3 - P(0) > .1471",
     "pass": true,
     "detail": ""
    },
    {
     "check": "S3 eps = sup |L P| <= rho + sum D V < 3.328667e-22 (certificate.tex:209-212), exact rational; 6 eps/eta < 9.987e-13 < 1e-12",
     "pass": true,
     "detail": "eps <= 3.328666321e-22 (rho 1.278e-67, products 3.329e-22)"
    },
    {
     "check": "S4 the |P - P^ref| table: Delta_0 < 3.517e-17, Delta_1/R < 1.233e-14, Delta_2/R^2 < 4.327e-12, <=236: < 1.902e-11, <=220: < 7e-11",
     "pass": true,
     "detail": "3.517e-17, 1.233e-14, 4.327e-12, 1.902e-11, 6.199e-11"
    },
    {
     "check": "S4 chart jets |grad p_i| < .592, |D^2 p_i| < 3.05 and degree-40 truncation < .0002 (value), < .006 (gradient), both charts",
     "pass": true,
     "detail": "P780 grad 0.5913 hess 3.0490 tail40 1.44e-04/5.91e-03; P420 grad 0.5913 hess 3.0490 tail40 1.44e-04/5.91e-03"
    },
    {
     "check": "S4 reference chart (degree 220) on the disk: absolute sum < .414 and k^2-weighted sum < 6 (certificate.tex:624-627); 861 coefficients to degree 40",
     "pass": true,
     "detail": "sum 0.41225, k^2 sum 5.4214, degree-40 sum 0.41211"
    }
   ],
   "sources": {
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/main.tex": "6565fb64c02e0064feb1bc689de2ca6bf3a696f9042a9eb73621a6392eda3818",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/introduction.tex": "d8cdaf01471b51ac7b88fe576916180bd751fb8438717e155bab00d9b1b9f011",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/tex/certificate.tex": "0bc2a5d291eee02b5ebeb7cf01cd631a6bc55bfbd895001e2ccbbbd1a5fdf184",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/tex/certificate-w.tex": "6ca02b67f05819fda1822c85fb0ae1dce215712aca0952c38a9700e937660276",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/tex/barrier-coefficients.tex": "bc60dce8d3eb17992dabf7babfc7662ae57b7a513240cdb453ddd5c4c3900ac6",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/tex/reproducibility.tex": "4ea3881ad583d48304424acb1479c1401aeb740a4e1943a166e15430a88d7a82",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/build/rectangular/07-certificates.tex": "218f32283e84bd5f6d2d1771eaf87d17537285627d73ec3601d77a55eedcff9f",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/verification/certificate/inputs/newc22": "41d42a58936b9da0f01a98a24fc4e998a00b83c9f4032a4ae166b7ef11073912",
    "preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex": "c396f2580a711b8b9e51ecf1265cb0e691acd745943f81bfc14ec0df209e1028",
    "preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/README.md": "1857fc06d04a43588207ba2f726679c50a418299a5fd0313423a354680a783a7",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/verification/certificate/inputs/qP420_gmp": "7dc037ea127cece2216f3219eeeca4cdab0288319a10111a78d272586b776f0b",
    "preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/verification/certificate/inputs/qP780_gmp": "f23c0480f1109509545fdd17bb4de81602f50952411554b337105defbe7a070c"
   },
   "forges": [
    {
     "forge": "barrier constant term 1.098619383270680538 -> 1.0986122 (below log 3 + 7.094e-6)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the last completion of the dual script +3 -> +2",
     "verdict": "REFUTED"
    },
    {
     "forge": "the zero-word fiber 19 -> 18 in the curve histogram",
     "verdict": "REFUTED"
    },
    {
     "forge": "one coefficient of qP420_gmp changed by one unit (entry 50000)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the H_W recursion stopped at d = 108 instead of 110 (the residual is no longer zero through degree 110)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 590.6
  },
  {
   "id": "F-214",
   "index": 214,
   "family": "107",
   "title": "Staggered extraction for exact matrix multiplication over every field",
   "tier": "A",
   "dir": "Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026",
   "leanChallenges": [
    "MatrixFields",
    "MatrixMultiplication"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the parameter certificate — the five integer arrays, decoded by the printed rules and contracted by the printed finite formulas, give every printed table value and enclosure and 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006746 with every feasibility condition; NOT the extraction theory that turns this ratio into a bound on omega",
   "decider": "instruments/openaimath/finite/f214_staggered_extraction.py",
   "decides": "a finite component: the parameter certificate — the five integer arrays, decoded by the printed rules and contracted by the printed finite formulas, give every printed table value and enclosure and 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006746 with every feasibility condition; NOT the extraction theory that turns this ratio into a bound on omega",
   "value": {
    "H0": "[1.841147374210963999970680, 1.841147374210963999970681]",
    "Bst": "[1.158066972073907565308106, 1.158066972073907565308107]",
    "T": "[2.345955062997053532856369, 2.345955062997053532856370]",
    "Cst": "[3.771836581848272756855089, 3.771836581848272756855090]",
    "yld": "[5.612983956059236756825770, 5.612983956059236756825771]",
    "S0": "[0.031704071407964669154688, 0.031704071407964669154689]",
    "S1": "[0.882362235981496244138529, 0.882362235981496244138530]",
    "S2": "[11.680704014205215449067291, 11.680704014205215449067292]",
    "Sst": "[12.594770321594676362360509, 12.594770321594676362360510]",
    "ratio": "[2.371054886006745684861111, 2.371054886006745684861112]",
    "gap": "[0.000004676829725968327301, 0.000004676829725968327302]",
    "TB": "[0.468025165562828972929130, 0.468025165562828972929131]",
    "beta": "[0.533798154491618238625488, 0.533798154491618238625489]",
    "max_width": "4.226e-64",
    "F_arguments": 223,
    "seconds": 0.7
   },
   "checks": [
    {
     "check": "the headline as printed (abstract and Theorem thm:main)",
     "pass": true,
     "detail": "main.tex:43-45, introduction.tex:57-63"
    },
    {
     "check": "the definitions the decider evaluates are printed where cited",
     "pass": true,
     "detail": "parameters.tex:14, staggering.tex:122,192, leaf-volumes.tex:33, joint-extraction.tex:132, parameters.tex:327"
    },
    {
     "check": "the TeX appendix arrays equal verification/data/parameters.json",
     "pass": true,
     "detail": "lengths [17, 118, 54, 75, 36]"
    },
    {
     "check": "array lengths 17 / 118 / 54 / 75 / 36 as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "statistic coding re-derived from the 49 length-2 words of CW_5: r = (0,1,2,2,3,4), D = (1,10,25,2,10,1), kappa = (5,4,2,3,1,0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "kappa is an involution with D_kappa(i) = D_i and r_kappa(i) = 4 - r_i",
     "pass": true,
     "detail": ""
    },
    {
     "check": "shape counts: 30 sorted, 21 positive sorted, 21 positive ordered size-8, 24 size-8 with a zero",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the decoding rules as written consume every array exactly",
     "pass": true,
     "detail": "{'P': (118, 118), 'p': (54, 54), 'd': (75, 75), 'z': (36, 36)}"
    },
    {
     "check": "the authors' decoding table parameter-assignments.tsv agrees row for row with this decoding",
     "pass": true,
     "detail": "450 rows here, 450 in the TSV, same order"
    },
    {
     "check": "223 distinct F-arguments (implicit zero included), all in [-120770, 45052]",
     "pass": true,
     "detail": "223 in [-120770, 45052]"
    },
    {
     "check": "2^-23 < F(n) < 2^23 for every argument (exact integer comparison of Nnum^64 with DEN^64 2^23)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "d range: 0 <= d <= 0.110106, both ends attained",
     "pass": true,
     "detail": ""
    },
    {
     "check": "printed examples d_{(1,3,4),(0,2,2)} = 92966/10^6 and d_{(1,3,4),(1,1,2)} = 1204/10^6",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every split law satisfies q_s(b) = q_s(s-b) exactly (equal integer weights)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every zero-coordinate child (Stage A t, Stage B u) has its two other sides matched by kappa, two-zero shapes included",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:capacities: all 30 partial capacities within 4.795e-10 of the printed values",
     "pass": true,
     "detail": ""
    },
    {
     "check": "all 30 partial capacity entries are positive",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:scalars: H_0, B_*, T, S_0, S_1, S_2 within 3e-8 of the printed values",
     "pass": true,
     "detail": "H0 [1.841147374210, 1.841147374211]; Bst [1.158066972073, 1.158066972074]; T [2.345955062997, 2.345955062998]; S0 [0.031704071407, 0.031704071408]; S1 [0.882362235981, 0.882362235982]; S2 [11.680704014205, 11.680704014206]"
    },
    {
     "check": "printed enclosure 5.612983956059236756 < H_0 + C_* < 5.612983956059236757",
     "pass": true,
     "detail": "[5.612983956059236756825770, 5.612983956059236756825771]"
    },
    {
     "check": "printed enclosure 12.594770321594676362 < S_* < 12.594770321594676363",
     "pass": true,
     "detail": "[12.594770321594676362360509, 12.594770321594676362360510]"
    },
    {
     "check": "printed enclosure 2.371054886006745684 < 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006745685",
     "pass": true,
     "detail": "[2.371054886006745684861111, 2.371054886006745684861112]"
    },
    {
     "check": "printed enclosure 0.000004676829725968 < H_0 + C_* + (Omega/3) S_* - 8 log 7 at Omega = 2.371056 < 0.000004676829725969",
     "pass": true,
     "detail": "[0.000004676829725968327301, 0.000004676829725968327302]"
    },
    {
     "check": "printed enclosure 0.468025165562828972 < T log 2 - B_* < 0.468025165562828973",
     "pass": true,
     "detail": "[0.468025165562828972929130, 0.468025165562828972929131]"
    },
    {
     "check": "printed L_C row: each entry in (listed, listed + 1e-15)",
     "pass": true,
     "detail": "[1.11534382195434982, 1.11534382195434983]; [0.96211864242411813, 0.96211864242411814]; [1.39673845184325473, 1.39673845184325474]"
    },
    {
     "check": "printed balance numerator b row: each entry in (listed, listed + 1e-15)",
     "pass": true,
     "detail": "[0.51074831568238671, 0.51074831568238672]; [0.66397349521261839, 0.66397349521261840]; [0.22935368579348180, 0.22935368579348181]"
    },
    {
     "check": "printed lambda row: each entry in (listed, listed + 1e-15)",
     "pass": true,
     "detail": "[0.36376129124608113, 0.36376129124608114]; [0.47289016636113967, 0.47289016636113968]; [0.16334854239277919, 0.16334854239277920]"
    },
    {
     "check": "L_C by C_* - (L_A+L_B) agrees with L_C by the lambda formula (staggering.tex:213-220)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "beta = 2C_* - min L_A - min(L_A+L_B) - min(L_B+L_C) - min L_C in (0.533798153, 0.533798155), < 1",
     "pass": true,
     "detail": "[0.533798154491618, 0.533798154491619]"
    },
    {
     "check": "feasibility: L_A, L_B, L_C entrywise positive; balance numerators and fractions positive; T log 2 > B_*; S_* > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "rounded tables: (12 + Omega) 4e-8 = 5.7484224e-7, and the margin from the printed centres minus it exceeds 4.1011e-6",
     "pass": true,
     "detail": "centre margin [0.0000046759671522, 0.0000046759671523], less the uncertainty [0.0000041011249122, 0.0000041011249123]"
    },
    {
     "check": "printed error constants: tau < 3.120348e-32, 2048 tau < 6.391e-29, (68/3 + 9 Omega + 8) eps < 3.324e-27",
     "pass": true,
     "detail": ""
    },
    {
     "check": "THE HEADLINE INEQUALITY: 3(8 log 7 - H_0 - C_*)/S_* < 2.371054886006746 (< 2.371054887)",
     "pass": true,
     "detail": "ratio in [2.371054886006745684861111, 2.371054886006745684861112]"
    }
   ],
   "sources": {
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/main.tex": "10df2f1e4f98720f3d67e670e588c2d5873d0139ee313ef4ba28814c8f4badc1",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/introduction.tex": "e3127684f7901c346b02ebd596bc94adafc5fc3d2472ff2c86e8188435fcfb99",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/parameters.tex": "1973b01415786e51cb222ec4d83dcb0af1cad61a983b0a0d27649705b90d2a0c",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/parameter-tables.tex": "0e46bba625593b2a122cc093db0d0764647af670a7a9aebaa85aaed94b81e447",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/staggering.tex": "7298c89a21b2d30090f8103a805d8efec771d498952375d3717146cee90a8a23",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/leaf-volumes.tex": "bff4a264faae58036793e07edcde69631a1743ea07b7c5d411b7ca680a0b33ae",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/build/sections/joint-extraction.tex": "c9db330d781ad3c1b10c160edd0d3128370335d1e8cd76237ed5e9f81382e326",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/verification/data/parameters.json": "538deff25fb30b68b01f7ff43b2a62ce0719afba25891ff38fda7027a31c2b65",
    "preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/verification/data/parameter-assignments.tsv": "b65c55be68d38ad1f8340c963d128ccd088632ef409270d1a28242debf53af94"
   },
   "forges": [
    {
     "forge": "claimed bound lowered to 2.3710548860067455 (below the true ratio)",
     "verdict": "REFUTED"
    },
    {
     "forge": "one initial weight changed by one unit (a_8 = 38593 -> 38594; ratio [2.371054886165415247083620, 2.371054886165415247083621])",
     "verdict": "REFUTED"
    },
    {
     "forge": "one Stage B integer of a reachable parent changed by one unit (p[0] = -39594 -> -39593)",
     "verdict": "REFUTED"
    },
    {
     "forge": "z_t normalised over unordered pairs (an off-diagonal weight counted once; parameters.tex:76-80 warns against it)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the zero-amount parent (6,0,2) skipped in the z traversal (parameters.tex:86-88): caught by exact consumption only, since every zero-amount parent is at the tail of its traversal or reads no entry, so no number moves",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.7
  },
  {
   "id": "F-232",
   "index": 232,
   "family": "118",
   "title": "Additive hardness and unbounded configuration gaps in bin packing",
   "tier": "C",
   "dir": "Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026",
   "leanChallenges": [
    "BinPackingGap"
   ],
   "word": "NEEDS DATA",
   "why": "preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/build/source/consequences.tex l.9-24 (proof of Theorem thm:unbounded-gap), /build/source/instance.tex l.26-60 (parameters eq:parameters and geometry). Instance itself NOT PUBLISHED (only its prescription). Lean: lean/ComparatorChallenges/BinPackingGap.lean; proof lean/OAI/Computability/BinPacking/ (284 files).",
   "decider": null
  },
  {
   "id": "F-234",
   "index": 234,
   "family": "119",
   "title": "Hellinger contraction with arbitrary Boolean output bias",
   "tier": "A",
   "dir": "Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026",
   "leanChallenges": [
    "CourtadeKumar",
    "SoftChannel204"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the scalar certificates of the proof — the induction parameters, the 17 profiles' five Bernstein sign checks, the three-coordinate base table, the 21-box pair interval certificate, and every finite comparison and Bernstein/table entry of the central, intermediate and large-bias bands; NOT the theorem, whose induction, continuation and norm-inequality arguments are analytic",
   "decider": "instruments/openaimath/finite/f234_hellinger_bias.py",
   "decides": "a finite component: the scalar certificates of the proof — the induction parameters, the 17 profiles' five Bernstein sign checks, the three-coordinate base table, the 21-box pair interval certificate, and every finite comparison and Bernstein/table entry of the central, intermediate and large-bias bands; NOT the theorem, whose induction, continuation and norm-inequality arguments are analytic",
   "value": {
    "pair": {
     "1/0": {
      "certified": true,
      "boxes": 5881,
      "deepest": 7,
      "printed_depth": 8
     },
     "1/1": {
      "certified": true,
      "boxes": 9809,
      "deepest": 7,
      "printed_depth": 8
     },
     "1/2": {
      "certified": true,
      "boxes": 9881,
      "deepest": 7,
      "printed_depth": 8
     },
     "1/3": {
      "certified": true,
      "boxes": 8513,
      "deepest": 6,
      "printed_depth": 7
     },
     "1/4": {
      "certified": true,
      "boxes": 10577,
      "deepest": 7,
      "printed_depth": 8
     },
     "1/5": {
      "certified": true,
      "boxes": 14625,
      "deepest": 7,
      "printed_depth": 8
     },
     "1/6": {
      "certified": true,
      "boxes": 20745,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/0": {
      "certified": true,
      "boxes": 32361,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/1": {
      "certified": true,
      "boxes": 27769,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/2": {
      "certified": true,
      "boxes": 29913,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/3": {
      "certified": true,
      "boxes": 45153,
      "deepest": 6,
      "printed_depth": 7
     },
     "2/4": {
      "certified": true,
      "boxes": 38601,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/5": {
      "certified": true,
      "boxes": 38481,
      "deepest": 7,
      "printed_depth": 8
     },
     "2/6": {
      "certified": true,
      "boxes": 102297,
      "deepest": 7,
      "printed_depth": 8
     },
     "3/0": {
      "certified": true,
      "boxes": 197129,
      "deepest": 13,
      "printed_depth": 14
     },
     "3/1": {
      "certified": true,
      "boxes": 34801,
      "deepest": 6,
      "printed_depth": 7
     },
     "3/2": {
      "certified": true,
      "boxes": 22577,
      "deepest": 6,
      "printed_depth": 7
     },
     "3/3": {
      "certified": true,
      "boxes": 22233,
      "deepest": 6,
      "printed_depth": 7
     },
     "3/4": {
      "certified": true,
      "boxes": 26969,
      "deepest": 6,
      "printed_depth": 7
     },
     "3/5": {
      "certified": true,
      "boxes": 36169,
      "deepest": 6,
      "printed_depth": 7
     },
     "3/6": {
      "certified": true,
      "boxes": 64009,
      "deepest": 6,
      "printed_depth": 7
     }
    },
    "runtime_s": 528.0
   },
   "checks": [
    {
     "check": "the theorem as stated: H(m) - E H(T_rho f) <= 1 - H(rho) (main.tex Theorem thm:main)",
     "pass": true,
     "detail": "main.tex:54-61"
    },
    {
     "check": "Table ind:tab:parameters parsed: 7 rows, consecutive s-intervals covering [0, .84]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "r_0^2 <= delta <= r_1^2: 4r_0^2(1-r_0^2) <= s_lo^2, 4r_1^2(1-r_1^2) >= s_hi^2, r_1^2 < 1/2 on every row",
     "pass": true,
     "detail": ""
    },
    {
     "check": "U < 0 < V and L + U >= .488 on every row (min L + U = .488)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "root-derivative sign (ind:eq:root-derivative): a > 0 and a^2 - (s_hi/2L)^2 mu/Lambda >= 12133/44180000 on every row",
     "pass": true,
     "detail": "minimum 12133/44180000 (= the printed bound: True)"
    },
    {
     "check": "b(m) = twice the top-(1-|m|)/2 integral of the law of Z (induction.tex:94-117), on a 1/64 grid of [-1,1]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the knots (sc:knots) trace b(m) exactly (piecewise affine between them); b(m) <= 1 - |m|",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the 17 profiles printed in scalar.tex equal the 17 lines of data/profile_tables.txt",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the depth/base table printed in scalar.tex equals the second block of data/profile_tables.txt",
     "pass": true,
     "detail": ""
    },
    {
     "check": "every profile interval lies in one parameter group (no straddling)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "17 profiles x 5 sign checks (P, E_B, J_s1, V_s0, V_s1) certified by exact Bernstein coefficients within the printed depths; B(0) = 1, B(+-1) = 0",
     "pass": true,
     "detail": "depths used (P,E,J,V0,V1) 5:10034 17:10023 39:10011 70:10012 108:00011 160:00001 220:00000 300:00000 400:00000 490:00000 570:02000 630:03000 700:14000 750:15010 800:15010 827:16011 840:16011; all 85 equal the printed depths"
    },
    {
     "check": "base column of Table sc:tab:profile-depths: 17 rows x 128 subintervals x 3 sets x 2 signs, interval lower bound - B(eta m_D) > 0 and floor(10^4 .) >= the printed entry",
     "pass": true,
     "detail": "recomputed/printed 5:7/7 17:7/7 39:8/8 70:10/10 108:7/7 160:14/14 220:78/78 300:69/69 400:146/146 490:224/224 570:233/233 630:255/255 700:236/236 750:222/222 800:191/191 827:177/177 840:176/176"
    },
    {
     "check": "majority: E H(T_rho f)/s = (s sqrt(3+s^2) + 3 sqrt(8-5s^2+s^4))/8 (both H^2 identities), (8-5u+u^2) - (2 + 3(1-u)/4)^2 = 7(1-u)^2/16, 2s^2 + 3(2 + 3(1-s^2)/4) = 33/4 - s^2/4 >= 8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "s = 0 limits of E G/s for {7}, {7,6}, {7,6,5}: (sqrt3/4, 3/4), (sqrt2/2, 1), ((1+2sqrt2)/4, (3+sqrt2)/4)",
     "pass": true,
     "detail": "[[{3: Fraction(1, 4)}, {1: Fraction(3, 4)}], [{2: Fraction(1, 2)}, {1: Fraction(1, 1)}], [{2: Fraction(1, 2), 1: Fraction(1, 4)}, {1: Fraction(3, 4), 2: Fraction(1, 4)}]]"
    },
    {
     "check": "pair certificate (sc:interval): all 21 initial boxes certified within the depths of Table sc:tab:interval-depths",
     "pass": true,
     "detail": "798493 boxes visited; deepest level per case/group 1/0:7 1/1:7 1/2:7 1/3:6 1/4:7 1/5:7 1/6:7 2/0:7 2/1:7 2/2:7 2/3:6 2/4:7 2/5:7 2/6:7 3/0:13 3/1:6 3/2:6 3/3:6 3/4:6 3/5:6 3/6:6"
    },
    {
     "check": "the box test as evaluated here (augmented evaluation first) agrees with the paper's order (plain value first) on 54 sampled level-5/6 boxes",
     "pass": true,
     "detail": ""
    },
    {
     "check": "continuation: s >= .84 gives 0 < r = 1 - s^2 <= .2944 = 184/625",
     "pass": true,
     "detail": ""
    },
    {
     "check": "central domain: (.999)^2 < 1 - R_0^4/4; .999 - 1 + .84 = .839; (.91595)^2 < .839; s - b = m^2/(1+h) <= R_0^4/(4*1.999) < .001",
     "pass": true,
     "detail": ""
    },
    {
     "check": "calibration (cen:constants): alpha_0 series (6 terms) > 1.08834; 1/(b_0 a_0^2) - 1 < .006257 < .0063; P_0 < .018641 < .019; P_0/((1-b_0^2)a_0^2) < .053154 < .0532; R_0^3/(4*1.999)(1 + 1/(.839*.84)) < .007720 < .008",
     "pass": true,
     "detail": "1.0883484 0.0062569 0.0186410 0.0531535 0.0077190"
    },
    {
     "check": "central-band Bernstein row minima floor(1000 min_j beta_ij): P_1 order (5,12) and P_2 order (6,7) equal the printed rows, all > 0",
     "pass": true,
     "detail": "P_1 [134, 353, 365, 278, 214, 46], P_2 [179, 364, 559, 765, 983, 1214, 1457]"
    },
    {
     "check": "Lemma cen:k-bounds: E_1/2 = (1-t)^2 t(2t^2-2t-1), E_3/4 = (1-t)^2 (2t+1)(3t^2-3t+1)/2, the three derivative numerators, (5t+1)(5t-2)^2/25 = 5t^3-3t^2+.16, t^2(5t-3)+(1-t) = 5t^3-3t^2-t+1; -3t^3+6t^2+4t+2 > 0 on [0,1] (Bernstein)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(cen:T-ratio) (1 - R_0)(2/R_0 - R_0) > 4.5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "alpha < 1.1 (1 + c + .75c^2 at c = 1 - .915 = .085); (1-c)(1+c+3c^2/4) = 1 - c^2/4 - 3c^3/4; 1.21*.001/.84 < .004; .20*4.5 - .88 - .004 = .016",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(cen:resources) 1 - .0063 - .008 = .9857 > .985; (cen:mean-bound) (R_0(.543)/2 + .431/2)/.9924 < .298 < .31 with .543^2 > R_0, .431^2 > .0063/.034, .9924^2 < .985",
     "pass": true,
     "detail": ""
    },
    {
     "check": "sqrt 3 < 1.73206; 4.32 > c_0 = 2 sqrt3 + .62 and 4.32^2 > 1 + 4.32 c_0; (cen:absorption-constant) = 444890776/228150625 < 39/20 < 2",
     "pass": true,
     "detail": "444890776/228150625"
    },
    {
     "check": "intermediate small range: (184/625)^2 < 87/1000; (1-m^2) - (1 - 51m^2/100)^2 = m^2(1/50 - 2601m^2/10000), positive at m = 1/5; L(3/50) = 220141/265000 > 4/5 > 10032241/12775350; L(1/5) = 3953/6000 > 16/25 > 4504/7245",
     "pass": true,
     "detail": "10032241/12775350 4504/7245"
    },
    {
     "check": "Q_* identities (1 - Q_* = v(2-s)/(s(1+s)), 6/(1+s) - 2/s, 3v - rQ_*/2 = r/s, Q_*' numerator) at 7 rational s; Q_*(21/25) = 425/483 > .879",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Q_0(1/5)^2 = (5087/2400)^2/6 and Q_0(2/3)^2 = (4761/2500)^2/5 at a_0 = 87/1000; 25435^2 < 6*10392^2; 4761^2 < 5*2130^2; 2(m^2-m+1) >= 3/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R_0: (1+m)^2 > 3(1-m) at m = 2/3; (87/1000)(37/50)(27/8) < 1/4; 1 - (37/50)^2 > 4/9; 3 > (5/3)^2; (87/50)^5 < 36; 3(913/1000)(5/3)/(2*6) = 4565/12000 > 1/3; R_0(2/3, 87/1000)^2 = (4761/2500)^2/5 (terms 9/(5 sqrt5), 3/sqrt5)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "margins: (1+s)/2 >= 23/25; .92(.879 - .866) - .01 = 49/25000 > 0; .92(.879 - .852) - .02 = 121/25000 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "angles: pi < 22/7 (Machin enclosure); sin(21/100) >= .21 - .21^3/6 > 1/5; sin(21/25) >= .84 - .84^3/6 > 37/50; sqrt2 - 1 < 21/50; (11/7 + 21/100)/4 < 9/20; (11/7 + 21/25)/4 < 603/1000; tan(9/20) < 1/2, tan(603/1000) < 7/10",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T bounds: 1/24 < (21/100)^2, 4/5 < (9/10)^2, 37^2/1131 < (111/100)^2; small-rectangle q >= (27/50)/(5/4) = 54/125 > (13/20)^2; 1/sqrt(2(1+37/50)) = 5/sqrt87, 25/87 > (53/100)^2, 53/100 > (18/25)^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "root coefficients: 1/(2(1+13/20)^2) = 200/1089 < 19/100, 1 + 2/(1+13/20)^2 = 1889/1089 < 7/4, 1/(2(1+18/25)^2) = 625/3698 < 17/100, 1 + 2/(1+18/25)^2 = 3099/1849 < 42/25",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4 sqrt q - q^2 - 3 = -(sqrt q - 1)^2 (q + 2 sqrt q + 3) (i.e. -(q-1)^2 (1 + 2/(1+sqrt q)^2)); the root identity (mid:root-identity); (1 - 33y/10)^2(1+y) - (1-3y)^2 = y(2/5 - 471y/100 + 1089y^2/100), 2/5 - (471/100)(441/10000) > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "positive series: (1-y)^(-1/4), C(4j,2j)/16^j (1-y)^(-j-1/4) through y^2, averages u^2 -> 1/3, u^4 -> 1/5, t/sin t = 1 + Y/6 + 3Y^2/40 + ..., giving P_0 = 1 + G^2/2 + Y(173/360 + 11G^2/24 + 7G^4/32)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table mid:certificate-table: the six minima floor(C min B_ij) recomputed equal the printed 90, 27, 293, 120, 53, 26 (degrees within the printed orders); every nonzero term of S has i+j even >= 2",
     "pass": true,
     "detail": "U1-1:90 U2-1:293 U11:27 U21:120 L1/100:53 L1/50:26"
    },
    {
     "check": "c_1..c_10 = 1/2, 1/8, 1/16, 5/128, 7/256, 21/1024, 33/2048, 429/32768, 715/65536, 2431/262144 (recurrence and C(2k,k)/((2k-1)4^k)); remaining mass 46189/262144",
     "pass": true,
     "detail": ""
    },
    {
     "check": "log brackets: S_15(51/25) + (51/25)^16/(16!(1 - (51/25)/17)) < 7691/1000 < 100/13 < 7698/1000 < S_15(2041/1000); .361^2 > .13; (3/2)/(1 + (184/625)/2) = 625/478 > 1.3; 1 - 2s - s^2 < 0 at s = .84",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table lb:arithmetic-table: ell, R, Bbar, Sbar, Tbar recomputed equal the printed integers on all four intervals; C <= the rounded total = the printed fraction < .982, .969, .954, .940 < 1",
     "pass": true,
     "detail": "746521967/60049597/2525240 748896408/56019174/1084482 750555827/51401364/283275 751524751/46967364/36923"
    }
   ],
   "sources": {
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/induction.tex": "1a8ce2bbd7dd38bec9c688cbc9b626abcd21218065ff0caf06dc91842ee3d9cf",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/scalar.tex": "8b7936e2e2aead6a7f10a1d9fe283d2b9345be59ab93278c5e875c80cdff73e5",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/verification/data/profile_tables.txt": "0316dfd9df24557cbc73f3485ab9175d4224ebadc42b703a53541db8a743607e",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/central.tex": "4299dce6a51124a18754cbd964adfeeded192d145ff78d5518dce6067a63f860",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/intermediate.tex": "6449ecf10b7bd68bd4a7b8ea3eb0d640ddd176d9d37379695272ea28b02d4664",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/large-bias.tex": "12e31ef806052b4c6cddcb519e6bcb56d14439a9b2d90aca72077a87485a7d83",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/main.tex": "b286e05e0e9321c4a0cb925734a078b127e9d8694e8fccc5d4f6a484bfb0347b",
    "preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/build/sections/continuation.tex": "e66a6045882baf985dd9ea7569ffed62b9ee3b716c90a7f6f8f3b4c15bde5521"
   },
   "forges": [
    {
     "forge": "profile 5: c_17 = -28650 -> 471350 (a sign check fails within its depth)",
     "verdict": "REFUTED"
    },
    {
     "forge": "base column: row 220 printed 79 instead of 78",
     "verdict": "REFUTED"
    },
    {
     "forge": "central band: P_1 row minimum 46 printed as 47",
     "verdict": "REFUTED"
    },
    {
     "forge": "large bias: Bbar of [.84,.88] printed one larger",
     "verdict": "REFUTED"
    },
    {
     "forge": "intermediate table: U_{1,+} minimum printed 28 instead of 27",
     "verdict": "REFUTED"
    },
    {
     "forge": "pair certificate alone, Case 1 of parameter row 3 with mu doubled (1.024 -> 2.048), printed depth 7",
     "verdict": "REFUTED"
    },
    {
     "forge": "pair certificate alone, Case 1 of parameter row 0 with its printed depth 8 replaced by 3",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 528.0
  },
  {
   "id": "F-248",
   "index": 248,
   "family": "130",
   "title": "An explicit power saving for the exact discrete Fourier transform",
   "tier": "C",
   "dir": "An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026",
   "leanChallenges": [
    "ExactFourier",
    "UniformFourier"
   ],
   "word": "NEEDS DATA",
   "why": "Interface and lemma in build/sections/synthesis.tex lines 18-60 (Lemma syn:local-certificate at lines 39-46), built from the network of build/sections/network.tex (matrix eq. net:C at line 6; Prop. net:finite-interface at line 370; parameters m=10^6, W_*=2^71 at network.tex line 470 and introduction.tex line 71). The 2^b-coordinate word is never printed and is astronomically large.",
   "decider": null
  },
  {
   "id": "F-262",
   "index": 262,
   "family": "138",
   "title": "Subset Sum in Time $`O(2^{0.49n})`$",
   "tier": "A",
   "dir": "Subset-Sum-in-Time-2-power-0-49n-October-4-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: the numerical certificate of Proposition prop:filtering-moments (Appendix app:numerical) - every printed enclosure, error bound and comparison recomputed in directed rational interval arithmetic; the analytic trapezoid estimate, contour shifts and Parseval monotonicity are read, not run, and the O(2^0.49n) algorithm is theory",
   "decider": "instruments/openaimath/finite/f262_subset_sum.py",
   "decides": "a finite component: the numerical certificate of Proposition prop:filtering-moments (Appendix app:numerical) - every printed enclosure, error bound and comparison recomputed in directed rational interval arithmetic; the analytic trapezoid estimate, contour shifts and Parseval monotonicity are read, not run, and the O(2^0.49n) algorithm is theory",
   "value": {
    "Z_f": "[1.19636252831, 1.19636252831]",
    "Z_g": "[1.2877406567, 1.2877406567]",
    "Z_fk": "[1.21107898762, 1.21107898762]",
    "Z_gk": "[1.30381114921, 1.30381114921]",
    "T64(2^2r_b)": "[2.49015139011, 2.49015139011]",
    "T64(2^2s_b)": "[2.26072002809, 2.26072002809]",
    "H^": "[0.0158910314229, 0.0158910314234]",
    "mean X^": "[1.42896843477, 1.42896843478]",
    "mean X'^": "[1.42884700931, 1.42884700932]",
    "scale": "S = 10^60",
    "runtime_s": 0.42
   },
   "checks": [
    {
     "check": "every bound decided below is printed in the paper as quoted (35 statements)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the published object read: 6 + 6 coefficients, 6 normalizer centers, rows 0..32 of the node table, 17 cosines",
     "pass": true,
     "detail": "r = ['851/1000', '233/500', '13/40', '253/1000', '28/125', '-157/500'], s = ['1281/1000', '-469/500', '731/1000', '-559/1000', '391/1000', '11/100'], kappa = 2063/2000"
    },
    {
     "check": "Delta = eta - tau = .02776 and Gamma = 2 tau - eta = .46220 (filtering.tex:63-65)",
     "pass": true,
     "detail": "347/12500, 2311/5000"
    },
    {
     "check": "the real sup bounds |r| <= 2.433, |s| <= 4.010 (filtering.tex:137): sum |r_j| and sum |s_j|, exactly",
     "pass": true,
     "detail": "2433/1000, 401/100"
    },
    {
     "check": "pi/4 = arctan(1/2) + arctan(1/3) through degree 159 gives 3.1415926535897 < pi < 3.1415926535899",
     "pass": true,
     "detail": "width 8.5e-51"
    },
    {
     "check": "the log recipe gives .693147180559 < ln 2 < .693147180561",
     "pass": true,
     "detail": "[0.69314718056, 0.69314718056]"
    },
    {
     "check": "the 17 printed cosines cos(2 pi j/64), j = 0..16, lie within 3e-10 of the enclosures (numerical.tex:138-146)",
     "pass": true,
     "detail": "max enclosure width 1.0e-13; off: []"
    },
    {
     "check": "every exponential argument of the finite tables has |t| <= 8 (numerical.tex:167)",
     "pass": true,
     "detail": "max |t| = 5.9384 over 1036 evaluations"
    },
    {
     "check": "first table: Zf = 95709/80000 within 5e-7",
     "pass": true,
     "detail": "computed [1.19636252831, 1.19636252831], |center - printed| = 2.83e-08"
    },
    {
     "check": "first table: Zg = 12877407/10000000 within 5e-7",
     "pass": true,
     "detail": "computed [1.2877406567, 1.2877406567], |center - printed| = 4.33e-08"
    },
    {
     "check": "first table: Zfk = 1211079/1000000 within 5e-7",
     "pass": true,
     "detail": "computed [1.21107898762, 1.21107898762], |center - printed| = 1.24e-08"
    },
    {
     "check": "first table: Zgk = 13038111/10000000 within 5e-7",
     "pass": true,
     "detail": "computed [1.30381114921, 1.30381114921], |center - printed| = 4.92e-08"
    },
    {
     "check": "first table: Tr = 12450757/5000000 within 5e-6",
     "pass": true,
     "detail": "computed [2.49015139011, 2.49015139011], |center - printed| = 9.89e-09"
    },
    {
     "check": "first table: Ts = 28259/12500 within 5e-6",
     "pass": true,
     "detail": "computed [2.26072002809, 2.26072002809], |center - printed| = 2.81e-08"
    },
    {
     "check": "node table column P^_j: all 33 printed centers within 4e-6 of the enclosures",
     "pass": true,
     "detail": "largest |computed - printed| = 4.94e-08 at j = 0; rows not passing: []"
    },
    {
     "check": "node table column Y^_j: all 33 printed centers within 2e-5 of the enclosures",
     "pass": true,
     "detail": "largest |computed - printed| = 4.97e-08 at j = 14; rows not passing: []"
    },
    {
     "check": "node table column X^_j: all 33 printed centers within .0004 of the enclosures",
     "pass": true,
     "detail": "largest |computed - printed| = 4.88e-06 at j = 19; rows not passing: []"
    },
    {
     "check": "node table column X^'_j: all 33 printed centers within .0004 of the enclosures",
     "pass": true,
     "detail": "largest |computed - printed| = 4.95e-06 at j = 13; rows not passing: []"
    },
    {
     "check": "all four columns invariant under j -> 64 - j (enclosures at j and 64 - j overlap, j = 1..31)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "X^_32 = X'^_32 = 0 exactly (M(pi) = 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:hatted-average-bound, the paper's way: (1/64)(weighted table centers of X^) + .0004 = 1.429368 < 2859/2000",
     "pass": true,
     "detail": "9147957/6400000"
    },
    {
     "check": "eq:hatted-average-bound, the paper's way: (1/64)(weighted table centers of X'^) + .0004 = 1.429247 < 2859/2000",
     "pass": true,
     "detail": "9147181/6400000"
    },
    {
     "check": "eq:hatted-average-bound directly: (1/64) sum_{j<64} X^_j < 2859/2000",
     "pass": true,
     "detail": "[1.42896843477, 1.42896843478]"
    },
    {
     "check": "eq:hatted-average-bound directly: (1/64) sum_{j<64} X'^_j < 2859/2000",
     "pass": true,
     "detail": "[1.42884700931, 1.42884700932]"
    },
    {
     "check": "strip |Im z| <= .45: |r(z)| <= coefficient sum < 6.50",
     "pass": true,
     "detail": "[6.48269504689, 6.48269504689]"
    },
    {
     "check": "strip |Im z| <= .45: |s(z)| <= coefficient sum < 8.71",
     "pass": true,
     "detail": "[8.69639812025, 8.69639812025]"
    },
    {
     "check": "strip |Im z| <= .45: |2 r_beta(z)| <= coefficient sum < 18.4",
     "pass": true,
     "detail": "[18.2361485492, 18.2361485492]"
    },
    {
     "check": "strip |Im z| <= .45: |2 s_beta(z)| <= coefficient sum < 22.4",
     "pass": true,
     "detail": "[22.2082207442, 22.2082207442]"
    },
    {
     "check": "eq:inner-quadrature-errors, Z_f, Z_g, Z_{f,k}, Z_{g,k} (B_0 = 2^{kappa 8.71}): 2 B_0/(e^{28.8} - 1) < 4e-10",
     "pass": true,
     "detail": "[3.14711254279e-10, 3.14711254279e-10]"
    },
    {
     "check": "eq:inner-quadrature-errors, P(x_j) (B_0 = 2^{6.50+8.71}): 2 B_0/(e^{28.8} - 1) < 3e-8",
     "pass": true,
     "detail": "[2.35513144413e-08, 2.35513144413e-08]"
    },
    {
     "check": "eq:inner-quadrature-errors, Y(x_j) (B_0 = 15.21 x 2^{15.21}): 2 B_0/(e^{28.8} - 1) < 5e-7",
     "pass": true,
     "detail": "[3.58215492653e-07, 3.58215492653e-07]"
    },
    {
     "check": "eq:inner-quadrature-errors, int 2^{2 r_beta}, int 2^{2 s_beta} (B_0 = 2^{22.4}): 2 B_0/(e^{28.8} - 1) < 5e-6",
     "pass": true,
     "detail": "[3.43891007689e-06, 3.43891007689e-06]"
    },
    {
     "check": "the four normalizer strip bound covers F, G, F^k, G^k: max(6.50, 8.71, k 6.50, k 8.71) = k 8.71",
     "pass": true,
     "detail": ""
    },
    {
     "check": "P^_j > 1.06789 at every node",
     "pass": true,
     "detail": "min lower endpoint 1.067899371"
    },
    {
     "check": "eq:node-score-errors |L - L^| < 1e-7 (propagated: 4.146e-08)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:node-score-errors |B - B^| < 2e-6 (propagated: 4.737e-07)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:node-score-errors |H - H^| < 5e-9 (propagated: 1.879e-09)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "true/hatted first integrand <= 2^{8.5(1e-7 + .0315 x 2e-6 + 5e-9)} < 1.00001",
     "pass": true,
     "detail": "[1.00000098981, 1.00000098981]"
    },
    {
     "check": "true/hatted second integrand <= 2^{8.3((1+2a)1e-7 + |a(2-k)-(k-1)| 2e-6 + (1+a)5e-9)} < 1.00001",
     "pass": true,
     "detail": "[1.00000098177, 1.00000098177]"
    },
    {
     "check": "eq:shifted-l2: Z_f^-2 int 2^{2 r_beta} <= (T64 + 5e-6)/(Z^_f - 4e-10)^2 < 1.761",
     "pass": true,
     "detail": "1.739807"
    },
    {
     "check": "eq:shifted-l2: Z_g^-2 int 2^{2 s_beta} < 1.38",
     "pass": true,
     "detail": "1.363298"
    },
    {
     "check": "eq:Q-strip: .761 x .38 < .54^2 (so |Q - 1| < .54, |Q| > .46)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "-log_2 .46 < 1.121",
     "pass": true,
     "detail": "[1.12029423372, 1.12029423372]"
    },
    {
     "check": "strip |Im| <= .19: |r| <= sum |r_j| cosh(.19 j) < 2.96",
     "pass": true,
     "detail": "[2.94938055377, 2.94938055377]"
    },
    {
     "check": "strip |Im| <= .19: |s| <= sum |s_j| cosh(.19 j) < 4.67",
     "pass": true,
     "detail": "[4.66277270584, 4.66277270584]"
    },
    {
     "check": "eq:B-strip: log Z_f + log Z_g > 0 and 1.761 x 1.38 x (2.96 + 4.67 + log Z_f + log Z_g)^2 < 13^2",
     "pass": true,
     "detail": "squared bound 165.5444"
    },
    {
     "check": "H < .016 (H^ + 5e-9; H^ = [0.0158910314229, 0.0158910314234])",
     "pass": true,
     "detail": "0.015891036"
    },
    {
     "check": "8.5(.02776 + 1.121 + .0315 x 13/.46 + .016) < 17.6, with .02776 = Delta and .0315 = kappa - 1",
     "pass": true,
     "detail": "17.46731"
    },
    {
     "check": "av - u = -(1+2a)L + (a(2-k)-(k-1))B + (1+a)H (coefficients of a(v) - u with u, v from the score identities), and 8.3(...) < 17.6",
     "pass": true,
     "detail": "16.81826"
    },
    {
     "check": "2 + 2 cosh(.38) < 4.16",
     "pass": true,
     "detail": "[4.14614599865, 4.14614599865]"
    },
    {
     "check": "4.16 x 2^17.6 < 2^20, i.e. 4.16^5 < 2^12",
     "pass": true,
     "detail": "1245.8525720576"
    },
    {
     "check": "2^21/(e^{64 x .38} - 1) < .00007",
     "pass": true,
     "detail": "[5.7489444081e-05, 5.7489444081e-05]"
    },
    {
     "check": "1.4295(1.00001) + .00007 = 1.429584295 exactly",
     "pass": true,
     "detail": "285916859/200000000"
    },
    {
     "check": "2^{.5168} > 1.4307781 > 1.429584295 (so both filtering moments are < 2^{.5168})",
     "pass": true,
     "detail": "[1.4307781583, 1.4307781583]"
    }
   ],
   "sources": {
    "preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/build/sections/numerical.tex": "a4cc12248568d90368197466f562944f119b8f68535f72c576dd2f342b33b6dd",
    "preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/build/sections/filtering.tex": "17443991a51fa09b86c8523ec4cada86cb32833d37b9e9fef0b38b56eb814b8e"
   },
   "forges": [
    {
     "forge": "node table: P^_5 printed 1.7323079 -> 1.7323179 (moved by 1e-5, allowed error 4e-6)",
     "verdict": "REFUTED"
    },
    {
     "forge": "published coefficient r_6 = -.314 -> -.315",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed cos x_5 = .8819212643 -> .8819212653 (moved by 1e-9, allowed 3e-10)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.4
  },
  {
   "id": "F-287",
   "index": 287,
   "family": "153",
   "title": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions",
   "tier": "A",
   "dir": "Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: the root location of the degree-31 polynomial (all 31 roots isolated, beta in (1.8392, 1.8393) the only real root, alpha the only other expanding pair, beta not Pisot) and every printed integer of the rational moment certificate with its coefficient and trigonometric inputs; NOT the singularity of nu_{1/beta}, which rests on the analytic argument",
   "decider": "instruments/openaimath/finite/f287_bernoulli_non_pisot.py",
   "decides": "a finite component: the root location of the degree-31 polynomial (all 31 roots isolated, beta in (1.8392, 1.8393) the only real root, alpha the only other expanding pair, beta not Pisot) and every printed integer of the rational moment certificate with its coefficient and trigonometric inputs; NOT the singularity of nu_{1/beta}, which rests on the analytic argument",
   "value": {
    "schur_cohn_counts": {
     ".994^2": 28,
     "1.0002089": 28,
     "1.0002094": 30,
     "1.8392^2": 30,
     "1.8393^2": 31
    },
    "B_numerators": [
     10861284406435197401851134503,
     14731866010915969694738277129,
     3155364376927094122833206055
    ],
    "rounded_products": 4758,
    "R": 753,
    "c~": "0.000211815537786",
    "v~": "0.00014392088425",
    "T~": "0.000105303788239"
   },
   "checks": [
    {
     "check": "A. the polynomial as printed in nonpisot.tex:15 and certificate.tex:21",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. (t^2+t+1) p(t) = t^33 - t^32 - t^31 - t^30 - 1 (nonpisot.tex:20)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. p is monic in Z[t] with |p(0)| = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. (not claimed; a second route to non-Pisot) p mod 67 is irreducible (Rabin), so p is irreducible over Q",
     "pass": true,
     "detail": "the paper avoids irreducibility (nonpisot.tex:95)"
    },
    {
     "check": "B. the table gives 16 centres, one real, hence 31 disks",
     "pass": true,
     "detail": "16 centres"
    },
    {
     "check": "B. Rouche (exact Taylor coefficients): exactly one root in each of the 31 disks",
     "pass": true,
     "detail": "31 / 31 disks pass"
    },
    {
     "check": "B. the 31 disks are pairwise disjoint (min centre distance > 2 delta)",
     "pass": true,
     "detail": "min |z - z'| = 1.847e-01"
    },
    {
     "check": "B. printed: |z - z'| > .01 for distinct centres, including conjugates (certificate.tex:72-73)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. the 30 nonreal disks miss the real axis (|Im z| > delta)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. printed (cert:residual): |z| < 2, ||p'(z)||_inf > 1, 2e35 ||p(z)||_1 < ||p'(z)||_inf at every centre",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. printed: sum_j |p_j| 3^j < 1e17 (certificate.tex:84)",
     "pass": true,
     "detail": "1061131219257037"
    },
    {
     "check": "B. printed (cert:center-bounds): 1.83928 < z_beta < 1.83929",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. printed (cert:center-bounds): 1.0002090 < |z_alpha|^2 < 1.0002093",
     "pass": true,
     "detail": "1.0002092099"
    },
    {
     "check": "B. printed (cert:center-bounds): |z| < .9939 at the 14 other centres",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Prop np:roots: 1.8392 < beta < 1.8393 (beta in the real disk)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Prop np:roots: 1.0002089 < |alpha|^2 < 1.0002094 (|z_a|^2 - 4d > lower, |z_a|^2 + 4d + d^2 < upper)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Prop np:roots: |alpha|^-1 < .9999 (|alpha|^2 > 1.0002089 > 1/.9999^2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Prop np:roots: Im alpha > 0 (z_alpha has Im > delta)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Prop np:roots: the other 28 roots have modulus < .994 (|z| + delta < .994)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. outside the unit circle exactly beta, alpha, conj(alpha) (|z| - delta > 1 for those three)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. the paper's non-Pisot step: 1.0002094 * .994 < 1 (nonpisot.tex:65)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. self-test of both tools on (2t-1)(t-3)(5t+1)(4t^2+1): Sturm finds 3 real roots, 1 in (0, 1]; Schur-Cohn finds 4 roots with |r| < 1, 1 with |r|^2 < 1/5, 4 with |r|^2 < 1/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. Sturm: p is squarefree (last chain term a nonzero constant) and has exactly one real root",
     "pass": true,
     "detail": "chain length 11"
    },
    {
     "check": "C. Sturm: that root lies in (1.8392, 1.8393]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. Schur-Cohn on the Graeffe square: #{r : |r|^2 < .994^2} = 28",
     "pass": true,
     "detail": "28"
    },
    {
     "check": "C. Schur-Cohn on the Graeffe square: #{r : |r|^2 < 1.0002089} = 28",
     "pass": true,
     "detail": "28"
    },
    {
     "check": "C. Schur-Cohn on the Graeffe square: #{r : |r|^2 < 1.0002094} = 30",
     "pass": true,
     "detail": "30"
    },
    {
     "check": "C. Schur-Cohn on the Graeffe square: #{r : |r|^2 < 1.8392^2} = 30",
     "pass": true,
     "detail": "30"
    },
    {
     "check": "C. Schur-Cohn on the Graeffe square: #{r : |r|^2 < 1.8393^2} = 31",
     "pass": true,
     "detail": "31"
    },
    {
     "check": "D. cert:exact-output: |R| = 753 equals the printed integer",
     "pass": true,
     "detail": "computed 753, printed 753"
    },
    {
     "check": "D. cert:exact-output: |X|+|Y|+sum|w| = 198442946/W equals the printed integer",
     "pass": true,
     "detail": "computed 198442946, printed 198442946"
    },
    {
     "check": "D. cert:exact-output: |X|+|Y|+sum_{k<50}|w| = 11816095/W equals the printed integer",
     "pass": true,
     "detail": "computed 11816095, printed 11816095"
    },
    {
     "check": "D. cert:exact-output: W S c~ equals the printed integer",
     "pass": true,
     "detail": "computed 211815537786494774284357596751188225, printed 211815537786494774284357596751188225"
    },
    {
     "check": "D. cert:exact-output: W^2 S v~ equals the printed integer",
     "pass": true,
     "detail": "computed 143920884250294630908494564120361905955993456, printed 143920884250294630908494564120361905955993456"
    },
    {
     "check": "D. cert:exact-output: W^2 S T~ equals the printed integer",
     "pass": true,
     "detail": "computed 105303788239450118397416374596041413014338334, printed 105303788239450118397416374596041413014338334"
    },
    {
     "check": "D. the printed small-row table (certificate.tex:558-562) equals the computed nonzero rows with k < 50",
     "pass": true,
     "detail": "[(32, 0, -536932), (36, 1, 1265024), (37, 0, 331701), (39, 1, 47119), (40, 0, 1483338), (43, 0, 436861), (44, 1, 458216), (47, 1, 756904)]"
    },
    {
     "check": "D. cert:array-bounds: max |c_A(m)| < .847 and .8460 < max |c_A(m)| < .8462",
     "pass": true,
     "detail": "0.846098"
    },
    {
     "check": "D. cert:array-bounds: max_{m >= -20} |c_0(m)| < .018",
     "pass": true,
     "detail": "0.017264"
    },
    {
     "check": "D. cert:array-bounds: max_A |u'_A| < .0087 (squared moduli)",
     "pass": true,
     "detail": "0.008596"
    },
    {
     "check": "D. stored pairs: |a_z| < 1 and |v_z| <= 1 (certificate.tex:268-270)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. |a'| > .99 (certificate.tex:353)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. rounded angles |x| <= 13, |y| <= 170 on every table entry (certificate.tex:299)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. cert:B-output: max final numerator of B2 equals the printed integer",
     "pass": true,
     "detail": "computed 10861284406435197401851134503, printed 10861284406435197401851134503"
    },
    {
     "check": "D. cert:B-output: max final numerator of B1 equals the printed integer",
     "pass": true,
     "detail": "computed 14731866010915969694738277129, printed 14731866010915969694738277129"
    },
    {
     "check": "D. cert:B-output: max final numerator of B0 equals the printed integer",
     "pass": true,
     "detail": "computed 3155364376927094122833206055, printed 3155364376927094122833206055"
    },
    {
     "check": "D. B-output x 1e6/S <= 10862, 14732, 3156, hence B_2 < .011, B_1 < .015, B_0 < .0032 (cert:B-bounds)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. 211810 < 1e9 c~ < 211820, 143910 < 1e9 v~ < 143930, 105300 < 1e9 T~ < 105310 (certificate.tex:493-495)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. |G| <= weight sum < .199 (nonpisot.tex:219-221)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. E Re(Z G) >= c~ - .199 Delta > .0002108205 > .000210 (certificate.tex:502)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. E|G|^2 <= v~ + .199^2 Delta < .000144119 < .000145 (certificate.tex:503)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. T <= T~ + .199^2 Delta < .000116 and .011^2 > .000116, so sqrt T < .011 (certificate.tex:521)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. small-row absolute weight 11816095/W < .012 (certificate.tex:523)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the final sum .000116(.011) + 2(.012)(.011)(.015) + .012^2(.0032) = .0000056968 < .000007 (certificate.tex:667-672)",
     "pass": true,
     "detail": "7121/1250000000"
    },
    {
     "check": "E. Machin: |pi' - (16 A_30(1/5) - 4 A_10(1/239))| + 16/(61 5^61) + 4/(21 239^21) < 1/S (certificate.tex:129-135)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. |p''(z)| <= sum j(j-1)|p_j| 2.001^(j-2) < 4e13 for |z| < 2.001 (certificate.tex:263)",
     "pass": true,
     "detail": "1.0637e+12"
    },
    {
     "check": "E. 4e13 * 1e-35 <= 4e-22 (p' moves less than 4e-22 within a disk, certificate.tex:265)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 1 - .9999^2 > .00019 (certificate.tex:363)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 3.142 * 4 * .00871 < .11 (certificate.tex:379)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. P <= .11^2/(2(1-.9999^2)) + .11^4/(12(1-.9999^4)) + .11^6/(45(1-.9999^6)) < 31 (certificate.tex:384-386)",
     "pass": true,
     "detail": "30.2821"
    },
    {
     "check": "E. .1 (.11)^8 / (1 - .9999^8) < 3e-6 (certificate.tex:419)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. log-cos lemma: h0 + h0^2/2 + h0^3/3 = y/2 + y^2/12 + y^3/45 + (terms of degree >= 4 with sum|coef| < .03)",
     "pass": true,
     "detail": "11304991/1119744000"
    },
    {
     "check": "E. log-cos lemma: h^4/(4(1-h)) < .016 y^4 (h <= y/2 <= .00605) and (1 + h + h^2)/40320 < .00004",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. log-cos lemma: .016 + .00004 + .03 <= .1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. negative end: .994^2996 < 3e-8 (certificate.tex:331)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. negative end: 3.142^2 200^2/2 * .994^5994/(1 - .994^2) < 5e-8 (certificate.tex:328-329)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. negative end: two coefficients with three shifts each, 2 * 3 * 32 <= 200 (certificate.tex:321-323)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. positive end: 3.142 * 8 * .545^100 / (1 - .545) < 1e-20 (certificate.tex:337)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. positive end: 1/beta < 1/1.8392 < .545",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. finite factors: 2 * 5100 * 1e-15 * (1+1e-15)^5100 + 5100 * (1/2S) * (1+1e-15)^5100 < 3e-11 (certificate.tex:312-316)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the stated errors 3e-11 + 5e-8 + 1e-20 + 4e-9 + 3e-6 < 3.055e-6 < Delta = 5e-6 (certificate.tex:424-425)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. a paired factor from table entries within 1e-16: 1e-16 (2 + 2(1 + 1e-16)) + 1/(2S) < 1e-15",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. tangent bound: u = 3.142(.018 + 7e-20), tan u <= u/(1 - u^2/2) <= t = .06 (certificate.tex:533-537)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. B-factor constants: (1-t^2)^(-1/2) <= 1.002, (1-t^2)^(-1) <= 1.004, 1 + t^2 = 1.0036, 2t = .12 (certificate.tex:577-578)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. own pi enclosure: |pi' - pi| < 1/S and pi < 3.142",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. stored pairs vs the true roots: base errors < 2/S, coefficient errors < 1e-21 (certificate.tex:266-268)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. the exact coefficients have modulus < 1.001 (certificate.tex:270)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. cert:coefficient-errors: |c_0(m) - C_0(m)| < 7e-20 on all 7111 computed entries",
     "pass": true,
     "detail": "max 9.916e-29"
    },
    {
     "check": "F. cert:coefficient-errors: |c_1(m) - C_1(m)| < 2.1e-19 on all 7107 computed entries",
     "pass": true,
     "detail": "max 1.668e-28"
    },
    {
     "check": "F. |a| > .99 for a = 1/alpha (certificate.tex:353)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. |u_A| <= .00871 for both types (certificate.tex:354)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. |u'_A - u_A| < 5e-21 (certificate.tex:353-354)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. sum_{r in I} |1/p'(r)| <= 32, so |C_0(m)| <= 32(.994)^(-1-m) for m < 0 (certificate.tex:319)",
     "pass": true,
     "detail": "1.3466"
    },
    {
     "check": "F. beta: base 1/beta < .545, type-0 and type-1 coefficients of modulus <= 4 (certificate.tex:334-335)",
     "pass": true,
     "detail": "|coef0| = 7.109e-09"
    },
    {
     "check": "F. max_{-20 <= n <= 2000} |C_0(n)| < .018 (true values; the tangent bound for big rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "F. every table entry g_A(m), h_A(m) is within 1e-16 of cos, sin(pi C_A(m)) (certificate.tex:308-309)",
     "pass": true,
     "detail": "max 5.241e-28"
    },
    {
     "check": "F. rounded angles of the B-product inputs (sums of up to four coefficients): |x| <= 13, |y| <= 170 (certificate.tex:297-299)",
     "pass": true,
     "detail": "max |x| = 5.078, |y| = 25.784"
    },
    {
     "check": "F. every input of the B products: |g(V'(m))| and |h(V'(m))| within 1e-15 of |cos|, |sin|(pi V(m)) (certificate.tex:579)",
     "pass": true,
     "detail": "11836 inputs, max 3.715e-28"
    }
   ],
   "sources": {
    "preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/build/certificate.tex": "7d71cb685bcf4b33dd0ce8a9ae34dd0be0afdff28596d77f2515f5ee638ea47d",
    "preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/build/nonpisot.tex": "e8d6e8537dc60f4142f629ac4c4e400efe74f6aed98a90f52630533b8955fb1a"
   },
   "forges": [
    {
     "forge": "the real centre moved by 1e-34 (ten radii) along the axis",
     "verdict": "REFUTED"
    },
    {
     "forge": "the alpha centre moved by 1e-33 in its imaginary part",
     "verdict": "REFUTED"
    },
    {
     "forge": "W S c~ printed one unit larger",
     "verdict": "REFUTED"
    },
    {
     "forge": "the k = 40 small-row weight printed as 1483339",
     "verdict": "REFUTED"
    },
    {
     "forge": "the B_0 numerator printed one unit smaller",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 25.4
  },
  {
   "id": "F-294",
   "index": 294,
   "family": "157",
   "title": "A counterexample to Hadwiger's conjecture",
   "tier": "C",
   "dir": "A-counterexample-to-Hadwigers-conjecture-September-23-2026",
   "leanChallenges": [
    "HadwigerCounterexample",
    "ListHadwiger"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. Theorem thm:main in build/sections/01-introduction.tex lines 20-27; sampling of raw vertices in build/sections/02-geometry.tex lines 258-320; parameter order in build/sections/14-parameters.tex.",
   "decider": null
  },
  {
   "id": "F-295",
   "index": 295,
   "family": "157",
   "title": "A counterexample to the Colin de Verdière chromatic conjecture",
   "tier": "C",
   "dir": "A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026",
   "leanChallenges": [
    "HadwigerCounterexample",
    "ListHadwiger"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. Theorem thm:main at build/sections/01-introduction.tex line 42; base graphs Theorem thm:base-graphs at line 147; 'm=2^{1000gN} is the sampled graph order' at lines 315-317.",
   "decider": null
  },
  {
   "id": "F-297a",
   "index": 297,
   "family": "158",
   "title": "The Euclidean plane is not five-colorable",
   "tier": "A",
   "dir": "The-Euclidean-plane-is-not-five-colorable-September-23-2026",
   "leanChallenges": [
    "EuclideanFiveColor",
    "PlaneColoring"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Lemma angular-certificate (all seven placed Moser-spindle vertices strictly in the region 0<l<4, P<P'), the eleven unit edges and non-3-colourability, and every number of the paper's rational verification. NOT the headline (the plane is not 5-colourable), which rests on the transfer theorem and the measurable/angular theory.",
   "decider": "instruments/openaimath/finite/f297a_plane_seven_points.py",
   "decides": "a finite component: Lemma angular-certificate (all seven placed Moser-spindle vertices strictly in the region 0<l<4, P<P'), the eleven unit edges and non-3-colourability, and every number of the paper's rational verification. NOT the headline (the plane is not 5-colourable), which rests on the transfer theorem and the measurable/angular theory.",
   "value": {
    "isolation_digits": 8,
    "lemma_failed": false,
    "margins": {
     "0": {
      "l": "1.700816",
      "P": "0.866124",
      "P'": "1.38897"
     },
     "A": {
      "l": "1.287637",
      "P": "0.0895503",
      "P'": "0.119773"
     },
     "B": {
      "l": "0.095637",
      "P": "1.68999e-06",
      "P'": "0.00069832"
     },
     "T": {
      "l": "0.682458",
      "P": "0.0221255",
      "P'": "0.0265819"
     },
     "uA": {
      "l": "2.735008",
      "P": "9.72524",
      "P'": "19.4763"
     },
     "uB": {
      "l": "0.459246",
      "P": "0.00199268",
      "P'": "0.025071"
     },
     "uT": {
      "l": "2.493438",
      "P": "12.5744",
      "P'": "13.0226"
     }
    }
   },
   "checks": [
    {
     "check": "the headline as stated (main.tex abstract, introduction.tex Theorem thm:main)",
     "pass": true,
     "detail": "NOT decided here; this row decides the seven-point certificate only"
    },
    {
     "check": "the lemma, the region, the vertex set and the placement as printed (angular.tex:484-550, 577-580)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the three printed tables parse to 7 rows each, and seven squared-radius intervals",
     "pass": true,
     "detail": "7 7 7 7"
    },
    {
     "check": "|w|^2 = 1 for w = (35+12i)/37 (35^2 + 12^2 = 37^2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|u|^2 = (25+11)/36 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|1-u|^2 = (1+11)/36",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|T-uT|^2 = 3|1-u|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "printed coordinates of uA (angular.tex:665-669)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "printed coordinates of uB (angular.tex:665-669)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "printed coordinates of uT (angular.tex:665-669)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the figure draws eleven distinct edges",
     "pass": true,
     "detail": "0-A 0-B 0-uA 0-uB A-B A-T B-T T-uT uA-uB uA-uT uB-uT"
    },
    {
     "check": "unit edge 0-A: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge 0-B: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge 0-uA: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge 0-uB: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge A-B: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge A-T: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge B-T: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge T-uT: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge uA-uB: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge uA-uT: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "unit edge uB-uT: |g-h|^2 = 1 and |z_g - z_h|^2 = 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the seven points are distinct, and exactly 11 of the 21 pairs are at distance 1",
     "pass": true,
     "detail": "unit pairs 11"
    },
    {
     "check": "every proper 3-colouring of the rhombus 0,A,B,T has c(0) = c(T) (all 81 tried)",
     "pass": true,
     "detail": "6 proper"
    },
    {
     "check": "the eleven-edge graph has no proper 3-colouring (all 3^7 = 2187 tried)",
     "pass": true,
     "detail": "0 proper"
    },
    {
     "check": "Lemma angular-certificate at z_0: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_0: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_0: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_A: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_A: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_A: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_B: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_B: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_B: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_T: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_T: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_T: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uA: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uA: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uA: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uB: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uB: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uB: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uT: 0 < |z_g|^2",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uT: |z_g|^2 < 4",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "Lemma angular-certificate at z_uT: P < P'",
     "pass": true,
     "detail": "exact sign +1, isolation sign +1 at 8 digits"
    },
    {
     "check": "tab:angular-coordinates 0: |xi - (-6453/5000)| < h exactly",
     "pass": true,
     "detail": "xi ~ -1.2905946"
    },
    {
     "check": "tab:angular-coordinates 0: |upsilon - (469/2500)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 0.1875676"
    },
    {
     "check": "tab:angular-coordinates 0: P < P_+ = 9/10 exactly",
     "pass": true,
     "detail": "P ~ 0.866"
    },
    {
     "check": "tab:angular-coordinates 0: P'_- = 27/20 < P' exactly",
     "pass": true,
     "detail": "P' ~ 1.389"
    },
    {
     "check": "tab:angular-coordinates 0: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates A: |xi - (-1267/2000)| < h exactly",
     "pass": true,
     "detail": "xi ~ -0.6335435"
    },
    {
     "check": "tab:angular-coordinates A: |upsilon - (4707/5000)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 0.9414136"
    },
    {
     "check": "tab:angular-coordinates A: P < P_+ = 19/200 exactly",
     "pass": true,
     "detail": "P ~ 0.0896"
    },
    {
     "check": "tab:angular-coordinates A: P'_- = 11/100 < P' exactly",
     "pass": true,
     "detail": "P' ~ 0.1198"
    },
    {
     "check": "tab:angular-coordinates A: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates B: |xi - (-773/2500)| < h exactly",
     "pass": true,
     "detail": "xi ~ -0.3092192"
    },
    {
     "check": "tab:angular-coordinates B: |upsilon - (-9/2000)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ -0.0045323"
    },
    {
     "check": "tab:angular-coordinates B: P < P_+ = 1/500000 exactly",
     "pass": true,
     "detail": "P ~ 1.69e-06"
    },
    {
     "check": "tab:angular-coordinates B: P'_- = 13/20000 < P' exactly",
     "pass": true,
     "detail": "P' ~ 0.0006983"
    },
    {
     "check": "tab:angular-coordinates B: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates T: |xi - (1739/5000)| < h exactly",
     "pass": true,
     "detail": "xi ~ 0.3478318"
    },
    {
     "check": "tab:angular-coordinates T: |upsilon - (7493/10000)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 0.7493138"
    },
    {
     "check": "tab:angular-coordinates T: P < P_+ = 23/1000 exactly",
     "pass": true,
     "detail": "P ~ 0.0221"
    },
    {
     "check": "tab:angular-coordinates T: P'_- = 13/500 < P' exactly",
     "pass": true,
     "detail": "P' ~ 0.02658"
    },
    {
     "check": "tab:angular-coordinates T: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates uA: |xi - (-5799/5000)| < h exactly",
     "pass": true,
     "detail": "xi ~ -1.1597561"
    },
    {
     "check": "tab:angular-coordinates uA: |upsilon - (1179/1000)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 1.1789713"
    },
    {
     "check": "tab:angular-coordinates uA: P < P_+ = 49/5 exactly",
     "pass": true,
     "detail": "P ~ 9.73"
    },
    {
     "check": "tab:angular-coordinates uA: P'_- = 19 < P' exactly",
     "pass": true,
     "detail": "P' ~ 19.48"
    },
    {
     "check": "tab:angular-coordinates uA: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates uB: |xi - (-1833/5000)| < h exactly",
     "pass": true,
     "detail": "xi ~ -0.3665946"
    },
    {
     "check": "tab:angular-coordinates uB: |upsilon - (57/100)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 0.5699600"
    },
    {
     "check": "tab:angular-coordinates uB: P < P_+ = 11/5000 exactly",
     "pass": true,
     "detail": "P ~ 0.00199"
    },
    {
     "check": "tab:angular-coordinates uB: P'_- = 3/125 < P' exactly",
     "pass": true,
     "detail": "P' ~ 0.02507"
    },
    {
     "check": "tab:angular-coordinates uB: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tab:angular-coordinates uT: |xi - (-1179/5000)| < h exactly",
     "pass": true,
     "detail": "xi ~ -0.2357561"
    },
    {
     "check": "tab:angular-coordinates uT: |upsilon - (7807/5000)| < h exactly",
     "pass": true,
     "detail": "upsilon ~ 1.5613637"
    },
    {
     "check": "tab:angular-coordinates uT: P < P_+ = 253/20 exactly",
     "pass": true,
     "detail": "P ~ 12.6"
    },
    {
     "check": "tab:angular-coordinates uT: P'_- = 129/10 < P' exactly",
     "pass": true,
     "detail": "P' ~ 13.02"
    },
    {
     "check": "tab:angular-coordinates uT: P_+ < P'_-",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1.73205^2 < 3 < 1.73206^2 and 3.31662^2 < 11 < 3.31663^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a_* + b_*) eps + eps^2 = 0.000025243425 < 0.000026",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the product error bound bounds |sqrt3 sqrt11 - a_* b_*| (from the radical bounds)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "0.000026/6 < eps",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the rotation has absolute row sum 47/37",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of 0 by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table 0: d_x = 3840000",
     "pass": true,
     "detail": "recomputed 3840000"
    },
    {
     "check": "residual table 0: d_y = -23040000",
     "pass": true,
     "detail": "recomputed -23040000"
    },
    {
     "check": "residual table 0: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of A by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table A: d_x = -29520000",
     "pass": true,
     "detail": "recomputed -29520000"
    },
    {
     "check": "residual table A: d_y = 10176000",
     "pass": true,
     "detail": "recomputed 10176000"
    },
    {
     "check": "residual table A: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of B by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table B: d_x = -12240000",
     "pass": true,
     "detail": "recomputed -12240000"
    },
    {
     "check": "residual table B: d_y = -22464000",
     "pass": true,
     "detail": "recomputed -22464000"
    },
    {
     "check": "residual table B: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of T by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table T: d_x = 25440000",
     "pass": true,
     "detail": "recomputed 25440000"
    },
    {
     "check": "residual table T: d_y = 10752000",
     "pass": true,
     "detail": "recomputed 10752000"
    },
    {
     "check": "residual table T: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of uA by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table uA: d_x = 32043244",
     "pass": true,
     "detail": "recomputed 32043244"
    },
    {
     "check": "residual table uA: d_y = -19212795",
     "pass": true,
     "detail": "recomputed -19212795"
    },
    {
     "check": "residual table uA: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of uB by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table uB: d_x = 4763244",
     "pass": true,
     "detail": "recomputed 4763244"
    },
    {
     "check": "residual table uB: d_y = -27212795",
     "pass": true,
     "detail": "recomputed -27212795"
    },
    {
     "check": "residual table uB: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "midpoint substitution moves each coordinate of uT by at most eps (exact)",
     "pass": true,
     "detail": "excess sign -1"
    },
    {
     "check": "residual table uT: d_x = 32966488",
     "pass": true,
     "detail": "recomputed 32966488"
    },
    {
     "check": "residual table uT: d_y = -23385590",
     "pass": true,
     "detail": "recomputed -23385590"
    },
    {
     "check": "residual table uT: |d_x|, |d_y| < 35520000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "35520000 = 710400000000/20000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(47/37) eps + 0.00005 < 0.00011 = h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors 0: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors 0: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors 0: y_u < b_y = 353/10000",
     "pass": true,
     "detail": "y_u = 0.0352350441"
    },
    {
     "check": "factors 0: H < b_H = 4963/1000",
     "pass": true,
     "detail": "H = 4.9626444122"
    },
    {
     "check": "factors 0: a = 17/10 < l_l <= l_u < b = 171/100",
     "pass": true,
     "detail": "[1.7005169402, 1.7011673482]"
    },
    {
     "check": "factors 0: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors 0: b_y b_H^2 < P_+ = 9/10",
     "pass": true,
     "detail": "0.8694873257"
    },
    {
     "check": "factors 0: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 27/20",
     "pass": true,
     "detail": "1.378219325"
    },
    {
     "check": "squared-radius interval 0: [l_l, l_u] and the exact |z_g|^2 lie in (17/10, 171/100)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors A: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors A: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors A: y_u < b_y = 887/1000",
     "pass": true,
     "detail": "y_u = 0.8864410801"
    },
    {
     "check": "factors A: H < b_H = 319/1000",
     "pass": true,
     "detail": "H = 0.3183580322"
    },
    {
     "check": "factors A: a = 32/25 < l_l <= l_u < b = 129/100",
     "pass": true,
     "detail": "[1.2872097562, 1.2879027122]"
    },
    {
     "check": "factors A: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors A: b_y b_H^2 < P_+ = 19/200",
     "pass": true,
     "detail": "0.090262007"
    },
    {
     "check": "factors A: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 11/100",
     "pass": true,
     "detail": "0.111392325632"
    },
    {
     "check": "squared-radius interval A: [l_l, l_u] and the exact |z_g|^2 lie in (32/25, 129/100)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors B: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors B: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors B: y_u < b_y = 11/500000",
     "pass": true,
     "detail": "y_u = 2.12521e-05"
    },
    {
     "check": "factors B: H < b_H = 287/1000",
     "pass": true,
     "detail": "H = 0.2869987562"
    },
    {
     "check": "factors B: a = 19/200 < l_l <= l_u < b = 12/125",
     "pass": true,
     "detail": "[0.0955559002, 0.0956939282]"
    },
    {
     "check": "factors B: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors B: b_y b_H^2 < P_+ = 1/500000",
     "pass": true,
     "detail": "1.812118e-06"
    },
    {
     "check": "factors B: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 13/20000",
     "pass": true,
     "detail": "0.000683844714368"
    },
    {
     "check": "squared-radius interval B: [l_l, l_u] and the exact |z_g|^2 lie in (19/200, 12/125)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors T: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors T: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors T: y_u < b_y = 281/500",
     "pass": true,
     "detail": "y_u = 0.5616153481"
    },
    {
     "check": "factors T: H < b_H = 199/1000",
     "pass": true,
     "detail": "H = 0.1989503398"
    },
    {
     "check": "factors T: a = 341/500 < l_l <= l_u < b = 683/1000",
     "pass": true,
     "detail": "[0.6821739922, 0.6826567162]"
    },
    {
     "check": "factors T: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors T: b_y b_H^2 < P_+ = 23/1000",
     "pass": true,
     "detail": "0.022255762"
    },
    {
     "check": "factors T: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 13/500",
     "pass": true,
     "detail": "0.026433649589671347"
    },
    {
     "check": "squared-radius interval T: [l_l, l_u] and the exact |z_g|^2 lie in (17/25, 69/100)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uA: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uA: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uA: y_u < b_y = 1391/1000",
     "pass": true,
     "detail": "y_u = 1.3903003921"
    },
    {
     "check": "factors uA: H < b_H = 2647/1000",
     "pass": true,
     "detail": "H = 2.6463919922"
    },
    {
     "check": "factors uA: a = 273/100 < l_l <= l_u < b = 137/50",
     "pass": true,
     "detail": "[2.7346625282, 2.7356916002]"
    },
    {
     "check": "factors uA: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uA: b_y b_H^2 < P_+ = 49/5",
     "pass": true,
     "detail": "9.746193119"
    },
    {
     "check": "factors uA: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 19",
     "pass": true,
     "detail": "19.1818593033795"
    },
    {
     "check": "squared-radius interval uA: [l_l, l_u] and the exact |z_g|^2 lie in (273/100, 137/50)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uB: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uB: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uB: y_u < b_y = 163/500",
     "pass": true,
     "detail": "y_u = 0.3250254121"
    },
    {
     "check": "factors uB: H < b_H = 79/1000",
     "pass": true,
     "detail": "H = 0.0786540602"
    },
    {
     "check": "factors uB: a = 459/1000 < l_l <= l_u < b = 23/50",
     "pass": true,
     "detail": "[0.4590895322, 0.4595016362]"
    },
    {
     "check": "factors uB: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uB: b_y b_H^2 < P_+ = 11/5000",
     "pass": true,
     "detail": "0.002034566"
    },
    {
     "check": "factors uB: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 3/125",
     "pass": true,
     "detail": "0.024955647752214"
    },
    {
     "check": "squared-radius interval uB: [l_l, l_u] and the exact |z_g|^2 lie in (459/1000, 23/50)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uT: |xi_0|, |upsilon_0| > h",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uT: (x_l,x_u,y_l,y_u) = ((m-11)^2,(m+11)^2,(n-11)^2,(n+11)^2)/10^10, m, n integers",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uT: y_u < b_y = 61/25",
     "pass": true,
     "detail": "y_u = 2.4383134801"
    },
    {
     "check": "factors uT: H < b_H = 2273/1000",
     "pass": true,
     "detail": "H = 2.2716641518"
    },
    {
     "check": "factors uT: a = 2493/1000 < l_l <= l_u < b = 499/200",
     "pass": true,
     "detail": "[2.4931762402, 2.4939670082]"
    },
    {
     "check": "factors uT: (a, b) lies in (0, 4) and avoids 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "factors uT: b_y b_H^2 < P_+ = 253/20",
     "pass": true,
     "detail": "12.60633076"
    },
    {
     "check": "factors uT: a^3 (1-b/4) min(|a-1|,|b-1|)^2 > P'_- = 129/10",
     "pass": true,
     "detail": "12.994601315953503"
    },
    {
     "check": "squared-radius interval uT: [l_l, l_u] and the exact |z_g|^2 lie in (249/100, 5/2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the worked row: 2.44 (2.273)^2 < 12.65 and (2.493)^3 (1 - 2.495/4)(1.493)^2 > 12.9",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/angular.tex": "63dbb1f99bfabfbae18ddd3f3b6c254057154ed089d6b4222f826992958e0dde",
    "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/moser-placement.tex": "fc922e1b0363f164b5d078d2c6929b327b486dd7445e5210bfd97a216c008e9e",
    "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/main.tex": "c6ab6417b431f9e4453cc36d913fddc4d16cdd3e44c7edabd757f573eeb00857",
    "preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/introduction.tex": "41222c5b4e8be3352a08fa554116285007268c841216432fa8aab62208a96e1a"
   },
   "forges": [
    {
     "forge": "tab:angular-coordinates uT: P'_- printed 13.0 instead of 12.9",
     "verdict": "REFUTED"
    },
    {
     "forge": "residual table A: d_x off by one",
     "verdict": "REFUTED"
    },
    {
     "forge": "placement translation moved by +0.1i (lemma fails at: Lemma angular-certificate at z_T: P < P')",
     "verdict": "REFUTED"
    },
    {
     "forge": "u = (3+4i)/5 instead of (5+i sqrt11)/6",
     "verdict": "REFUTED"
    },
    {
     "forge": "edge T-uT dropped (ten edges): 12 proper 3-colourings",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.0
  },
  {
   "id": "F-297b",
   "index": 297,
   "family": "158",
   "title": "The Euclidean plane is not five-colorable",
   "tier": "C",
   "dir": "The-Euclidean-plane-is-not-five-colorable-September-23-2026",
   "leanChallenges": [
    "EuclideanFiveColor",
    "PlaneColoring"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. build/sections/introduction.tex lines 31-36 ('an unrestricted lower bound can also be established without first displaying a finite graph witnessing it') and lines 309-313 ('Theorem~\\ref{thm:main} and graph compactness ... give a finite unit-distance graph H that is not five-colorable'). The proof goes through weak measurable colorings (build/sections/transfer.tex) and spectral/ergodic arguments.",
   "decider": null
  },
  {
   "id": "F-300",
   "index": 300,
   "family": "161",
   "title": "A counterexample to Sidorenko's conjecture",
   "tier": "C",
   "dir": "A-counterexample-to-Sidorenkos-conjecture-September-23-2026",
   "leanChallenges": [
    "SidorenkoCounterexample"
   ],
   "word": "NEEDS DATA",
   "why": "H: build/sections/complex.tex Table tab:complex at lines 19-56 (Theorem thm:main in build/sections/introduction.tex lines 18-26; Lean ComparatorChallenges/SidorenkoCounterexample.lean def faces). Host: NOT PUBLISHED. build/sections/kernel.tex lines 19, 566 and 673-676 choose D and q 'sufficiently large'; the Lean solution module is OAI.Combinatorics.Sidorenko.RandomHost (existential).",
   "decider": null
  },
  {
   "id": "F-301",
   "index": 301,
   "family": "162",
   "title": "Balanced counterexamples to Ryser's conjecture at prime orders",
   "tier": "C",
   "dir": "Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026",
   "leanChallenges": [
    "BalancedRyser",
    "RyserCovering",
    "RyserOddExtensions"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. Random choices in build/candidates.tex lines 75-140; the random selection procedure in build/selection.tex lines 20-35 ('All subsequent assertions concern sufficiently large q', line 93).",
   "decider": null
  },
  {
   "id": "F-302",
   "index": 302,
   "family": "162",
   "title": "A counterexample to Ryser's covering conjecture",
   "tier": "C",
   "dir": "A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026",
   "leanChallenges": [
    "BalancedRyser",
    "RyserCovering",
    "RyserOddExtensions"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. build/selection.tex lines 1-60 (independent random shifts; Lemma sel-domination at line 52).",
   "decider": null
  },
  {
   "id": "F-336",
   "index": 336,
   "family": "187",
   "title": "Snaky in 21 Maker moves",
   "tier": "A",
   "dir": "Snaky-in-21-Maker-moves-September-25-2026",
   "leanChallenges": [
    "SnakyCertificate",
    "SnakyConditional",
    "SnakyTwentyOne"
   ],
   "word": "CERTIFIED",
   "why": "the whole headline (Theorem thm:main, 21 Maker moves on Z^2) and Corollary cor:finite-board, twice: (I) the certificate evaluates to A_727 = {}, h_727 = 21 under the paper's combination rule — sound only with Lemma lem:combination, proved in prose; (II) a rule-free search shows the certificate-guided policy wins within 21 claims against every Breaker line, given three stated monotonicity/locality reductions",
   "decider": "instruments/openaimath/finite/f336_snaky.py",
   "decides": "the whole headline (Theorem thm:main, 21 Maker moves on Z^2) and Corollary cor:finite-board, twice: (I) the certificate evaluates to A_727 = {}, h_727 = 21 under the paper's combination rule — sound only with Lemma lem:combination, proved in prose; (II) a rule-free search shows the certificate-guided policy wins within 21 claims against every Breaker line, given three stated monotonicity/locality reductions",
   "value": {
    "A_727": [],
    "T_727": 251,
    "h_727": 21,
    "nodes": 1620,
    "reply_classes": 37042,
    "search_worst_case_claims": 21,
    "search_states": 2166,
    "search_replies_examined": 44456
   },
   "checks": [
    {
     "check": "the paper states the theorem, the combination rule and the bases as read here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the policy searched in II is the paper's (strategy.tex:14-29) and the corollary's boards are as read (strategy.tex:64-67)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the appendix prints the literal file (\\lstinputlisting of ../verification/certificate.txt)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "SHA-256 of the decided bytes equals the one printed (verification.tex:75)",
     "pass": true,
     "detail": "3fa12d36a6d4dbb1"
    },
    {
     "check": "the file has 38,367 bytes (census) ",
     "pass": true,
     "detail": "38367"
    },
    {
     "check": "the target S read from eq:target",
     "pass": true,
     "detail": "[(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1)]"
    },
    {
     "check": "the bit rule reproduces the printed table eq:symmetries, and the 8 maps are the 8 signed permutations",
     "pass": true,
     "detail": ""
    },
    {
     "check": "grammar: every line parses, nonempty child lists, every reference backward and s in 0..7",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exactly 722 lines, numbered 6..727 in order (prop:certificate)",
     "pass": true,
     "detail": "722"
    },
    {
     "check": "4,089 references, inline ones included (verification.tex:16,22)",
     "pass": true,
     "detail": "4089"
    },
    {
     "check": "898 inline combinations (verification.tex:23)",
     "pass": true,
     "detail": "898"
    },
    {
     "check": "1,620 combination nodes (verification.tex:24)",
     "pass": true,
     "detail": "1620"
    },
    {
     "check": "every card and node has A in T and p in T; every child envelope inside its parent",
     "pass": true,
     "detail": ""
    },
    {
     "check": "heights of all 728 cards at most 21",
     "pass": true,
     "detail": "21"
    },
    {
     "check": "37,042 local reply classes over all nodes (verification.tex:44)",
     "pass": true,
     "detail": "37042"
    },
    {
     "check": "every local reply class leaves a child whose envelope avoids it and whose requirement is owned",
     "pass": true,
     "detail": "0 bad"
    },
    {
     "check": "card 6 = eq:first-card (A_6 = {(0,y):0<=y<=4}, T_6 = A_6 + (1,3),(1,4),(1,5), h_6 = 2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "card 7 = the printed A_7, T_7, h_7 = 3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "final card: p = (8,8), A = {}, |T| = 251, h = 21 (eq:final-card)",
     "pass": true,
     "detail": "(8, 8) [] 251 21"
    },
    {
     "check": "the printed final values are those of eq:final-card",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T_727 inside {0..16}^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the claimed bound 21 is the final height",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:top: all six rows (A_j, |T_j|, h_j, placements, remaining cells) recomputed",
     "pass": true,
     "detail": "648:okokokokok 708:okokokokok 712:okokokokok 713:okokokokok 725:okokokokok 726:okokokokok"
    },
    {
     "check": "each of the 32 placed child requirements of card 727 is exactly {(8,8)}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the intersection of the 32 child envelopes is exactly {(8,8)}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the nonlocal example: 708:100 avoids (7,8), pivot (8,7); its child 684:011 avoids (8,6), pivot (11,8)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "II. rule-free search: the certificate policy wins on Z^2 against every Breaker line within 21 actual claims",
     "pass": true,
     "detail": "worst case 21 claims, 2166 states"
    },
    {
     "check": "cross-check I/II: in every state the search reaches, the actual Maker cells contain the certificate's A and Breaker owns no envelope cell",
     "pass": true,
     "detail": "599 of 2166 states hold Maker cells beyond A"
    },
    {
     "check": "II. the policy's root envelope is T_727 (so every pivot and completing cell lies on the 251-cell board)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/verification/certificate.txt": "3fa12d36a6d4dbb185e3f2808c8dfde13d85ee309afbca030d6ad17ae9fe3d04",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/introduction.tex": "56e35be69cac7fd8905f9bef385f770f20c90394676bfb7c960fad28525d6403",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/templates.tex": "d78a592bc2628a898b8b43bf237dfbdf4a1565c9bc11fd945db73e04a3b618f7",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/certificate-proof.tex": "4f38b93893ae50cbdbec687a243c9eafda6e7b6cabd46d5c023f8ae3381be5b4",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/strategy.tex": "6499d0bc16bddacf0f778a232aaf1bd8d9a54533375b9d95f384ff511b4864bc",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/verification.tex": "7edcabcedf3e6551a54169ce390e8ba3c32a82264a9848dd03ba47396268f433",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/certificate-appendix.tex": "cbc040a864882a0335a378d231541efed77a17eb83e48fc53aefbcb2ab943185",
    "preprints/Snaky-in-21-Maker-moves-September-25-2026/build/paper.tex": "864956989aba1c861c0a99dcd6355a584b4321bc1b57a0ffefe04644a18bcf22"
   },
   "forges": [
    {
     "forge": "card 727 pivot moved (8,8) -> (8,7) -> fails: 37,042 local reply classes over all nodes (verification.tex:; final card: p = (8,8), A = {}, |T| = 251, h = 21 (eq:final-c; II. rule-free search: the certificate policy wins on Z^2 aga || search: line fails: Breaker reply (8, 8) leaves no child; claim (8, 7) reply (8, 8)",
     "verdict": "REFUTED"
    },
    {
     "forge": "card 727 last child 726:6FF dropped -> fails: 4,089 references, inline ones included (verification.tex:16,; 37,042 local reply classes over all nodes (verification.tex:; final card: p = (8,8), A = {}, |T| = 251, h = 21 (eq:final-c; Table tab:top: all six rows (A_j, |T_j|, h_j, placements, re; the intersection of the 32 child envelopes is exactly {(8,8); II. rule-free search: the certificate policy wins on Z^2 aga || search: line fails: Breaker reply (9, 9) leaves no child; claim (8, 8) reply (9, 9)",
     "verdict": "REFUTED"
    },
    {
     "forge": "card 708: child 684:011 transposed (684:111) -> fails: final card: p = (8,8), A = {}, |T| = 251, h = 21 (eq:final-c; Table tab:top: all six rows (A_j, |T_j|, h_j, placements, re; each of the 32 placed child requirements of card 727 is exac; the intersection of the 32 child envelopes is exactly {(8,8); the nonlocal example: 708:100 avoids (7,8), pivot (8,7); its; II. rule-free search: the certificate policy wins on Z^2 aga || search: line fails: Breaker reply (8, 9) leaves no child; claim (8, 8) reply (8, 9)",
     "verdict": "REFUTED"
    },
    {
     "forge": "card 311: inline pivot (4,1) -> (4,2) -> fails: 37,042 local reply classes over all nodes (verification.tex:; final card: p = (8,8), A = {}, |T| = 251, h = 21 (eq:final-c; Table tab:top: all six rows (A_j, |T_j|, h_j, placements, re; each of the 32 placed child requirements of card 727 is exac; II. rule-free search: the certificate policy wins on Z^2 aga || search: line fails: Breaker reply (5, 7) leaves no child; claim (8, 8) reply (7, 8) go card 708 placed s=1 t=(0, 0) claim (8, 7) reply (8, 6) go card 684 placed s=0 t=(1, 1) claim (11, 8) reply a cell outside the envelope go inline node claim (8, 9) reply (9, 8) go inline node claim (8, 10) reply (9, 10) go card 485 placed s=3 t=(4, 8) claim (5, 8) reply a cell outside the envelope go card 313 placed s=3 t=(0, 5) claim (9, 7) reply (9, 6) go card 312 placed s=5 t=(4, 0) claim (7, 7) reply (7, 6) go card 311 placed s=4 t=(0, 2) claim (6, 7) reply (6, 8) go inline node claim (5, 6) reply (5, 7)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the bound claimed as 20 instead of 21 -> fails: the claimed bound 20 is the final height; II. rule-free search: the certificate policy wins on Z^2 aga || search: worst case 21 claims, 2166 states",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.2
  },
  {
   "id": "F-338",
   "index": 338,
   "family": "189",
   "title": "Cycle--clique Ramsey numbers",
   "tier": "A",
   "dir": "Cycle-clique-Ramsey-numbers-September-25-2026",
   "leanChallenges": [
    "CycleCliqueRamsey"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the 3099 published deduction traces of Proposition finite:verified exclude all 3099 (pattern, k, t) instances by the paper's stated rules, applied correctly (the rules' soundness and the reduction of R(C_m,K_n) to these instances are the paper's, not decided here)",
   "decider": "instruments/openaimath/finite/f338_cycle_clique_ramsey.py",
   "decides": "a finite component: the 3099 published deduction traces of Proposition finite:verified exclude all 3099 (pattern, k, t) instances by the paper's stated rules, applied correctly (the rules' soundness and the reduction of R(C_m,K_n) to these instances are the paper's, not decided here)",
   "value": {
    "pairs": 42,
    "instances": 3099,
    "records_checked": 3099,
    "round_entries_checked": 2259,
    "initial_flags_equal": 3099,
    "witness_weights_equal_bound": 3099,
    "outputs_by_k": {
     "5": [
      0,
      4,
      3
     ],
     "6": [
      0,
      8,
      6
     ],
     "7": [
      0,
      21,
      5
     ],
     "8": [
      0,
      30,
      4
     ],
     "9": [
      0,
      57,
      6
     ],
     "10": [
      0,
      71,
      4
     ],
     "11": [
      0,
      134,
      8
     ],
     "12": [
      0,
      161,
      3
     ],
     "13": [
      0,
      305,
      6
     ],
     "14": [
      0,
      315,
      0
     ],
     "15": [
      0,
      597,
      4
     ],
     "16": [
      0,
      468,
      0
     ],
     "17": [
      0,
      878,
      1
     ]
    },
    "failing": [],
    "trace_seconds": 31.0,
    "sample": null
   },
   "checks": [
    {
     "check": "the theorem and the finite proposition as printed",
     "pass": true,
     "detail": "01-introduction.tex:10-16, 09-finite-patterns.tex:11-27"
    },
    {
     "check": "the rule statements read are the ones cited",
     "pass": true,
     "detail": "09-finite-rules.tex, 09-finite-packing.tex"
    },
    {
     "check": "(a) 42 parameter pairs in (finite:domain), as printed (09-finite-patterns.tex:121)",
     "pass": true,
     "detail": "42 pairs; printed 42"
    },
    {
     "check": "(a) the enumeration emits canonical, pairwise distinct patterns",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a) 3099 instances, as printed (09-finite-patterns.tex:122)",
     "pass": true,
     "detail": "3099 enumerated; printed 3099"
    },
    {
     "check": "(a) a(r, m) of (finite:chain-count) equals the direct count of compositions up to reversal (r <= 7, m <= 9)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a) the generating function (finite:count-series) gives the same count for every pair",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a) Table finite:counts (09-finite-patterns.tex:163-175) equals the enumeration, pair by pair",
     "pass": true,
     "detail": "42 table cells"
    },
    {
     "check": "(a) summary.json coverage (patterns and per-amount counts) equals the enumeration",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a) summary.json total_patterns = 3099, unresolved = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(a) certificates.jsonl sha256 equals the one summary.json records",
     "pass": true,
     "detail": "a5646def0211a8b4393167d76aa09c30647fda73043756a362ede9773d3cab39"
    },
    {
     "check": "(a) the certificate file has one record per line, 3099 lines",
     "pass": true,
     "detail": "3099"
    },
    {
     "check": "(a) every enumerated instance is in the file exactly once, and nothing else is",
     "pass": true,
     "detail": "missing 0, extra 0, repeated 0"
    },
    {
     "check": "(a) every recorded pattern is in the canonical form (normalized components, nondecreasing)",
     "pass": true,
     "detail": "3099 of 3099"
    },
    {
     "check": "(b) every recorded initial flag is justified by Lemma finite:extension-rule, and every trace step and final contradiction checks (3099 records)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(b) the recorded initial flags EQUAL the recomputed union of extension intervals",
     "pass": true,
     "detail": "3099 of 3099"
    },
    {
     "check": "(b) every round entry re-justified by a trial of Lemma finite:strengthening on its round-start matrix",
     "pass": true,
     "detail": "2259 of 2259 round entries"
    },
    {
     "check": "(c) for every output-2 record, an exact search finds NO packing contradiction on the initial matrix (so output 1 is not the right class)",
     "pass": true,
     "detail": "failing records []"
    },
    {
     "check": "(c) the same search finds a packing on every output-1 initial matrix (it is not blind)",
     "pass": true,
     "detail": "missed in records []"
    },
    {
     "check": "(c) all final witnesses are packing contradictions, as printed (10-implementation.tex:104)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(c) per-k outputs 0/1/2 recomputed equal Table finite:results (09-finite-results.tex:17-31)",
     "pass": true,
     "detail": "{\"5\": [0, 4, 3], \"6\": [0, 8, 6], \"7\": [0, 21, 5], \"8\": [0, 30, 4], \"9\": [0, 57, 6], \"10\": [0, 71, 4], \"11\": [0, 134, 8], \"12\": [0, 161, 3], \"13\": [0, 305, 6], \"14\": [0, 315, 0], \"15\": [0, 597, 4], \"16\": [0, 468, 0], \"17\": [0, 878, 1]}"
    },
    {
     "check": "(c) totals 0 / 3049 / 50 as printed (09-finite-results.tex:31)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(c) expected-output.txt and summary.json counts equal Table finite:results",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(c) \"one record for each of the 3099 parameter-pattern instances\" (10-implementation.tex:92-93)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the worked example: k=5, t=3, P=((),(2)): Q={0,1,4}; x=0,y=2,z=0,w=4,E=empty forbids d=1,2",
     "pass": true,
     "detail": "Q=[0, 1, 4] interval=(1, 2)"
    }
   ],
   "sources": {
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/09-finite-patterns.tex": "72a8ce1cb0e509f4184c170b3f2411941230583696d1638f7c3b3b5dbdb01134",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/09-finite-rules.tex": "8a98dd5599ff6443dc96a593133f54dfe2fe48be61a44908d95bc0ee95548008",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/09-finite-packing.tex": "fd3bb6f8b5ed5f60dc898d574a97f79f0f20b10512812421c2267d390c3aab67",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/09-finite-results.tex": "5a2daad2b0c7ce4d013702ca80b64757363a692735a828ff395aff992c22d3ad",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/10-implementation.tex": "f5792bf8398110772a9ad58c53b60393ba68f9ddcae4dce756e13f4e1c290836",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/build/sections/01-introduction.tex": "388a0ab5c80170203e839fbf27f40fb7198f55a15e3325262ce3d9115bfa3bff",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/verification/data/expected-output.txt": "4cc194d9611dc742e09672bb9cd714ae0bb5005fdd52187d002f9db383ddb285",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/verification/data/summary.json": "d8da1cb27bfdf50a17cad752d9699a703e46f1a93bb9967dc5ac9da6f124a2b9",
    "preprints/Cycle-clique-Ramsey-numbers-September-25-2026/verification/data/certificates.jsonl": "a5646def0211a8b4393167d76aa09c30647fda73043756a362ede9773d3cab39"
   },
   "forges": [
    {
     "forge": "record 0: parameter 0 (no edge) added to the clique pair (0,1) of the initial flags",
     "verdict": "REFUTED"
    },
    {
     "forge": "record 1: final witness weight of option [0, 1, 3] raised by one (total adjusted)",
     "verdict": "REFUTED"
    },
    {
     "forge": "record 2 (k=5,t=3,pattern=[[], [2]]): last round of prohibitions removed",
     "verdict": "REFUTED"
    },
    {
     "forge": "record 2: the first round entry's parameter d flipped",
     "verdict": "REFUTED"
    },
    {
     "forge": "the last record (k=17) deleted from the file",
     "verdict": "REFUTED"
    },
    {
     "forge": "record 5 replaced by a copy of record 4",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 31.3
  },
  {
   "id": "F-339",
   "index": 339,
   "family": "190",
   "title": "Polynomial removal fails for ordered binary matrices",
   "tier": "B",
   "dir": "Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026",
   "leanChallenges": [
    "MatrixRemoval"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Theorem thm:main at h = 1 and h = 2 (A_1 776 x 776, A_2 3,096 x 3,096), both inequalities decided exactly (N_H(A_h) = 2 m^130 by an exhaustive ordered-copy search; dist_H(A_h) >= eps_h by the sampling argument with every finite step checked, for h <= 6), and the finite lemmas of Section 3; the theorem for every h (the headline) rests on the paper's uniform proofs and is not decided here",
   "decider": "instruments/openaimath/finite/f339_ordered_removal.py",
   "decides": "a finite component: Theorem thm:main at h = 1 and h = 2 (A_1 776 x 776, A_2 3,096 x 3,096), both inequalities decided exactly (N_H(A_h) = 2 m^130 by an exhaustive ordered-copy search; dist_H(A_h) >= eps_h by the sampling argument with every finite step checked, for h <= 6), and the finite lemmas of Section 3; the theorem for every h (the headline) rests on the paper's uniform proofs and is not decided here",
   "value": {
    "h=1": {
     "n": 776,
     "quotient": [
      389,
      389
     ],
     "N_H": 2722258935367507707706996859454145691648,
     "N_H = 2 m^130": true,
     "copies in quotient": 2,
     "search nodes": 396,
     "seconds": 0.2,
     "N_H after zeroing L_1": 2722258935367507707706996859454145691648
    },
    "h=2": {
     "n": 3096,
     "quotient": [
      779,
      779
     ],
     "N_H": 3705346855594118253554271520278013051304639509300498049262642688253220148477952,
     "N_H = 2 m^130": true,
     "copies in quotient": 4,
     "search nodes": 800,
     "seconds": 0.7
    },
    "runtime_s": 12.5
   },
   "checks": [
    {
     "check": "the theorem and the construction as printed",
     "pass": true,
     "detail": "01-introduction.tex:33-47, 02-construction.tex:11-37, 159-162, 03b-count.tex:16-21, 04-distance.tex:12-17"
    },
    {
     "check": "L. Lemma lem:anchor-properties: distinct rows and columns",
     "pass": true,
     "detail": "min row ones 58, min column ones 48"
    },
    {
     "check": "L. Lemma lem:anchor-properties: at most one zero per column in rows 1..32, per row in columns 1..32",
     "pass": true,
     "detail": "min row ones 58, min column ones 48"
    },
    {
     "check": "L. Lemma lem:anchor-properties: every row >= 58 ones, every column >= 48 ones",
     "pass": true,
     "detail": "min row ones 58, min column ones 48"
    },
    {
     "check": "L. Lemma lem:anchor-properties: rows 33..64 on columns 60..64 are the 32 five-bit words",
     "pass": true,
     "detail": "min row ones 58, min column ones 48"
    },
    {
     "check": "N. H has pairwise distinct rows and pairwise distinct columns (needed for the run reduction)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. n = d_h m with d_h = 6hs + 2h + 2 = 386h + 2, on both axes, h = 1..6",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the printed orders n_1 = 776 and n_2 = 3,096",
     "pass": true,
     "detail": "computed 776, 3096; printed 776, 3096"
    },
    {
     "check": "C. the four order equivalences eq:tree-orders hold on the built axis orders, h = 1..5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the entry rules assign every cell once (no conflicting assignment), h = 1, 2, 3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the entry between leaf blocks p, q is 1[p <= q]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "L. Lemma lem:mode-incidences: every variable position lies in <= 4 modes' designated sets and has <= 4 ones against the anchor groups, h = 1..6",
     "pass": true,
     "detail": "[(4, 4), (4, 4), (4, 4), (4, 4), (4, 4), (4, 4)]"
    },
    {
     "check": "L. Lemma lem:nonanchor-traces: on every 5 non-anchor column blocks the non-anchor rows realize <= 27 < 32 words, h = 1..3",
     "pass": true,
     "detail": "max words [5, 9, 11] (non-anchor column blocks [5, 11, 23])"
    },
    {
     "check": "N. h = 1: every ordered copy of H in A_1 maps entry (65,65) into the leaf diagonal L_1 (Prop prop:copy-count)",
     "pass": true,
     "detail": "2 copies in the 389 x 389 quotient, N_H = 2722258935367507707706996859454145691648, off-leaf placements 0"
    },
    {
     "check": "N. h = 1: N_H(A_1) <= m n^(2k-2) (eq:copy-bound)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. h = 1: N_H(A_1) / n^132 <= eps_h 2^-h, exactly (the theorem's count inequality)",
     "pass": true,
     "detail": "N_H = 2^131"
    },
    {
     "check": "N. h = 1: Lemma lem:anchor-rigidity: every ordered copy of S uses the 64 row and 64 column groups of one mode, in order",
     "pass": true,
     "detail": "6 copies of S in the quotient (6 modes), weighted count 6 m^128"
    },
    {
     "check": "N. h = 1: Remark rem:hitting-not-repair: after the m cells of L_1 are set to 0 (which meets every original copy), H still occurs, and no new copy puts (65,65) on L_1",
     "pass": true,
     "detail": "N_H after the change = 2722258935367507707706996859454145691648 (= 2^131), entry (65,65) now at [\"[('var', 0, '+', 0)]\"]"
    },
    {
     "check": "N. h = 2: every ordered copy of H in A_2 maps entry (65,65) into the leaf diagonal L_2 (Prop prop:copy-count)",
     "pass": true,
     "detail": "4 copies in the 779 x 779 quotient, N_H = 3705346855594118253554271520278013051304639509300498049262642688253220148477952, off-leaf placements 0"
    },
    {
     "check": "N. h = 2: N_H(A_2) <= m n^(2k-2) (eq:copy-bound)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. h = 2: N_H(A_2) / n^132 <= eps_h 2^-h, exactly (the theorem's count inequality)",
     "pass": true,
     "detail": "N_H = 2^261"
    },
    {
     "check": "N. h = 2: Lemma lem:anchor-rigidity: every ordered copy of S uses the 64 row and 64 column groups of one mode, in order",
     "pass": true,
     "detail": "12 copies of S in the quotient (12 modes), weighted count 12 m^128"
    },
    {
     "check": "D. every forced body completes the selected anchors to H (66 x 66 submatrix = H off the body, on protected cells), h = 1..6",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. for every leaf z the \"no body equals P\" constraints over the free cells are UNSATISFIABLE, h = 1..6 (exact DPLL; full enumeration at h = 1)",
     "pass": true,
     "detail": "max clauses / free cells per z: [(1, 4, 3), (2, 8, 7), (3, 12, 11), (4, 16, 15), (5, 20, 19), (6, 24, 23)]"
    },
    {
     "check": "D. every protected cell is a selected pair with probability <= 1/m^2 (exact), h = 1..6",
     "pass": true,
     "detail": "[(1, '1/4'), (2, '1/16'), (3, '1/64'), (4, '1/256'), (5, '1/1024'), (6, '1/4096')]"
    },
    {
     "check": "D. hence dist_H(A_h) >= m^2/n^2 = (386h+2)^-2 = eps_h, h = 1..6",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/01-introduction.tex": "88d6ccc55361ba0409a9d532a8b075f3f688c44568a376c5f5b27f437bf110b2",
    "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/02-construction.tex": "19dbe8eea7abc5165a27f450bb4fab0d9c2f59caf40eaec47c6ace73819916ec",
    "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/03-pinning.tex": "bda96d50abbf389ef99fdc0a8888b3068c6d25d79c213f15edf7d66907c89467",
    "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/03b-count.tex": "11c3c67a6c042ef1d03387ac2c3e76a89703d508efbb24480f7d1ada0f6f47be",
    "preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/build/sections/04-distance.tex": "4f7e9ca6e5f9a88d3a091758acf76dade80096f06a978089b3b232d355125acb"
   },
   "forges": [
    {
     "forge": "S(40,61) flipped (two five-bit words collide)",
     "verdict": "REFUSED"
    },
    {
     "forge": "eq:variable-entries line 2 printed as 1[p <= q] instead of 1[p < q]",
     "verdict": "REFUSED"
    },
    {
     "forge": "the printed order n_1 = 777",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 12.5
  },
  {
   "id": "F-345",
   "index": 345,
   "family": "196",
   "title": "A Torsion-Free Group Algebra with Zero Divisors",
   "tier": "C",
   "dir": "A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026",
   "leanChallenges": [
    "TorsionFreeZeroDivisors"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED. build/sections/factors.tex lines 31-35: 'The construction is probabilistic: it proves that suitable finite matchings exist, without specifying matchings from which a concrete presentation and zero-divisor factors can be read off.' The Lean solution (OAI.Algebra.GroupRing) is existential.",
   "decider": null
  },
  {
   "id": "F-346",
   "index": 346,
   "family": "197",
   "title": "A Torsion-Free Group Algebra That Is Not Directly Finite",
   "tier": "C",
   "dir": "A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026",
   "leanChallenges": [
    "GroupRingDeterminant",
    "KaplanskyDirectFiniteness",
    "KaplanskyFinitelyPresented",
    "OddKaplansky"
   ],
   "word": "NEEDS DATA",
   "why": "NOT PUBLISHED; the construction is probabilistic. Section sections/random.tex (input at build/paper.tex line 44); build/sections/topology.tex line 3 ('The probabilistic argument has produced finite immersed graphs'); build/sections/assembly.tex line 7 ('sufficiently large').",
   "decider": null
  },
  {
   "id": "F-347",
   "index": 347,
   "family": "197",
   "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two",
   "tier": "C",
   "dir": "A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026",
   "leanChallenges": [
    "GroupRingDeterminant",
    "KaplanskyDirectFiniteness",
    "KaplanskyFinitelyPresented",
    "OddKaplansky"
   ],
   "word": "NEEDS DATA",
   "why": "Not in checkable form. build/sections/01-introduction.tex lines 8-15 (theorem and 'terminating prescription'); build/sections/04-hypergraph.tex lines 8-20 and 53 (parameters, random slabs, 'for all sufficiently large h').",
   "decider": null
  },
  {
   "id": "F-349",
   "index": 349,
   "family": "197",
   "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic",
   "tier": "C",
   "dir": "A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026",
   "leanChallenges": [
    "GroupRingDeterminant",
    "KaplanskyDirectFiniteness",
    "KaplanskyFinitelyPresented",
    "OddKaplansky"
   ],
   "word": "NEEDS DATA",
   "why": "Specified but infeasible. Theorem op:main in build/sections/01-introduction.tex lines 14-30; finite data in build/sections/02-finite-data.tex; explicit idempotent eq op:explicit-e in build/sections/03-modular.tex line 191.",
   "decider": null
  },
  {
   "id": "F-351",
   "index": 351,
   "family": "199",
   "title": "An explicit counterexample to the Auslander-Reiten conjecture",
   "tier": "A",
   "dir": "An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026",
   "leanChallenges": [
    "AuslanderReiten",
    "Tachikawa"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the ten-dimensional algebra C, its trivial extension T and trace, the 179-entry cochain (cocycle and boundary identities, evaluation q^3), the resolution of s over C for every i, and the finite identities of the lift — not the Ext vanishing over Lambda (F, Lambda, Z are not published)",
   "decider": "instruments/openaimath/finite/f351_auslander_reiten.py",
   "decides": "a finite component: the ten-dimensional algebra C, its trivial extension T and trace, the 179-entry cochain (cocycle and boundary identities, evaluation q^3), the resolution of s over C for every i, and the finite identities of the lift — not the Ext vanishing over Lambda (F, Lambda, Z are not published)",
   "value": {
    "C_product_rules": 12,
    "C_nonzero_radical_words_by_length": [
     8,
     12,
     7,
     1
    ],
    "T_nonzero_radical_words_by_length": [
     18,
     52,
     65,
     33,
     5
    ],
    "four_words": 104976,
    "composable_four_words": 15250,
    "p_on_zeta": "q^3",
    "C_dim": 10,
    "T_dim": 20,
    "A_dim": 400,
    "cochain_entries": 179
   },
   "checks": [
    {
     "check": "C: the corner table names ten distinct basis letters, two idempotents e, f in their own corners",
     "pass": true,
     "detail": "{'e': ('e', 'e'), 'x': ('e', 'e'), 'y': ('e', 'e'), 'z': ('e', 'e'), 'u': ('e', 'f'), 'v': ('e', 'f'), 't': ('f', 'e'), 'j': ('f', 'e'), 'f': ('f', 'f'), 'n': ('f', 'f')}"
    },
    {
     "check": "C: every printed product respects the corners (composable, value in the right corner)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: 1 = e + f is a two-sided unit",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: associative on all 10^3 = 1000 basis triples",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: dimension ten",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: the paper prints the grading u,t:1  x,y,n:2  v,j:3  z:4, idempotents 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: every product is homogeneous for that grading",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the 11 printed triples are exactly the composable radical triples of total degree <= 4",
     "pass": true,
     "detail": "printed ['ntu', 'tun', 'tut', 'txu', 'tyu', 'unt', 'utu', 'utx', 'uty', 'xut', 'yut'], computed ['ntu', 'tun', 'tut', 'txu', 'tyu', 'unt', 'utu', 'utx', 'uty', 'xut', 'yut']"
    },
    {
     "check": "C: the printed associativity table — (ab)c = a(bc) = the printed value for all 11",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the span N of the eight non-idempotent letters is a two-sided ideal",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: (rad C)^5 = 0 (no nonzero product of five radical letters) and N^4 != 0",
     "pass": true,
     "detail": "nonzero words by length [8, 12, 7, 1]"
    },
    {
     "check": "C: utut = q(1+q)z != 0 (Lemma alg:C, and (ut)^2 = (y+qx)^2 in 08-consequences)",
     "pass": true,
     "detail": "(q^2+q)z"
    },
    {
     "check": "C/N = k x k: e, f orthogonal idempotents",
     "pass": true,
     "detail": ""
    },
    {
     "check": "08-consequences: eu = u and ue = 0 (C noncommutative)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Remark: eCe = k<x,y>/(x^2, y^2, xy - q yx) with z = yx (relations hold; e,x,y,yx span eCe)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Remark: fCe has actions tx = j, ty = q^2 j, jx = jy = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Remark: t -> x + q^-1 y, j -> q^-1 z (scaled by q) is right R-linear onto the right ideal (x + q^-1 y)R, bijective",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Remark: (x + q^-1 y)R is the kernel of r -> (x+y)r on R (first syzygy of (x+y)R)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: twenty basis letters (lower case and their duals), dimension 20",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the printed transposition recurrences coc:transpose-recurrence hold for the dual actions alg:dual-actions",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the dual of a corner iCj lies in jLi (dual letters have swapped endpoints under the actions)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: associative on all 20^3 = 8000 basis triples",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "T: 1 = e + f is a two-sided unit",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: lambda(a,phi) = phi(1) is symmetric on all 400 letter pairs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the trace-dual of each letter is its starred letter, lambda(a b*) = delta_ab (Gram = permutation)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the trace form is nondegenerate (Gram rank 20 over F_2(q))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the span r of the 18 letters other than e, f is a two-sided ideal; T/r = k x k",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: r^6 = 0 (no nonzero product of six radical letters)",
     "pass": true,
     "detail": "nonzero words by length [18, 52, 65, 33, 5]"
    },
    {
     "check": "T: deg(a*) = 5 - deg(a) makes every product homogeneous; all radical degrees in 1..5",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "h_H (fix C, scale DC by H) is an algebra automorphism of T, H an indeterminate (400 pairs)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "p: the table has the printed number of entries (179)",
     "pass": true,
     "detail": "179 parsed, 179 printed"
    },
    {
     "check": "p: distinct three-letter inputs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "p: every entry has composable radical inputs and an output with the same two endpoints (an I-bimodule map)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "p: weight -1 — one more capital input letter than capital output letters, in every entry",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "p: 18^4 = 104,976 radical four-words, exactly 15,250 composable (printed counts)",
     "pass": true,
     "detail": "computed 15250 composable"
    },
    {
     "check": "p: the normalized Hochschild cocycle equation holds on all 104,976 radical four-words, every output coefficient",
     "pass": true,
     "detail": "0 failing words []"
    },
    {
     "check": "p: the support {u,v,t,E,X,Y,Z,U,V} of z_H (03-algebra) equals that of gamma (09-cochain)",
     "pass": true,
     "detail": "['E', 'U', 'V', 'X', 'Y', 'Z', 't', 'u', 'v']"
    },
    {
     "check": "p: the boundary identity coc:boundary, sum_w p(a,b,h(w)) w* = a z_H(b) + z_H(ab) + z_H(a) h^-1(b), for all 18^2 = 324 pairs with H an indeterminate",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "p: the two printed coefficient forms coc:boundary-constant and coc:boundary-linear hold for every (a,b,v)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "p: p(t,x,J) = 0 and p(t,y,J) = q^3 f, as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "zeta: tx = j, ty = q^2 j, xJ = T, yJ = q^2 T in T",
     "pass": true,
     "detail": ""
    },
    {
     "check": "zeta = q^2[t|x|J] + [t|y|J] has endpoints f on both sides and is a cycle in s^r (x)_T P (x)_T s",
     "pass": true,
     "detail": "{}"
    },
    {
     "check": "zeta: the simple-valued cochain pairs to q^3 != 0 with zeta",
     "pass": true,
     "detail": "q^3"
    },
    {
     "check": "p: every entry with output f (the only output the character of s sees) has exactly one capital input",
     "pass": true,
     "detail": "5 entries with output f"
    },
    {
     "check": "res: the bases of Ce, Cf (and eC, fC) are the letters with that right (left) idempotent, as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: the printed table of a u and a l_i on Ce holds as identities in F_2[q][Q], Q = q^i (all i >= 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: the printed table of l_i a on eC holds as identities in F_2[q][Q] (all i >= 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: left multiplication by u sends (f,t,j,n) to (u, y+qx, z, v), four independent vectors",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: right multiplication by u on Ce has image <u,v,n> = rad(Cf) and kernel <l_0,z,j>",
     "pass": true,
     "detail": "rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3]"
    },
    {
     "check": "res: right multiplication by l_i on Ce has image <l_i,z,j> and kernel <l_(i+1),z,j>, for EVERY i >= 0",
     "pass": true,
     "detail": "rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3]"
    },
    {
     "check": "res: ker(u) = im(l_0) and ker(l_i) = im(l_(i+1)) for every i >= 0 — R is exact at every index",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: l_0 on eC (exceptional): kernel <qx+y,z,u,v> = image of u., image <l_0,z>",
     "pass": true,
     "detail": "rank f [2] (generic 2, exceptional {}), rank S [2], rank [f|S] [2], kernel vectors killed True, rank K [4]"
    },
    {
     "check": "res: l_i on eC for EVERY i >= 1: kernel <l_(i-1),z,v>, image <l_i,z,v>",
     "pass": true,
     "detail": "rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3]"
    },
    {
     "check": "res: Hom complex cohomology — 0 in degree 0 (u. injective) and 1 (ker l_0. = im u.); in degree 2 ker(l_1.) = <l_0,z,v> over im(l_0.) = <l_0,z> is the line of v; ker(l_(i+1).) = im(l_i.) for every i >= 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: v f = v, v e = 0, and vt = qz is the only nonzero radical product (a boundary): the line is s^r",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lift: for every lower-case letter w, w w* = left(w)* and w* w = right(w)*",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lift: row dimensions of C are 6,4 and column dimensions 6,4",
     "pass": true,
     "detail": "([6, 4], [6, 4])"
    },
    {
     "check": "lift: pi(xi_H) = sum_w h(w) w* = 0 in characteristic two (H an indeterminate)",
     "pass": true,
     "detail": "0"
    },
    {
     "check": "lift: the endpoint identity lift:endpoint, sum over radical w with left(w) = r of h(w) w* = r*, r = e, f",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lift: the twisted coevaluation h(a) xi_H = xi_H a holds in T (x)_I T for all 20 letters a",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lift: lift:projected-casimir, sum a h(w) (x) w* + sum h(w) (x) w* h^-1(a) = a (x) r*, for all 18 radical a",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lift: in xi_H only w = f* has chi(w*) != 0, with value h(f*) = H f*",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dims: dim C = 10, dim T = 20, dim A = 400, dim Lambda = 800 + dim F (800 = 2 dim A)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/03-algebra.tex": "f3da85912c346824c1532feb9b410e169a0b520e8261ebbea2a95e8527908af5",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/09-cochain.tex": "838a489943b88b3501248a02d4835d09124a6b0002834de6fd380057948d340a",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/04-resolution.tex": "d19c8a962c5883d1f53d920604c137229b01c9c8370a17ca6fbfa0abbdbf2636",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/06-lift.tex": "1005f67f0188b8a02c43c1ef67062b785476ad648363eecd1704999ba6a21e39",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/08-consequences.tex": "79abd114831c82382ca0f241368932b7fe7728f5eff88310d7fa327f6cbf72ce",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/01-introduction.tex": "ea50dfba92fe6182d24af373a7680a24afbe8f30fd54f9f8ca95b4b949ccf0bb"
   },
   "forges": [
    {
     "forge": "C: the printed product ty = q^2 j changed to ty = q j",
     "verdict": "REFUTED"
    },
    {
     "forge": "p: the entry xXzz moved from coefficient q to coefficient q^2 (count, weight, endpoints kept)",
     "verdict": "REFUTED"
    },
    {
     "forge": "C: the printed associativity value for uty changed from q^2 z to q z",
     "verdict": "REFUTED"
    },
    {
     "forge": "res: the printed x l_i = q^(i+1) z changed to q^i z",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.3
  },
  {
   "id": "F-352",
   "index": 352,
   "family": "199",
   "title": "A counterexample to Tachikawa's second conjecture",
   "tier": "A",
   "dir": "A-counterexample-to-Tachikawas-second-conjecture-September-23-2026",
   "leanChallenges": [
    "AuslanderReiten",
    "Tachikawa"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the ten-dimensional algebra C (and its identity with the companion paper's), T and its symmetrizing form, E = T (x) T symmetric nondegenerate, the corner table of B, the exact sequences of Prop a:dual-dimensions, the resolution of s for every i, the socle element and the finite trace computation — not the Ext vanishing over A (Y, F, Lambda, Z, A, M are not published)",
   "decider": "instruments/openaimath/finite/f352_tachikawa.py",
   "decides": "a finite component: the ten-dimensional algebra C (and its identity with the companion paper's), T and its symmetrizing form, E = T (x) T symmetric nondegenerate, the corner table of B, the exact sequences of Prop a:dual-dimensions, the resolution of s for every i, the socle element and the finite trace computation — not the Ext vanishing over A (Y, F, Lambda, Z, A, M are not published)",
   "value": {
    "C_nonzero_radical_words_by_length": [
     8,
     12,
     7,
     1
    ],
    "T_nonzero_radical_words_by_length": [
     18,
     52,
     65,
     33,
     5
    ],
    "B_corner_dims": [
     [
      16,
      8,
      8,
      4
     ],
     [
      8,
      8,
      4,
      4
     ],
     [
      8,
      4,
      8,
      4
     ],
     [
      4,
      4,
      4,
      4
     ]
    ],
    "pairing_det": 4,
    "C_dim": 10,
    "T_dim": 20,
    "B_dim": 100,
    "E_dim": 400
   },
   "checks": [
    {
     "check": "C: the corner table names ten distinct letters, e and f in their own corners",
     "pass": true,
     "detail": "{'e': ('e', 'e'), 'x': ('e', 'e'), 'y': ('e', 'e'), 'z': ('e', 'e'), 'u': ('e', 'f'), 'v': ('e', 'f'), 't': ('f', 'e'), 'j': ('f', 'e'), 'f': ('f', 'f'), 'n': ('f', 'f')}"
    },
    {
     "check": "C: the multiplication table is complete, 10 x 10, rows and columns the ten letters",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: the unit is 1 = e + f (two-sided, from the table rows and columns e, f)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: the table respects the corners (every nonzero product composable, value in the corner)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the table equals, entry for entry (100 entries), the multiplication of the companion paper [OpenAIAR2026]",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: associative on all 10^3 = 1000 basis triples",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the printed grading covers the ten letters (e,f:0 u,t:1 x,y,n:2 v,j:3 z:4)",
     "pass": true,
     "detail": "{'e': 0, 'f': 0, 'u': 1, 't': 1, 'x': 2, 'y': 2, 'n': 2, 'v': 3, 'j': 3, 'z': 4}"
    },
    {
     "check": "C: the table is homogeneous for that grading",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the printed triples (two arrays, 5 + 6) are exactly the composable positive-degree triples of degree <= 4",
     "pass": true,
     "detail": "printed ['ntu', 'tun', 'tut', 'txu', 'tyu', 'unt', 'utu', 'utx', 'uty', 'xut', 'yut'] computed ['ntu', 'tun', 'tut', 'txu', 'tyu', 'unt', 'utu', 'utx', 'uty', 'xut', 'yut']"
    },
    {
     "check": "C: (ab)c = a(bc) = the printed value for all 11 printed triples",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "C: the positive-degree span is a two-sided ideal with fifth power zero; quotient ke + kf = k^2 (e, f orthogonal idempotents)",
     "pass": true,
     "detail": "nonzero radical words by length [8, 12, 7, 1]"
    },
    {
     "check": "C: eCe = k<x,y>/(x^2, y^2, yx - q^-1 xy): x^2 = y^2 = 0, q yx = xy, and e, x, y, yx = z span eCe",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: tx = j, ty = q^2 j, jx = jy = 0 (right-linearity of t -> x + q^-1 y, j -> q^-1 z)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: fCe = (x + q^-1 y)eCe as right eCe-modules (the map, scaled by q, is right-linear and its two images are a basis of the ideal)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C: (x + q^-1 y)eCe is the kernel of r -> (x+y)r on eCe (first syzygy of (x+y)eCe)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the paper's dual action (a phi b)(c) = phi(bca) holds for all 1000 (a, phi, b) in the constructed T",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "T: dimension 20, associative on all 20^3 = 8000 basis triples, unit e + f",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "T: t_T((a,phi)(b,psi)) = phi(b) + psi(a) on all 400 letter pairs; symmetric",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: the Gram matrix of t_T is a permutation matrix (a <-> a*), rank 20: t_T is nondegenerate",
     "pass": true,
     "detail": ""
    },
    {
     "check": "T: rad T = (rad C) + DC: an ideal, nilpotent (index 6), T/rad = k^2",
     "pass": true,
     "detail": "nonzero radical words by length [18, 52, 65, 33, 5]"
    },
    {
     "check": "T: h_H (fix C, scale DC by H) is an algebra automorphism, H an indeterminate",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B, E: dim B = 10^2 = 100 and dim E = 20^2 = 400, as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E: t_E = t_T (x) t_T has Gram G (x) G on the 400 x 400 basis pairs — a symmetric permutation matrix: symmetric, nondegenerate",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B: the printed corner-dimension matrix (16 8 8 4 / 8 8 4 4 / 8 4 8 4 / 4 4 4 4) is computed exactly; all entries even; products of the corner dimensions 4,2,2,2 of C",
     "pass": true,
     "detail": "computed [[16, 8, 8, 4], [8, 8, 4, 4], [8, 4, 8, 4], [4, 4, 4, 4]]"
    },
    {
     "check": "dual: the printed ordered bases of eC and Ce are the letters with left / right idempotent e",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dual: the z-coefficient pairing eC x Ce has exactly the printed nonzero entries ez,xy,yx,ze,uj,vt with coefficients 1,q,1,1,1,q",
     "pass": true,
     "detail": "computed {'ez': 1, 'xy': 2, 'yx': 1, 'ze': 1, 'uj': 1, 'vt': 2}"
    },
    {
     "check": "dual: its determinant is q^2 != 0, as printed",
     "pass": true,
     "detail": "4"
    },
    {
     "check": "dual: e -> z* gives D(eC) = Ce (a -> a z*) and D(Ce) = eC (a -> z* a), both bijective",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dual: Ce -> D(fC), e -> j*, sends (e,x,y,z,t,j) to the printed (j*, t*, q^2 t*, 0, q n*, f*)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dual: eC -> D(Cf), e -> v*, sends (e,x,y,z,u,v) to the printed (v*, u*, u*, 0, n*, f*)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dual: both have rank four (onto D(fC), D(Cf)) with kernels <l_-2,z> and <l_0,z>",
     "pass": true,
     "detail": "rank f [4] (generic 4, exceptional {}), rank S [4], rank [f|S] [4], kernel vectors killed True, rank K [2] | rank f [4] (generic 4, exceptional {}), rank S [4], rank [f|S] [4], kernel vectors killed True, rank K [2]"
    },
    {
     "check": "dual: rho_{l_-2} on Ce has image <l_-2,z>, kernel <y+qx,z,t,j>; lambda_{l_0} on eC has image <l_0,z>, kernel <y+qx,z,u,v>",
     "pass": true,
     "detail": "rank f [2] (generic 2, exceptional {}), rank S [2], rank [f|S] [2], kernel vectors killed True, rank K [4] | rank f [2] (generic 2, exceptional {}), rank S [2], rank [f|S] [2], kernel vectors killed True, rank K [4]"
    },
    {
     "check": "dual: rho_t (f,u,v,n) -> (t, y+qx, qz, qj) and lambda_u (f,t,j,n) -> (u, y+qx, z, v) as printed; each four independent, spanning the kernel",
     "pass": true,
     "detail": ""
    },
    {
     "check": "dual: both printed sequences 0 -> Cf -> Ce -> Ce -> D(fC) -> 0 and 0 -> fC -> eC -> eC -> D(Cf) -> 0 are exact",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: the printed rho_u, rho_{l_i} (on Ce) and lambda_{l_i} (on eC) hold as identities in F_2[q][Q], Q = q^i (all i)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: rho_u has image <u,v,n> = rad(Cf), kernel <l_0,z,j>; rho_{l_i} has image <l_i,z,j>, kernel <l_(i+1),z,j>, for EVERY i >= 0",
     "pass": true,
     "detail": "rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3] | rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3]"
    },
    {
     "check": "res: each later image is the preceding kernel — the resolution R of s is exact at every index",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: the dual complex starts injectively, lambda_u with image <u,y+qx,z,v>",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: lambda_{l_0} has image <l_0,z>, kernel <l_-1,z,u,v>; lambda_{l_i} has image <l_i,z,v>, kernel <l_(i-1),z,v>, for EVERY i >= 1",
     "pass": true,
     "detail": "rank f [2] (generic 2, exceptional {}), rank S [2], rank [f|S] [2], kernel vectors killed True, rank K [4] | rank f [3] (generic 3, exceptional {}), rank S [3], rank [f|S] [3], kernel vectors killed True, rank K [3]"
    },
    {
     "check": "res: q l_-1 = qx + y, so ker(lambda_{l_0}) = im(lambda_u)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: the only cohomology is <l_0,z,v>/<l_0,z> = k v-bar in degree 2 (ker lambda_{l_(i+1)} = im lambda_{l_i} for i >= 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "res: vf = v, ve = 0, and vt = qz (in im lambda_{l_0}) is the only nonzero radical product: the line is Ds",
     "pass": true,
     "detail": ""
    },
    {
     "check": "socle: chi (f -> 1, every other letter -> 0) is a character of T: chi(ab) = chi(a) chi(b) on all 400 pairs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "socle: a f* b = chi(a) chi(b) f* for all 400 letter pairs of T (so a zeta b = chi(a) chi(b) zeta in E by tensoring)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "socle: f* lies in rad T with f* f = f* (zeta in (rad E) p_f); sigma_H(zeta) = h(f*) (x) h(f*) = H^2 zeta",
     "pass": true,
     "detail": ""
    },
    {
     "check": "trace: tr_k(L_a R_{p_f}) on B vanishes for all 100 basis vectors a of B (so for every a in B)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "trace: dim(p_v B p_f) = 4 for each of the four vertices v (the p_f column of the corner matrix); chi_B(p_f) = 1",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/algebra.tex": "db879f0e5ffa97aaf598fdd03cc131c0f5d0722ced10b894e0820c75b928487d",
    "preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/stable.tex": "82b867b1b8ae024df4bd4b6d9483a26e2f3b99cdb9aa9475c8c930ede1d70091",
    "preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/bimodule.tex": "d52a78f4e2887d2971da2c7e265d39d7551acef9deeffef31caf6a4076596b51",
    "preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/build/03-algebra.tex": "f3da85912c346824c1532feb9b410e169a0b520e8261ebbea2a95e8527908af5"
   },
   "forges": [
    {
     "forge": "C: the table entry t.y = q^2 j changed to q j",
     "verdict": "REFUTED"
    },
    {
     "forge": "B: the printed corner dimension dim(eeB ee) = 16 changed to 12",
     "verdict": "REFUTED"
    },
    {
     "forge": "dual: the printed pairing coefficient of vt changed from q to 1",
     "verdict": "REFUTED"
    },
    {
     "forge": "res: the printed lambda_{l_i}(u) = (1+q^i)v changed to (1+q^(i+1))v",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.0
  },
  {
   "id": "F-360",
   "index": 360,
   "family": "205",
   "title": "Universal Tensor Squares for Symmetric Groups",
   "tier": "A",
   "dir": "Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026",
   "leanChallenges": [
    "Saxl",
    "UniversalTensorSquares"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Proposition prop:finite-check for every eligible n <= 64 (61 of the 61 degrees), with witnesses supplied and certified here (the release records none: NEEDS DATA for its own certificate); the large-degree theory (n > 64) is not decided",
   "decider": "instruments/openaimath/finite/f360_tensor_squares.py",
   "decides": "a finite component: Proposition prop:finite-check for every eligible n <= 64 (61 of the 61 degrees), with witnesses supplied and certified here (the release records none: NEEDS DATA for its own certificate); the large-degree theory (n > 64) is not decided",
   "value": {
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     "38": 1.55,
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     "40": 0.91,
     "41": 2.41,
     "42": 3.6,
     "43": 4.55,
     "44": 3.04,
     "45": 1.72,
     "46": 9.43,
     "47": 11.7,
     "48": 6.55,
     "49": 5.62,
     "50": 21.85,
     "51": 27.94,
     "52": 20.67,
     "53": 18.25,
     "54": 44.45,
     "55": 11.67,
     "56": 54.12,
     "57": 39.36,
     "58": 124.2,
     "59": 36.79,
     "60": 145.06,
     "61": 116.55,
     "62": 243.21,
     "63": 116.81,
     "64": 311.25
    },
    "exact_seconds": 2.6,
    "strip_map_entries": 556733759,
    "published_witnesses": "NEEDS DATA: none recorded (exit statuses only)",
    "undecided_degrees": []
   },
   "checks": [
    {
     "check": "the proposition as printed (finite.tex:12-17)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the release records exit statuses, not witnesses (finite.tex:223-225, RESULTS.json witness_format)",
     "pass": true,
     "detail": "NEEDS DATA for the published certificate: no lambda_n is recorded for any n"
    },
    {
     "check": "RESULTS.json eligible_degrees = {1..64} minus {2,4,9}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "RESULTS.json pins the sha256 of the two character logs read here",
     "pass": true,
     "detail": "its README.md pin does not match the shipped README, which says so itself (\"identifies the pre-sanitization documentation\")"
    },
    {
     "check": "character-61-degrees.log: one status-0 line for each eligible degree, no other",
     "pass": true,
     "detail": "61 lines"
    },
    {
     "check": "|P_64| = 1741630 and sum_{s<=64} |P_s| = 12308139 (finite.tex:186-187)",
     "pass": true,
     "detail": "1741630, 12308139"
    },
    {
     "check": "1000000007 and 1000000009 are prime (finite.tex:36-37), by trial division here",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the moduli used here, 1073741789 and 1073741783, are prime (trial division)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(2) exact character tables n <= 16: column orthogonality (n <= 12) and n! | sum |C| chi chi chi",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(2) covering lambda over ALL partitions, n <= 12, equal the recorded covering_shapes; none at n = 2, 4, 9",
     "pass": true,
     "detail": "n=2,4,9: none; e.g. n=8: [[4, 2, 1, 1], [3, 3, 2]]"
    },
    {
     "check": "(1)=(2) Horner residues equal the exact vectors mod p1 for every self-conjugate lambda, n <= 16",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(1) a certified self-conjugate witness for every eligible n <= 64 (61 degrees)",
     "pass": true,
     "detail": "witnesses: {\"1\": [1], \"3\": [2, 1], \"5\": [3, 1, 1], \"6\": [3, 2, 1], \"7\": [4, 1, 1, 1], \"8\": [4, 2, 1, 1], \"10\": [4, 3, 2, 1], \"11\": [4, 3, 3, 1], \"12\": [5, 3, 2, 1, 1], \"13\": [5, 3, 3, 1, 1], \"14\": [6, 3, 2, 1, 1, 1], \"15\": [5, 4, 3, 2, 1], \"16\": [6, 4, 2, 2, 1, 1], \"17\": [6, 4, 3, 2, 1, 1], \"18\": [6, 5, 2, 2, 2, 1], \"19\": [7, 4, 3, 2, 1, 1, 1], \"20\": [7, 5, 2, 2, 2, 1, 1], \"21\": [6, 5, 4, 3, 2, 1], \"22\": [6, 5, 4, 4, 2, 1], \"23\": [7, 6, 3, 2, 2, 2, 1], \"24\": [7, 5, 4, 4, 2, 1, 1], \"25\": [8, 5, 4, 3, 2, 1, 1, 1], \"26\": [8, 5, 4, 4, 2, 1, 1, 1], \"27\": [7, 6, 5, 3, 3, 2, 1], \"28\": [7, 6, 5, 4, 3, 2, 1], \"29\": [8, 7, 4, 3, 2, 2, 2, 1], \"30\": [8, 7, 4, 4, 2, 2, 2, 1], \"31\": [9, 6, 5, 3, 3, 2, 1, 1, 1], \"32\": [9, 6, 5, 4, 3, 2, 1, 1, 1], \"33\": [10, 7, 4, 3, 2, 2, 2, 1, 1, 1], \"34\": [8, 7, 6, 4, 3, 3, 2, 1], \"35\": [9, 8, 5, 3, 3, 2, 2, 2, 1], \"36\": [8, 7, 6, 5, 4, 3, 2, 1], \"37\": [10, 7, 6, 3, 3, 3, 2, 1, 1, 1], \"38\": [9, 7, 6, 5, 4, 3, 2, 1, 1], \"39\": [11, 8, 5, 3, 3, 2, 2, 2, 1, 1, 1], \"40\": [10, 7, 6, 5, 4, 3, 2, 1, 1, 1], \"41\": [10, 7, 6, 5, 5, 3, 2, 1, 1, 1], \"42\": [10, 8, 6, 5, 4, 3, 2, 2, 1, 1], \"43\": [9, 8, 7, 5, 5, 3, 3, 2, 1], \"44\": [10, 9, 6, 5, 4, 3, 2, 2, 2, 1], \"45\": [9, 8, 7, 6, 5, 4, 3, 2, 1], \"46\": [11, 9, 6, 5, 4, 3, 2, 2, 2, 1, 1], \"47\": [10, 8, 7, 6, 5, 4, 3, 2, 1, 1], \"48\": [12, 9, 6, 5, 4, 3, 2, 2, 2, 1, 1, 1], \"49\": [11, 8, 7, 6, 5, 4, 3, 2, 1, 1, 1], \"50\": [12, 9, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1], \"51\": [11, 9, 7, 6, 5, 4, 3, 2, 2, 1, 1], \"52\": [12, 9, 8, 5, 4, 3, 3, 3, 2, 1, 1, 1], \"53\": [11, 10, 7, 6, 5, 4, 3, 2, 2, 2, 1], \"54\": [12, 9, 8, 6, 4, 4, 3, 3, 2, 1, 1, 1], \"55\": [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], \"56\": [12, 9, 8, 7, 4, 4, 4, 3, 2, 1, 1, 1], \"57\": [13, 10, 7, 6, 5, 4, 3, 2, 2, 2, 1, 1, 1], \"58\": [12, 10, 8, 7, 4, 4, 4, 3, 2, 2, 1, 1], \"59\": [12, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 1], \"60\": [12, 9, 8, 7, 6, 6, 4, 3, 2, 1, 1, 1], \"61\": [13, 10, 9, 6, 5, 4, 3, 3, 3, 2, 1, 1, 1], \"62\": [12, 10, 8, 7, 6, 6, 4, 3, 2, 2, 1, 1], \"63\": [12, 11, 8, 7, 6, 5, 4, 3, 2, 2, 2, 1], \"64\": [12, 11, 8, 7, 6, 6, 4, 3, 2, 2, 2, 1]}"
    }
   ],
   "sources": {
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/build/sections/finite.tex": "22d57c6d5ca787757deded19d3c4546ed31e3fe43a61cdfc5d69664ea25d30a7",
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/build/sections/introduction.tex": "704d9d663bde909f002c9b51879af5fef14117e238dd5d826f3ec18aa041be7a",
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/verification/computation/character-61-degrees.log": "fd083c2d5aa4cd35c20e5972ee9c94ece225c1da44a0fb2f4f238bcbfdb264f9",
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/verification/computation/character-independent-1-12.log": "5ad29aaedff13ba5069eb291b40bb6c6e3ea975787a0df50a94dd5f3c9aec0e2",
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/verification/computation/RESULTS.json": "39ecafb07c8605719ea36c4230bd144f2bbff56c234c734482782b43c31d0ce2",
    "preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/verification/computation/README.md": "c2f63c561bfe99b4d6345bc1427c5b9a1ff07e3feb4ad446f277c3a6cb81ce00"
   },
   "forges": [
    {
     "forge": "witness at n=16 replaced by the self-conjugate (8, 2, 1, 1, 1, 1, 1, 1) (exact table: some g = 0)",
     "verdict": "REFUTED"
    },
    {
     "forge": "degree 9 declared eligible with witness (3, 3, 3)",
     "verdict": "REFUTED"
    },
    {
     "forge": "degree 9 declared eligible with witness (5, 1, 1, 1, 1)",
     "verdict": "REFUTED"
    },
    {
     "forge": "witness at n=12 replaced by the non-self-conjugate (5,3,2,2)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1555.7
  },
  {
   "id": "F-369",
   "index": 369,
   "family": "210",
   "title": "Foulkes' conjecture for the sixth symmetric power",
   "tier": "A",
   "dir": "Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026",
   "leanChallenges": [
    "FoulkesHowe"
   ],
   "word": "REFUSED",
   "why": "decided in part (pre-registration amendment 5): the base range is decided for b = 6..12 in the ledger run (b = 6..15 in a separate run of this session); b = 13..25 (estimated 8-12 CPU-hours) and the partitions outside the band at b = 26..29 (the certificate/flag pipeline, not re-implemented) are not; the band at b = 26..29 is decided whole (467,068 coefficients, none negative)",
   "decider": "instruments/openaimath/finite/f369_foulkes_six.py",
   "decides": "a finite component: (A) the theorem at b = 6..12 completely (Q >= F at every partition, exact, by a route independent of the paper's programs) with those Table tab:base rows recomputed; (B) every band coefficient of Section sec:band for b = 26..29 (467068, the band part of the b = 26..29 step). NOT decided: b = 13..25 of the base interval (cost), the certificate pipeline and flags clearing non-band partitions at b = 26..29, the rectangle reduction, and b >= 30 (companion theory; the part the Lean covers)",
   "value": {
    "base_rows": {
     "6": [
      2428,
      2428
     ],
     "7": [
      4476,
      315
     ],
     "8": [
      6820,
      255
     ],
     "9": [
      6899,
      250
     ],
     "10": [
      6886,
      264
     ],
     "11": [
      6814,
      279
     ],
     "12": [
      6850,
      294
     ]
    },
    "base_seconds": {
     "6": 0.9,
     "7": 2.0,
     "8": 3.8,
     "9": 7.1,
     "10": 12.6,
     "11": 21.3,
     "12": 35.2
    },
    "KQ_seconds": 76.5,
    "band": {
     "tails": 1908,
     "E_I_degree15": true,
     "l1_bound_bits": 80,
     "L_seconds": 4.2,
     "S_seconds": 5.3,
     "MR_seconds": 8.4
    },
    "band_26_29": {
     "coefficients": 467068,
     "zero": 2114,
     "negative": 0
    },
    "full_band": null
   },
   "checks": [
    {
     "check": "the theorem as printed (introduction.tex:19-26)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the definition of U as printed (upper.tex:93-106)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:base (certificates.tex:39-50) = appendix-a.stdout = expected-results.json; printed totals 145435 / 9272",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:band (verification.tex:172-195) = appendix-a/b.stdout (row labels 24, 30, ..., 144)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=6: Q >= F for all 2432 partitions of 36 with <= 6 parts (the theorem at b=6, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=6: dimension identities for Q, F and each cycle-type product; Hermite (two rows); Q = F identically",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=6: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=6: Table tab:base row recomputed: 2428 comparisons (Q < U), 2428 equalities; printed (2428, 2428)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=7: Q >= F for all 4494 partitions of 42 with <= 6 parts (the theorem at b=7, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=7: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=7: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=7: Table tab:base row recomputed: 4476 comparisons (Q < U), 315 equalities; printed (4476, 315)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=8: Q >= F for all 7760 partitions of 48 with <= 6 parts (the theorem at b=8, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=8: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=8: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=8: Table tab:base row recomputed: 6820 comparisons (Q < U), 255 equalities; printed (6820, 255)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=9: Q >= F for all 12692 partitions of 54 with <= 6 parts (the theorem at b=9, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=9: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=9: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=9: Table tab:base row recomputed: 6899 comparisons (Q < U), 250 equalities; printed (6899, 250)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=10: Q >= F for all 19858 partitions of 60 with <= 6 parts (the theorem at b=10, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=10: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=10: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=10: Table tab:base row recomputed: 6886 comparisons (Q < U), 264 equalities; printed (6886, 264)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=11: Q >= F for all 29941 partitions of 66 with <= 6 parts (the theorem at b=11, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=11: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=11: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=11: Table tab:base row recomputed: 6814 comparisons (Q < U), 279 equalities; printed (6814, 279)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=12: Q >= F for all 43752 partitions of 72 with <= 6 parts (the theorem at b=12, decided exactly)",
     "pass": true,
     "detail": "min(Q-F) = 0"
    },
    {
     "check": "(A) b=12: dimension identities for Q, F and each cycle-type product; Hermite (two rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) b=12: F <= U everywhere (Proposition prop:upper on this degree)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "(A) b=12: Table tab:base row recomputed: 6850 comparisons (Q < U), 294 equalities; printed (6850, 294)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(A) the interlacing enumeration of h_b(X^r) s_mu equals the alternant determinant on 6434 small cases",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(B) 1908 decreasing four-coordinate tails with |eta| <= 27 (band.tex:190); E_I integer of degree 15; packed decoding bounded (l1 < 2^159)",
     "pass": true,
     "detail": "l1 bound 80 bits"
    },
    {
     "check": "(B)=(A) the band formula equals 720(Q-F) from the independent route at every lambda with |eta| <= b+1, b = 6..12 (15808 partitions)",
     "pass": true,
     "detail": "0 mismatches"
    },
    {
     "check": "(B) b=26..29: 467068 band coefficients, 2114 zero, 0 negative, 0 not divisible by 720; printed 467068 coefficients, 2114 zero (verification.tex:174)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(B) the counting formula 3b - ceil(|eta|/2) - eta_1 + 1 reproduces every band-coefficient count of Table tab:band and the total 57065668",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/introduction.tex": "f0447bbe5113462ce532fd1c5ee8f3cb241e6ddbaff2b103d29dd263b8229472",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/certificates.tex": "2bf9183c91f9bfc1277cd03e27e09e1a111b6856d5b605a787d24bdf31434356",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/verification.tex": "671d980ba5c3ce3a12a920997fd9bc86fc9b2982eaf8f23821e5d086f66c97c3",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/band.tex": "412495904d153d85b043e6a9d8b88baf240e837348578d8002894bdf61785538",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/build/sections/upper.tex": "94465d3b42d4f25924492ea9d5903a2aeeb8b417e3eadfa47880116f980b123f",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/verification/support/expected/appendix-a.stdout": "5e721cae45a9ea9b9a3a6e47b75a6e8e5cdf8698efac72aa1cd8abd3457f55af",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/verification/support/expected/appendix-b.stdout": "ceebd4910f364036869ae7a619fd67a0c7ffb96db3c37866ea24b7e4decf2f70",
    "preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/verification/support/expected-results.json": "fa405cf6652e4c63f449f72271c1971faaf20f5d99f2bb64561732c525b2dd3a"
   },
   "forges": [
    {
     "forge": "Table tab:base row b=8 printed with 256 equalities (255 recomputed)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table tab:band row 26-29 printed with 2115 zero differences (2114 recomputed)",
     "verdict": "REFUTED"
    },
    {
     "forge": "source multiplicity F raised by one at an equality partition (38, 2, 2, 0, 0, 0) of b=7",
     "verdict": "REFUTED"
    },
    {
     "forge": "band cross factor (1 - y/q) replaced by (1 - y q)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 179.3
  },
  {
   "id": "F-409",
   "index": 409,
   "family": "222",
   "title": "Microscopic jamming in the negative spherical perceptron",
   "tier": "A",
   "dir": "Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026",
   "leanChallenges": [
    "IsingFiniteness",
    "PerceptronFreeEnergy",
    "SphericalField"
   ],
   "word": "REFUSED",
   "why": "a finite component: stage (I) of the certificate — the two final spectral tests (.2936535023 < lambda(.2936533) < .2936535043, .2936533037 < lambda(.2936535) < .2936533057, hence lambda(a) - a changes sign on (.2936533, .2936535)) by own exact integer certificates of the profile and of the Riccati shooting, the numeric steps of the three error lemmas, and the exact exponent arithmetic from the bracket to the printed gamma and theta bounds. NOT stage (II) (the six interval routines: the [0, .410] -> [.285, .303] narrowing and the contraction that make the root unique and equal to the selected coefficient), NOT the theory; hence not the headline",
   "decider": "instruments/openaimath/finite/f409_perceptron_jamming.py",
   "decides": "a finite component: stage (I) of the certificate — the two final spectral tests (.2936535023 < lambda(.2936533) < .2936535043, .2936533037 < lambda(.2936535) < .2936533057, hence lambda(a) - a changes sign on (.2936533, .2936535)) by own exact integer certificates of the profile and of the Riccati shooting, the numeric steps of the three error lemmas, and the exact exponent arithmetic from the bracket to the printed gamma and theta bounds. NOT stage (II) (the six interval routines: the [0, .410] -> [.285, .303] narrowing and the contraction that make the root unique and equal to the selected coefficient), NOT the theory; hence not the headline",
   "value": {
    "tests": {
     "2936533/10000000": {
      "seed": 2936535033,
      "window": [
       "2936535023/10000000000",
       "2936535043/10000000000"
      ],
      "match_lo_mid_hi": [
       "-1.011025e-08",
       "-4.749736e-10",
       "9.160308e-09"
      ],
      "lambda_secant_estimate": "0.2936535033492",
      "M0": "0.4716479104952180870969518288138558",
      "dM0": "-0.5226079904138646272764052074508453",
      "newton_iterations": 7,
      "published_db": {
       "M0": "0.4716479104952180870969518288148288",
       "dM0": "-0.5226079904138646272764052074508064",
       "our_minus_published": [
        "-9.73e-31",
        "-3.90e-32"
       ]
      }
     },
     "587307/2000000": {
      "seed": 2936533047,
      "window": [
       "2936533037/10000000000",
       "2936533057/10000000000"
      ],
      "match_lo_mid_hi": [
       "-9.833606e-09",
       "-1.983135e-10",
       "9.436979e-09"
      ],
      "lambda_secant_estimate": "0.2936533047205",
      "M0": "0.4716479715284377920871515676207816",
      "dM0": "-0.5226080111933846901160172613081223",
      "newton_iterations": 7,
      "published_db": {
       "M0": "0.4716479715284377920871515676217469",
       "dM0": "-0.5226080111933846901160172613080866",
       "our_minus_published": [
        "-9.65e-31",
        "-3.57e-32"
       ]
      }
     }
    },
    "discrepancies": [
     "main.tex:8865 prints the theta upper endpoint as \"0.4231087030795289\\ldots\"; the exact endpoint 2a/(1-2a) - 1 at a = .2936535 is 174614/412693 = 0.42310870307952885074..., so the printed 16th digit is ROUNDED (it is the binary64 repr), not a truncation followed by further digits; the stated bounds 0.4231063 < theta < 0.4231088 are unaffected"
    ],
    "facts": {
     "release_files": [
      "Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf",
      "README.md",
      "build/main.tex"
     ],
     "certificate/original_certificate.py (cited :9466) present as a file": false,
     "certificate/original_certificate.py recoverable from the appendix listing (sha256 match)": true,
     "reproducibility record (cited :9475)": false
    },
    "timing_seconds": {
     "profile 2936533": [
      3.7,
      0.3
     ],
     "riccati 2936533": 1.7,
     "profile 2936535": [
      3.7,
      0.3
     ],
     "riccati 2936535": 1.6
    },
    "refused_cost": {
     "measured_exact_profile_plus_three_riccati_pairs_per_parameter_s": 5.6,
     "measured_exact_node_evaluation_s": "3.98e-05",
     "nodes_per_reference_profile (2 x 2400 steps x 50 samples x M,X,k,l + 2 x 2000 x 50 for Q)": 1160000,
     "exact_tabulation_per_reference_s": 46.2,
     "integration_grid_cells_per_integral (2048 per dyadic scale on [2^-38, 1-2^-38], split at 1/2)": 151552,
     "reference_profiles_in_stage_II": "47: the table db of Appendix app:code has 49 entries, 47 at AD = 10^6 for the coarse, nonlinear and local tests plus the two final ones (counted after this decider first ran)",
     "exact_tabulation_all_references_h (a LOWER bound: no integral, no inverse lookup, no test)": 0.6,
     "paper_float_run_s (Python 3.12.13 + NumPy 2.3.5, :9479-9482)": 338.52
    },
    "seconds": 11.5,
    "refused_because": [
     "stage (II) is specified by its program: the prose of :8181-8800 and :9208-9463 names arrays and routines of Appendix app:code (Env, adjusted with caps .1431/.1301, nonlinearfunctional with geometric interpolation, lm/lx/lz, shifts, cv, cp, den, the k-grids and their clipping) rather than defining them; a clean-room decider may not take them from the checking code, and the prose alone underdetermines them",
     "cost: exact tabulation alone — 1.16e6 node evaluations per reference profile at a measured ~1e-4 s each, for the 47 reference profiles of stage II — is ~1.5 h of one core before any integral (151,552-cell grids), inverse lookup or sign test (value['refused_cost']); the paper's own binary64 NumPy run took 338.52 s",
     "missing file: the reproducibility record (software versions, source hashes, saved output of the October 4, 2026 run) cited at :9475-9483 is not in the release (certificate/original_certificate.py is absent as a file but its text is the appendix listing, whose sha256 matches the printed hash)"
    ]
   },
   "checks": [
    {
     "check": "the claims as printed: gamma/theta bounds (abstract, Theorem thm:main), the root bracket (:310), the two eigenvalue signs (:7838), the seeds and windows (:9499-9504), the 16-digit theta endpoints (:8865), the tail coefficients (:7849-7850)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. M-error lemma: damping .18 - .0082 > .17; .41 * .8 <= .34 (aX); .34^2/.09 < 1.29 (negative part of 2 beta); .05 + .41 * .8 < .38 (beta on [0,.1]); 2 phi(0) = sqrt(2/pi) < .8 (pi > 3.125)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. M-error lemma: source |f| <= 1e-23 + (2p + 2.02a) 1e-22 = 1e-23 + 1.0082e-22 < 2.1e-22; |F(0)| < 2.4e-21; e^1.29 (2.4e-21 + 24 * 2.1e-22) < 4e-20 (rigorous e^x)",
     "pass": true,
     "detail": "|F(0)| <= 2.199e-21, |F| <= 2.703e-20"
    },
    {
     "check": "2. tail constants: |c(c-1)| < 1.215 (+ side) and <= 1/4 (- side) on the stated c ranges, hence |d| < 1.4, |e| < 24, K < 3066",
     "pass": true,
     "detail": "max |c(c-1)| 1.2141 / 0.2500; |e| <= 23.302; K <= 3065.33"
    },
    {
     "check": "2. tail lemma: exponential-coefficient residual cost x^6 [2 (.41 e^-px^2/(2px)) .071 + 2 (.41 e^-px^2)] < 8 at p = .09, x = 12, with x^2 = 144 > 3/p (x^6 e^-px^2 decreasing on x >= 12)",
     "pass": true,
     "detail": "cost 5.9496"
    },
    {
     "check": "2. tail lemma: |T| + K x^-7 < .071 at x = 12 (c < .81, |d| < 1.4, |e| < 24, K < 3066); the derivative-bound sum < .00514 < .0055",
     "pass": true,
     "detail": "|S| <= 0.06849; derivative sum 0.005132"
    },
    {
     "check": "2. shooting bounds: K/20^7 + 2/10^36 < 2.41e-6; e^2[2.41e-6 + 20(2e-9 + 3e-19)] < 2.9e-5 < .001; 2.41e-6 e^-6 + 20 e^2 (2e-9 + 3e-19) < 4.5e-7; 2.41e-6 e^-20 + 20 e^2 3e-19 < 1e-10",
     "pass": true,
     "detail": "2.395e-06, 1.81e-05, 3.015e-07, 5.012e-15"
    },
    {
     "check": "1. a = 2936533/10000000: the trial M~ on all 4800 cells has ODE residual < 1e-23 (sum of |integer coefficients| < AD D^2 / 10^23)",
     "pass": true,
     "detail": "max residual bound 2.717e-30",
     "trial": true
    },
    {
     "check": "1. a = 2936533/10000000: exact C^1 joins (integer sums carried; c_1 opposite at z = 0); |M~(24)| < 1e-23; |M~(-24) - 24| < 1e-23",
     "pass": true,
     "detail": "|M~(24)| = 2.37e-33, |M~(-24) - 24| = 1.26e-32",
     "trial": true
    },
    {
     "check": "1. a = 2936533/10000000: -1.01 <= M~' <= .01 on every cell; M~(0) inside the value barriers phi(0) <= M(0) <= 2 phi(0)",
     "pass": true,
     "detail": "M~' in [-1.000000, -3.009e-34]; M~(0) = 0.471647910495218; M~'(0) = -0.522607990413865",
     "trial": true
    },
    {
     "check": "2. a = 2936533/10000000: the M-error lemma instance: damping 2p - 2a max M~' > .17; |E| <= max(res/.17, boundary + Gaussian tail) < 1e-22; source < 2.1e-22",
     "pass": true,
     "detail": "damping 0.41269; |E| <= 1.598e-29; source 1.000e-22"
    },
    {
     "check": "2. a = 2936533/10000000: the tail lemma at each tested lambda and side: c ranges, the identity R(T) (exact Laurent polynomial), x^6|R(T)| <= 6.62|e| + d^2 + 1 and remainder <= .7pK at x >= 12, the room identity, |d| < 1.4, |e| < 24, K < 3066",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. a = 2936533/10000000: six Riccati trials (lambda = seed/1e10 and -+ 1e-9, both sides): residual < 1e-20; drift negative part > -2, total > 20, part on x >= 12.02 > 8; source < 3e-19; initial error < 2.41e-6; bootstrap < .001 closed (middle: whole window, < 2.9e-5 and < 4.5e-7 centrally); endpoint error at 0 < 1e-10",
     "pass": true,
     "detail": "lo+1: res 3.9e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 8.332e-08, glob 6.16e-07, zero 1.8e-16; lo-1: res 3.1e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.344e-07, glob 1.73e-06, zero 4.9e-16; mid+1: res 3.9e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 2.395e-06, glob 1.80e-05, zero 3.0e-07; mid-1: res 3.1e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.396e-06, glob 1.80e-05, zero 3.0e-07; hi+1: res 4.2e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 8.332e-08, glob 6.16e-07, zero 1.8e-16; hi-1: res 3.0e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.344e-07, glob 1.73e-06, zero 4.9e-16",
     "trial": true
    },
    {
     "check": "3. a = 2936533/10000000: the printed integer tests — trial match S~_+(0) + S~_-(0) < -1e-9 at lambda = 2936535023/10000000000 and > 1e-9 at 2936535043/10000000000, total error < 2e-10: the exact match changes sign, so lambda(a) is in the printed window",
     "pass": true,
     "detail": "match -1.0110e-08 / -4.7497e-10 / 9.1603e-09 at lo / mid / hi; errors 6.6e-16, 6.6e-16; slope ~ 9.6353 per unit lambda"
    },
    {
     "check": "1. a = 587307/2000000: the trial M~ on all 4800 cells has ODE residual < 1e-23 (sum of |integer coefficients| < AD D^2 / 10^23)",
     "pass": true,
     "detail": "max residual bound 2.676e-30",
     "trial": true
    },
    {
     "check": "1. a = 587307/2000000: exact C^1 joins (integer sums carried; c_1 opposite at z = 0); |M~(24)| < 1e-23; |M~(-24) - 24| < 1e-23",
     "pass": true,
     "detail": "|M~(24)| = 4.71e-33, |M~(-24) - 24| = 8.79e-32",
     "trial": true
    },
    {
     "check": "1. a = 587307/2000000: -1.01 <= M~' <= .01 on every cell; M~(0) inside the value barriers phi(0) <= M(0) <= 2 phi(0)",
     "pass": true,
     "detail": "M~' in [-1.000000, -3.762e-34]; M~(0) = 0.471647971528438; M~'(0) = -0.522608011193385",
     "trial": true
    },
    {
     "check": "2. a = 587307/2000000: the M-error lemma instance: damping 2p - 2a max M~' > .17; |E| <= max(res/.17, boundary + Gaussian tail) < 1e-22; source < 2.1e-22",
     "pass": true,
     "detail": "damping 0.41269; |E| <= 1.574e-29; source 1.000e-22"
    },
    {
     "check": "2. a = 587307/2000000: the tail lemma at each tested lambda and side: c ranges, the identity R(T) (exact Laurent polynomial), x^6|R(T)| <= 6.62|e| + d^2 + 1 and remainder <= .7pK at x >= 12, the room identity, |d| < 1.4, |e| < 24, K < 3066",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. a = 587307/2000000: six Riccati trials (lambda = seed/1e10 and -+ 1e-9, both sides): residual < 1e-20; drift negative part > -2, total > 20, part on x >= 12.02 > 8; source < 3e-19; initial error < 2.41e-6; bootstrap < .001 closed (middle: whole window, < 2.9e-5 and < 4.5e-7 centrally); endpoint error at 0 < 1e-10",
     "pass": true,
     "detail": "lo+1: res 4.2e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 8.332e-08, glob 6.16e-07, zero 1.8e-16; lo-1: res 3.2e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.344e-07, glob 1.73e-06, zero 4.9e-16; mid+1: res 4.7e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 2.395e-06, glob 1.80e-05, zero 3.0e-07; mid-1: res 3.1e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.396e-06, glob 1.80e-05, zero 3.0e-07; hi+1: res 4.1e-33, neg 0.000, total 202.6, x>=12.02 128.1, src 2.4e-20, E20 8.332e-08, glob 6.16e-07, zero 1.8e-16; hi-1: res 2.9e-33, neg -0.396, total 79.6, x>=12.02 52.3, src 2.4e-20, E20 2.344e-07, glob 1.73e-06, zero 4.9e-16",
     "trial": true
    },
    {
     "check": "3. a = 587307/2000000: the printed integer tests — trial match S~_+(0) + S~_-(0) < -1e-9 at lambda = 2936533037/10000000000 and > 1e-9 at 2936533057/10000000000, total error < 2e-10: the exact match changes sign, so lambda(a) is in the printed window",
     "pass": true,
     "detail": "match -9.8336e-09 / -1.9831e-10 / 9.4370e-09 at lo / mid / hi; errors 6.6e-16, 6.6e-16; slope ~ 9.6353 per unit lambda"
    },
    {
     "check": "4. the certified windows are the printed decimals .2936535023 < lambda(.2936533) < .2936535043 and .2936533037 < lambda(.2936535) < .2936533057",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. lambda(2936533/10000000) > 2936533/10000000 and lambda(587307/2000000) < 587307/2000000 (Section 7, :7838-7839), hence (continuity of lambda, theory) a root in the bracket",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. for a in (.2936533, .2936535): gamma = 1 - 2a in [412693/1000000, 2063467/5000000] and theta = 2a/(1-2a) - 1 in [873066/2063467, 174614/412693] lie inside the printed 0.4126930 < gamma < 0.4126934 and 0.4231063 < theta < 264443/625000 (strict, a_* being strictly inside); gamma = 1/(2+theta); c_tail,+ = 2a-1 and c_tail,- = theta in the printed intervals (:7849-7850)",
     "pass": true,
     "detail": "theta endpoints 4231063544994904207336..., 4231087030795288507437..."
    },
    {
     "check": "4. the 16-digit theta endpoints printed at :8865 (0.4231063544994904..., 0.4231087030795289...) are the exact endpoints rounded to 16 digits",
     "pass": true,
     "detail": "exact: 0.4231063544994904207336... and 0.4231087030795288507437..."
    },
    {
     "check": "the appendix listing (681 lines) hashes to the printed source sha256 7bd5dbd9...85dc7f (:9496-9497): the cited certificate/original_certificate.py, absent as a file, is published byte for byte in the TeX",
     "pass": true,
     "detail": "7bd5dbd918d9a48fe4da62044e295239b74f7a1607e2643337ad4d073a85dc7f"
    },
    {
     "check": "the published starting data of the final two tests (db, Appendix app:code: M(0), M'(0), seed) agree with the independently shot ones to < 1e-28, and the db seeds are the text's seeds",
     "pass": true,
     "detail": "49 db entries"
    }
   ],
   "sources": {
    "preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/build/main.tex": "4d46d60a438d2c7e07c56c5e52ab9c592f58a6af46f8f78365984dc7cdee2be5"
   },
   "forges": [
    {
     "forge": "the printed seed for lambda(.2936533) moved by +30 (window (.2936535053, .2936535073))",
     "verdict": "REFUTED"
    },
    {
     "forge": "the upper claim moved to lambda(.2936534) < .2936534 (window around the true lambda(.2936534))",
     "verdict": "REFUTED"
    },
    {
     "forge": "the printed theta upper bound .4231088 -> .4231080",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 11.5
  },
  {
   "id": "F-426",
   "index": 426,
   "family": "227",
   "title": "Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase",
   "tier": "A",
   "dir": "Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026",
   "leanChallenges": [
    "CriticalSK",
    "CriticalSKMixing",
    "SKBarriers",
    "SKHighTemperature",
    "SKRatio"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Lemma rcut:lem:scalar-data and Table rcut:tab:scalar-intervals of the appendix route (the independent beta < 1/2 proof) - the 205 quadrature terms, every finite enclosure, the whole error budget and every interval bound recomputed by the appendix's directed integer recipe; the trapezoid lemma, strip holomorphy and beta-monotonicity are read, not run, and the cutoff headline (all beta < 1) is proved elsewhere in the paper by a different argument",
   "decider": "instruments/openaimath/finite/f426_sk_cutoff.py",
   "decides": "a finite component: Lemma rcut:lem:scalar-data and Table rcut:tab:scalar-intervals of the appendix route (the independent beta < 1/2 proof) - the 205 quadrature terms, every finite enclosure, the whole error budget and every interval bound recomputed by the appendix's directed integer recipe; the trapezoid lemma, strip holomorphy and beta-monotonicity are read, not run, and the cutoff headline (all beta < 1) is proved elsewhere in the paper by a different argument",
   "value": {
    "finite_at_beta": {
     "7/20": [
      "[0.8901398309, 0.8901398309]",
      "[0.5538146628, 0.5538146628]",
      "[0.128291983, 0.128291983]",
      "[0.08441159333, 0.08441159333]",
      "[0.08133207943, 0.08133207943]",
      "[0.01498195839, 0.01498195839]",
      "[0.6590343965, 0.6590343965]",
      "[0.2375531759, 0.2375531759]"
     ],
     "2/5": [
      "[0.8606483369, 0.8606483369]",
      "[0.5457666687, 0.5457666687]",
      "[0.1555041021, 0.1555041021]",
      "[0.0930998528, 0.0930998528]",
      "[0.09575125135, 0.09575125135]",
      "[0.01840653516, 0.01840653516]",
      "[0.7295180369, 0.7295180369]",
      "[0.2475280132, 0.2475280132]"
     ],
     "89/200": [
      "[0.8323358215, 0.8323358215]",
      "[0.5369203716, 0.5369203716]",
      "[0.1793540037, 0.1793540037]",
      "[0.1014793043, 0.1014793043]",
      "[0.1073850431, 0.1073850431]",
      "[0.02198813031, 0.02198813031]",
      "[0.7863777658, 0.7863777658]",
      "[0.2527658308, 0.2527658308]"
     ],
     "19/40": [
      "[0.8126981467, 0.8126981467]",
      "[0.5301843572, 0.5301843572]",
      "[0.1946884925, 0.1946884925]",
      "[0.1074388722, 0.1074388722]",
      "[0.1143162599, 0.1143162599]",
      "[0.0245493575, 0.0245493575]",
      "[0.8206696534, 0.8206696534]",
      "[0.2546250259, 0.2546250259]"
     ],
     "1/2": [
      "[0.7959457344, 0.7959457344]",
      "[0.5240712591, 0.5240712591]",
      "[0.2070329196, 0.2070329196]",
      "[0.1126327237, 0.1126327237]",
      "[0.1195534925, 0.1195534925]",
      "[0.02674439089, 0.02674439089]",
      "[0.8469768177, 0.8469768177]",
      "[0.2553295781, 0.2553295781]"
     ]
    },
    "five_endpoint": [
     "[0.4277180448, 0.4277180448]",
     "[0.08985934829, 0.08985934829]",
     "[0.09257698763, 0.09257698763]",
     "[0.05228099106, 0.05228099106]",
     "[0.2288748067, 0.2288748067]"
    ],
    "E_var": "1.311000e-05",
    "E_sd": "1.638751e-04",
    "e_a": "4.600001e-07",
    "E_par": "9.384570e-07",
    "tail": "[2.611916927e-12, 2.611916927e-12]",
    "whole_line": "[1.170081156e-07, 1.170081156e-07]",
    "interval_rows": [
     [
      "0.5138190",
      "0.4376394",
      "0.8429338"
     ],
     [
      "0.5140748",
      "0.4319050",
      "0.8964691"
     ],
     [
      "0.5159021",
      "0.4290269",
      "0.8978703"
     ],
     [
      "0.5131886",
      "0.4260221",
      "0.9145028"
     ]
    ],
    "scale": "S = 10^50",
    "runtime_s": 0.05
   },
   "checks": [
    {
     "check": "every bound decided below is printed in the paper as quoted (55 statements)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the published object read: 5 x 8 finite values, 5 endpoint values, 5 x 8 moment centers, 5 + 5 strict bounds, 4 x 4 interval bounds, 4 parenthesized coefficients",
     "pass": true,
     "detail": "temperatures ['7/20', '2/5', '89/200', '19/40', '1/2']"
    },
    {
     "check": "every exponential argument satisfies |d| <= 36.125 (and so |d/32| < 1.13)",
     "pass": true,
     "detail": "max |d| = 289/8"
    },
    {
     "check": "Table scalar-finite-values column N(V): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 3.37e-07; []"
    },
    {
     "check": "Table scalar-finite-values column N(P): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 3.72e-07; []"
    },
    {
     "check": "Table scalar-finite-values column sqrt T(V): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 4.92e-07; []"
    },
    {
     "check": "Table scalar-finite-values column sqrt T(P): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 4.07e-07; []"
    },
    {
     "check": "Table scalar-finite-values column N(yV): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 4.93e-07; []"
    },
    {
     "check": "Table scalar-finite-values column N(rho): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 4.65e-07; []"
    },
    {
     "check": "Table scalar-finite-values column sqrt T(f): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 3.96e-07; []"
    },
    {
     "check": "Table scalar-finite-values column sqrt T(yV): the 5 printed centers within 1e-6",
     "pass": true,
     "detail": "largest |computed - printed| 4.22e-07; []"
    },
    {
     "check": "eq:scalar-five-finite-values: k(0) finite within 1e-6 of 213859/500000",
     "pass": true,
     "detail": "[0.4277180448, 0.4277180448]"
    },
    {
     "check": "eq:scalar-five-finite-values: k(1) finite within 1e-6 of 89859/1000000",
     "pass": true,
     "detail": "[0.08985934829, 0.08985934829]"
    },
    {
     "check": "eq:scalar-five-finite-values: d(0) finite within 1e-6 of 92577/1000000",
     "pass": true,
     "detail": "[0.09257698763, 0.09257698763]"
    },
    {
     "check": "eq:scalar-five-finite-values: d'(0) finite within 1e-6 of 52281/1000000",
     "pass": true,
     "detail": "[0.05228099106, 0.05228099106]"
    },
    {
     "check": "eq:scalar-five-finite-values: d(1) finite within 1e-6 of 1831/8000",
     "pass": true,
     "detail": "[0.2288748067, 0.2288748067]"
    },
    {
     "check": "every finite quantity within .00065 of its Table scalar-moments center (40 entries)",
     "pass": true,
     "detail": "largest 0.000496"
    },
    {
     "check": "every finite standard deviation above .08 (20 entries)",
     "pass": true,
     "detail": "smallest lower endpoint 0.084412"
    },
    {
     "check": "pi = 16 arctan(1/5) - 4 arctan(1/239), ten terms each: 3.14159 < pi < 3.14160",
     "pass": true,
     "detail": "[3.141592653589792, 3.141592653589793]"
    },
    {
     "check": "|Im(beta x)| <= .5 x 1.5 = .75 < pi/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|Lambda| >= .14 for |m| <= 1: Lambda has no zero in the closed disk (disc < 0, |root|^2 = 1.7856 > 1) and on |m| = 1, |Lambda|^2 = .0436 - .084c + .45c^2 >= 124/3125 > .14^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|m| <= 1 gives |v| <= 2, |1.75 - m| >= .75, |2 - m| >= 1, |g| <= 2/.75 = 8/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "|F_{a',G'}(beta x)| <= (2+1) .5 |x| + [2 beta^2 1 2 + .5 x 2 + .12 + 3 beta^2 (8/3)] = 1.5|x| + 4.12 at beta <= .5, using 0 <= beta^2 a'/l' <= .25/.45 <= 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "edge |x| <= |Re x| + 1.5: 4.12 + 1.5 x 1.5 = 6.37; Gaussian edge factor e^{1.5^2/2} = e^{1.125}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "e^{1.125} < 4 and sqrt(1.25) < 1.12, so e^{1.125}(1.5 sqrt 1.25 + 6.37)^2 < 4 (1.68 + 6.37)^2 < 260",
     "pass": true,
     "detail": "e^1.125 = [3.080216849, 3.080216849]; 4(8.05)^2 = 25921/100"
    },
    {
     "check": "M = 1000 covers every edge integral: 260 and e^{1.125} x max(8/3, 2 + 1.5(8/3)^2 = 38/3, 9/(16 x .14), (8/3)^2, 4, 1) < 1000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "tail series constants: 1.5 x 8.4 + 4.12 = 16.72, 8.4 - .5 = 7.9, 7.9^2/2 = 31.205, 16.72 + .6 = 17.32, (8.3^2 - 7.9^2)/2 = 3.24, and (1.5|x|+4.12)^2 dominates (|x| + 1.87)^2 and 38/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "q_* = e^{-3.24}(17.32/16.72)^2 < .042026",
     "pass": true,
     "detail": "[0.04202513409, 0.04202513409]"
    },
    {
     "check": "eq:scalar-tail-error: .8(16.72)^2 e^{-31.205}/(sqrt(2 pi)(1 - q_*)) < 2.612e-12",
     "pass": true,
     "detail": "[2.611916927e-12, 2.611916927e-12]"
    },
    {
     "check": "whole-line error 2M/(e^{2 pi t/Delta} - 1), 2 pi 1.5/.4 = 7.5 pi: 2000/(e^{7.5 pi} - 1) < 1.171e-7 (pi > 3.14159)",
     "pass": true,
     "detail": "[1.170081156e-07, 1.170081156e-07]"
    },
    {
     "check": "E_quad = 2.3e-7 >= 1.171e-7 + 2.612e-12",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E_var = E_2 + E_1(8 + E_1) < 1.312e-5",
     "pass": true,
     "detail": "1.311000e-05"
    },
    {
     "check": "E_sd = E_var/.08 < 1.64e-4",
     "pass": true,
     "detail": "1.638751e-04"
    },
    {
     "check": "e_a = 2E_quad/(1 - E_quad) < 4.61e-7",
     "pass": true,
     "detail": "4.600001e-07"
    },
    {
     "check": "E_par = (1/2 + .64/(4 .45^2) + 3/4) e_a < 1e-6, with .64/(4 .45^2) = (1/4)(.64)/(.45)^2",
     "pass": true,
     "detail": "9.384570e-07"
    },
    {
     "check": "E_sd + E_par < .000165 and .00065 + .000165 = .000815 < .001",
     "pass": true,
     "detail": "1.648135e-04"
    },
    {
     "check": "|F| <= 2|h| + 1.87 on the real line (.5 + .5 + .12 + .75) and ||F||_2 <= 2 sqrt(.25 + .0625) + 1.87 < 3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the exponential recipe: e^{1.13} < 4 (Taylor upper bound 3.095657), 36.125/32 < 1.13, (1 + 16(1.13)^21/21!)^32 - 1 < 1e-12",
     "pass": true,
     "detail": "1.305e-16"
    },
    {
     "check": "Lemma scalar-data: every computed finite enclosure, widened by its transfer error, lies within .001 of its Table scalar-moments center (40 entries)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:scalar-five-enclosures: k(0) in (1069/2500, 2139/5000), strictly, after the e_a transfer",
     "pass": true,
     "detail": "[0.4277180448, 0.4277180448]"
    },
    {
     "check": "eq:scalar-five-enclosures: k(1) in (449/5000, 9/100), strictly, after the e_a transfer",
     "pass": true,
     "detail": "[0.08985934829, 0.08985934829]"
    },
    {
     "check": "eq:scalar-five-enclosures: d(0) in (37/400, 927/10000), strictly, after the e_a transfer",
     "pass": true,
     "detail": "[0.09257698763, 0.09257698763]"
    },
    {
     "check": "eq:scalar-five-enclosures: d'(0) in (261/5000, 131/2500), strictly, after the e_a transfer",
     "pass": true,
     "detail": "[0.05228099106, 0.05228099106]"
    },
    {
     "check": "eq:scalar-five-enclosures: d(1) in (143/625, 229/1000), strictly, after the e_a transfer",
     "pass": true,
     "detail": "[0.2288748067, 0.2288748067]"
    },
    {
     "check": "the five scaled endpoint integrands are bounded by one on the real line: (1+V)^2/(16 Lambda) <= 4/(16 x .4464), V^2/(16 Lambda), V^2/(4(2-y)), V^2/(4(2-y)^2) <= 1/4, V(1+y)/4 <= (32/27)/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [7/20, 2/5]: U_0 = 0.5138190 > 256909/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [7/20, 2/5]: U_1 = 0.4376394 > 218819/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [7/20, 2/5]: 2 sqrt(U_0 U_1) > 948403/1000000 (4 U_0 U_1 > 948403/1000000^2)",
     "pass": true,
     "detail": "2 sqrt = 0.9484038"
    },
    {
     "check": "Table scalar-intervals [7/20, 2/5]: s_F-bar < 168587/200000",
     "pass": true,
     "detail": "[0.8429338062, 0.8429338062]"
    },
    {
     "check": "Table scalar-intervals [2/5, 89/200]: U_0 = 0.5140748 > 257037/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [2/5, 89/200]: U_1 = 0.4319050 > 13497/31250",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [2/5, 89/200]: 2 sqrt(U_0 U_1) > 235601/250000 (4 U_0 U_1 > 235601/250000^2)",
     "pass": true,
     "detail": "2 sqrt = 0.9424044"
    },
    {
     "check": "Table scalar-intervals [2/5, 89/200]: s_F-bar < 89647/100000",
     "pass": true,
     "detail": "[0.896469139, 0.896469139]"
    },
    {
     "check": "Table scalar-intervals [89/200, 19/40]: U_0 = 0.5159021 > 257951/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [89/200, 19/40]: U_1 = 0.4290269 > 214513/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [89/200, 19/40]: 2 sqrt(U_0 U_1) > 470463/500000 (4 U_0 U_1 > 470463/500000^2)",
     "pass": true,
     "detail": "2 sqrt = 0.9409269"
    },
    {
     "check": "Table scalar-intervals [89/200, 19/40]: s_F-bar < 897871/1000000",
     "pass": true,
     "detail": "[0.8978702804, 0.8978702804]"
    },
    {
     "check": "Table scalar-intervals [19/40, 1/2]: U_0 = 0.5131886 > 128297/250000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [19/40, 1/2]: U_1 = 0.4260221 > 213011/500000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table scalar-intervals [19/40, 1/2]: 2 sqrt(U_0 U_1) > 935157/1000000 (4 U_0 U_1 > 935157/1000000^2)",
     "pass": true,
     "detail": "2 sqrt = 0.9351571"
    },
    {
     "check": "Table scalar-intervals [19/40, 1/2]: s_F-bar < 114313/125000",
     "pass": true,
     "detail": "[0.9145027622, 0.9145027622]"
    },
    {
     "check": "the parenthesized coefficients a_-^2/l_+ - a_- - 1.5 G_+^2 exceed the printed .229787, .152558, .111290, .074675",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the monotonicity step applies: every a_+-, l_+- in [.795, .891], [.45, .489], where 2a > l (so a^2/l - a increases in a)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "uniform margins from the printed table: U_0 > .513, U_1 > .426, s_F < .920, 2 sqrt(U_0 U_1) > .930",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the absorption: .920 = (92/93) .930, and -(1/93)(U_0 A^2 + U_1 |b_0|^2) <= -(.426/93)|b|^2 since .513 > .426",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the radius-.0003 boxes about .4277, .0899, .0926, .0523, .2289 contain the five strict enclosures",
     "pass": true,
     "detail": ""
    },
    {
     "check": "envelope arithmetic: .45 + d(0) + k(0) <= .9709, .70 + k(1) + d(1) - d'(0) <= .9674, |d'(0) - .06| <= .0080, sum .9789, + .018 = .9969",
     "pass": true,
     "detail": "9709/10000, 4837/5000, 1/125"
    },
    {
     "check": "Lambda = .45 + .25m^2 - .06m = .4464 + .25(m - .12)^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "l = <Lambda> = .45 + (.25 - .06)(1 - a) = .64 - .19a",
     "pass": true,
     "detail": ""
    },
    {
     "check": "32/27 - (1-m)(1+m)^2 = (m - 1/3)^2 (m + 5/3) >= 0 on [-1, 1], so (1+m)v <= 32/27 and rho <= 16/27",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1.75 - m - (1 - m^2) = (m - 1/2)^2 + 1/2 > 0, so g = v/(1.75 - m) is in [0, 1]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "averaging g over the sign gives (1-u)(1.75+u)/(1.75^2-u), u = tanh^2|h| (cross-multiplied identity in s = tanh|h|)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "d/du (1-u)(1.75+u)/(1.75^2-u) = 4(64u^2 - 392u - 35)/(16u - 49)^2, and 64u^2 - 392u - 35 < 0 on [0, 1] (convex; -35 and -363 at the ends)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "d/da' [a'/(.64 - .19a')] = ((.64 - .19a') + .19a')/(.64 - .19a')^2 = .64/(.64 - .19a')^2",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/appendices/scalar-certificate.tex": "b6e8f0c8689eeff9954731c19bc14449604aff36ae2513e66d708dc745bcc538",
    "preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/appendices/scalar.tex": "847bafff28f156840aebc736758523c19570d913339774205029cb26c2f207fe",
    "preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/build/sections/01-introduction.tex": "da85ec3824aaf200eef450894bed91efbf0df938a9e2e6b5fb82b6d09fa1316d"
   },
   "forges": [
    {
     "forge": "Table scalar-finite-values: N(V) at beta = .4 printed .860648 -> .860658 (moved 1e-5, radius 1e-6)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table scalar-intervals: U_0 on [.350, .400] printed .513818 -> .523818 (not a lower bound)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table scalar-intervals: s_F-bar on [.475, .500] printed .914504 -> .904504 (not an upper bound)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.0
  },
  {
   "id": "F-433",
   "index": 433,
   "family": "229",
   "title": "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model",
   "tier": "A",
   "dir": "The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026",
   "leanChallenges": [
    "ThreeStateSupercritical",
    "ThreeStateTreeClauses"
   ],
   "word": "REFUSED",
   "why": "decided in part (pre-registration amendment 5): the product inequality (b) is decided only on C x S, S x C and C' x C' (about 0.4% of S x S by volume; the rest projected above 20 CPU-hours at ~5 s per cell); the scalar certificate and inequality (a) on all of S x S are decided",
   "decider": "instruments/openaimath/finite/f433_potts_four.py",
   "decides": "a finite component: the single-posterior certificate and the product inequality (a) on all of S x S — not the probabilistic reduction",
   "value": {
    "scalar": {
     "H+1/2": [
      "19/50",
      84
     ],
     "F": [
      "0",
      56
     ],
     "G": [
      "87/1000",
      680
     ],
     "(3-E)H": [
      "0",
      84
     ]
    },
    "authors_counts": {
     "a": [
      4479,
      19,
      166,
      2074,
      0
     ],
     "b": [
      1935,
      20,
      97,
      871,
      0
     ]
    },
    "row_a_R_at_vertex_pairs": {
     "00": "8136/125",
     "01": "291/25",
     "02": "16384/3375",
     "03": "0",
     "10": "291/25",
     "11": "2378/125",
     "12": "25471/10125",
     "13": "0",
     "20": "16384/3375",
     "21": "25471/10125",
     "22": "194104/164025",
     "23": "0",
     "30": "0",
     "31": "0",
     "32": "0",
     "33": "0"
    },
    "row_a": {
     "cells": 3090,
     "glob": 88,
     "cell": 1435,
     "maxdepth": 19,
     "cap": 0,
     "subtrees": 45,
     "cpu_seconds": 468.2,
     "wall_seconds": 468.5,
     "undecided_volume": "0",
     "undecided_cells": 0,
     "seconds": 483.0,
     "region": "all of S x S"
    },
    "seconds": 483.2
   },
   "checks": [
    {
     "check": "coefficient table eq:polynomial-table parsed (H, F, G; 9 columns each)",
     "pass": true,
     "detail": "{'H': [0, 0, 1000, -100, -150, 300, -20, 70, -510], 'F': [0, 4000, -3000, 700, -4000, -800, 0, 0, 0], 'G': [1000, -1610, -3100, 773, 4000, -3980, 0, 0, 0]}"
    },
    {
     "check": "H(e_i) = 13/100, H(u) = 0, H(1/2,1/2,0,0) = -3/25 (03-cone.tex:76-81)",
     "pass": true,
     "detail": "13/100, 0, -3/25"
    },
    {
     "check": "all 24 coordinate permutations act on x = Lp as signed permutations with sign product +1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table tab:scalar-coefficients parsed (4 rows)",
     "pass": true,
     "detail": "[('H+1/2', 6, 84, Fraction(19, 50)), ('F', 5, 56, Fraction(0, 1)), ('G', 14, 680, Fraction(87, 1000)), ('(3-\\\\mathcal E)H', 6, 84, Fraction(0, 1))]"
    },
    {
     "check": "scalar form H+1/2: degree 6, 84 coefficients, minimum 19/50 (printed 84, 19/50)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form H+1/2: all coefficients >= 0 (the inequality on S)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form F: degree 5, 56 coefficients, minimum 0 (printed 56, 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form F: all coefficients >= 0 (the inequality on S)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form G: degree 14, 680 coefficients, minimum 87/1000 (printed 680, 87/1000)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form G: all coefficients >= 0 (the inequality on S)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form (3-E)H: degree 6, 84 coefficients, minimum 0 (printed 84, 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "scalar form (3-E)H: all coefficients >= 0 (the inequality on S)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "row a: R >= 0 at the 16 vertex pairs of S x S (exact)",
     "pass": true,
     "detail": "min 0"
    },
    {
     "check": "row a: the global Taylor array is a sum over S4-symmetric U-monomials (grouping valid)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "row a: Bernstein array = direct formula at 3 rational points; per-cell matrix = direct bound (exact)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "row a: every cell of the subdivision accepted (R >= 0 on all of S x S)",
     "pass": true,
     "detail": "{'cells': 3090, 'glob': 88, 'cell': 1435, 'maxdepth': 19, 'cap': 0, 'subtrees': 45, 'cpu_seconds': 468.2, 'wall_seconds': 468.5}; undecided volume fraction 0"
    }
   ],
   "sources": {
    "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/build/sections/02-experiments.tex": "68099107ef8d7096841e53be71467f6f3c07710afe53a600c277fe6272a668d1",
    "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/build/sections/03-cone.tex": "c5a7a96a2c337de4ca67b9c6d40ac8b6051367220881ff7f3e786a51bce87250",
    "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/build/sections/05-polynomial-certificates.tex": "00ac158765605d801bbb1e8a51a8df32e2d95948273f6ac025a71d7652c7486b",
    "preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/build/sections/06-finite-verification.tex": "a28c28ca311b64494893f2b51eaf9e1e93bedea15a0ff55f57e3a769bc18ea8f"
   },
   "forges": [
    {
     "forge": "printed minimum of G changed to 88/1000",
     "verdict": "REFUSED"
    },
    {
     "forge": "D0 with 1/10 in place of 1/1000: R(S0,S0) = -7902/125",
     "verdict": "REFUTED"
    },
    {
     "forge": "H with B-coefficient 1300 in place of 1000",
     "verdict": "REFUSED"
    }
   ],
   "seconds": 483.2
  },
  {
   "id": "F-435",
   "index": 435,
   "family": "229",
   "title": "The exact reconstruction threshold for the three-state symmetric channel",
   "tier": "A",
   "dir": "The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026",
   "leanChallenges": [
    "ThreeStateSupercritical",
    "ThreeStateTreeClauses"
   ],
   "word": "REFUSED",
   "why": "decided in part (pre-registration amendment 5): the interval certificate is decided only for q <= 4/5; q > 4/5, which holds the near-uniform corner where P ~ 2e-5, is not (extending band 2 to q <= 9/10 alone took 84,610 boxes and 1,916 s); the pair certificate (all 14 polynomials) and the grid certificate (all 28 stopping iterations) are decided whole",
   "decider": "instruments/openaimath/finite/f435_potts_three.py",
   "decides": "a finite component: the pair coefficient certificate (14 polynomials); the grid certificate for case(s) all 28 (-1, 0..26); the interval inequality P > 0 of Lemma wei:positive on the part q <= 4/5 of its domain (max(p_2, p_3)/p_1 <= (4/5)^18), all three bands, all xi in [.03, 1] — not the analytic reductions",
   "value": {
    "pair": {
     "counts": [
      13672,
      13672,
      13674,
      13625,
      13625,
      13625,
      13625,
      13672,
      13672,
      13674,
      13625,
      13625,
      13625,
      13625
     ],
     "seconds": 82.1
    },
    "grid": {
     "-1": 162,
     "0": 104,
     "1": 103,
     "2": 103,
     "3": 103,
     "4": 102,
     "5": 102,
     "6": 102,
     "7": 102,
     "8": 102,
     "9": 101,
     "10": 101,
     "11": 101,
     "12": 101,
     "13": 101,
     "14": 101,
     "15": 100,
     "16": 100,
     "17": 100,
     "18": 100,
     "19": 100,
     "20": 100,
     "21": 100,
     "22": 99,
     "23": 99,
     "24": 99,
     "25": 99,
     "26": 99
    },
    "interval": {
     "qmax": "4/5",
     "boxes": [
      392,
      3504,
      20324
     ],
     "seconds": [
      1.8,
      13.0,
      74.1
     ]
    },
    "seconds": 1688.7
   },
   "checks": [
    {
     "check": "pair: printed nonzero-coefficient counts parsed",
     "pass": true,
     "detail": "[13672, 13672, 13674, 13625, 13625, 13625, 13625]"
    },
    {
     "check": "pair: the certified Lemma loc:pair as printed (coarse bounds 13/2, 79/250, 11)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "pair: E = 2Q (coloring edge e_i = Q - q_i)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "pair: sum_j T_j = PE in both reflection classes (tilt weights sum to one)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "pair: the parametrisation inverse gives p = 2h/h_2 (h = (7,4,3))",
     "pass": true,
     "detail": "[Fraction(14, 3), Fraction(8, 3), Fraction(2, 1)]"
    },
    {
     "check": "pair: reflection 1, polynomial 1: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 341943436826369851392, 13672 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 1: nonzero coefficients = printed 13672",
     "pass": true,
     "detail": "13672"
    },
    {
     "check": "pair: reflection 1, polynomial 2: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 339447499331286859776, 13672 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 2: nonzero coefficients = printed 13672",
     "pass": true,
     "detail": "13672"
    },
    {
     "check": "pair: reflection 1, polynomial 3: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 3166423329333420490752, 13674 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 3: nonzero coefficients = printed 13674",
     "pass": true,
     "detail": "13674"
    },
    {
     "check": "pair: reflection 1, polynomial 4: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 54910624891825815552, 13625 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 4: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 1, polynomial 5: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 11994027762626592768, 13625 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 5: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 1, polynomial 6: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 19967499960663932928, 13625 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 6: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 1, polynomial 7: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 143928333151519113216, 13625 nonzero"
    },
    {
     "check": "pair: reflection 1, polynomial 7: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 2, polynomial 1: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 341943436826369851392, 13672 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 1: nonzero coefficients = printed 13672",
     "pass": true,
     "detail": "13672"
    },
    {
     "check": "pair: reflection 2, polynomial 2: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 339447499331286859776, 13672 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 2: nonzero coefficients = printed 13672",
     "pass": true,
     "detail": "13672"
    },
    {
     "check": "pair: reflection 2, polynomial 3: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 3166423329333420490752, 13674 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 3: nonzero coefficients = printed 13674",
     "pass": true,
     "detail": "13674"
    },
    {
     "check": "pair: reflection 2, polynomial 4: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 54910624891825815552, 13625 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 4: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 2, polynomial 5: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 11994027762626592768, 13625 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 5: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 2, polynomial 6: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 19967499960663932928, 13625 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 6: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "pair: reflection 2, polynomial 7: degree 28, integer coefficients, all >= 0",
     "pass": true,
     "detail": "min coefficient 143928333151519113216, 13625 nonzero"
    },
    {
     "check": "pair: reflection 2, polynomial 7: nonzero coefficients = printed 13625",
     "pass": true,
     "detail": "13625"
    },
    {
     "check": "grid: printed outputs parsed (27 Poisson values and the regular 162)",
     "pass": true,
     "detail": "[104, 103, 103, 103, 102, 102, 102, 102, 102, 101, 101, 101, 101, 101, 101, 100, 100, 100, 100, 100, 100, 100, 99, 99, 99, 99, 99]; regular 162"
    },
    {
     "check": "grid: 469 sorted triples of sum 72",
     "pass": true,
     "detail": ""
    },
    {
     "check": "grid: overlap 881/200 - 1/(477/1000)^2 = 453049/45505800 > 0 (eq. grid:overlap)",
     "pass": true,
     "detail": "453049/45505800"
    },
    {
     "check": "grid: the bins [d_min/200, d_max/200] share endpoints and cover [4, 881/200]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "grid: magnitudes A (least with A^2 d_min >= 200*10^12) satisfy a_+ >= d^{-1/2} on the bin and a_+ <= 1/2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "grid: case -1 tables have sum_h w = D_tab for every ordered pair and modes 0, 2 symmetric",
     "pass": true,
     "detail": "{'entries': [3212035, 3133243], 'D': [373248, 31104]}"
    },
    {
     "check": "grid: case -1 stops at iteration 162 (printed 162)",
     "pass": true,
     "detail": "mu at stop = 1345824395183/27000000000000; 125.0 s"
    },
    {
     "check": "grid: case 0 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 500000, 'D': 62208, 'nweights': 26, 'entries': 3132537}"
    },
    {
     "check": "grid: case 0 stops at iteration 104 (printed 104)",
     "pass": true,
     "detail": "mu at stop = 3183755683819/64000000000000; 50.9 s"
    },
    {
     "check": "grid: case 1 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 499066, 'D': 15552000000, 'nweights': 26, 'entries': 3147153}"
    },
    {
     "check": "grid: case 1 stops at iteration 103 (printed 103)",
     "pass": true,
     "detail": "mu at stop = 3451773609311/69120000000000; 53.6 s"
    },
    {
     "check": "grid: case 2 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 498136, 'D': 3888000000, 'nweights': 26, 'entries': 3146720}"
    },
    {
     "check": "grid: case 2 stops at iteration 103 (printed 103)",
     "pass": true,
     "detail": "mu at stop = 86146094726117/1728000000000000; 50.5 s"
    },
    {
     "check": "grid: case 3 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 497212, 'D': 7776000000, 'nweights': 26, 'entries': 3146318}"
    },
    {
     "check": "grid: case 3 stops at iteration 103 (printed 103)",
     "pass": true,
     "detail": "mu at stop = 4779151524293/96000000000000; 51.7 s"
    },
    {
     "check": "grid: case 4 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 496292, 'D': 7776000000, 'nweights': 26, 'entries': 3145724}"
    },
    {
     "check": "grid: case 4 stops at iteration 102 (printed 102)",
     "pass": true,
     "detail": "mu at stop = 43182818298419/864000000000000; 51.3 s"
    },
    {
     "check": "grid: case 5 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 495378, 'D': 15552000000, 'nweights': 26, 'entries': 3145301}"
    },
    {
     "check": "grid: case 5 stops at iteration 102 (printed 102)",
     "pass": true,
     "detail": "mu at stop = 43129729919137/864000000000000; 53.1 s"
    },
    {
     "check": "grid: case 6 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 494469, 'D': 31104000000, 'nweights': 26, 'entries': 3144777}"
    },
    {
     "check": "grid: case 6 stops at iteration 102 (printed 102)",
     "pass": true,
     "detail": "mu at stop = 14360530486949/288000000000000; 53.7 s"
    },
    {
     "check": "grid: case 7 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 493564, 'D': 7776000000, 'nweights': 26, 'entries': 3144345}"
    },
    {
     "check": "grid: case 7 stops at iteration 102 (printed 102)",
     "pass": true,
     "detail": "mu at stop = 86063778215377/1728000000000000; 50.9 s"
    },
    {
     "check": "grid: case 8 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 492665, 'D': 6220800000, 'nweights': 26, 'entries': 3144039}"
    },
    {
     "check": "grid: case 8 stops at iteration 102 (printed 102)",
     "pass": true,
     "detail": "mu at stop = 85983360894119/1728000000000000; 50.4 s"
    },
    {
     "check": "grid: case 9 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 491771, 'D': 31104000000, 'nweights': 26, 'entries': 3143525}"
    },
    {
     "check": "grid: case 9 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 43174524025537/864000000000000; 53.2 s"
    },
    {
     "check": "grid: case 10 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 490881, 'D': 31104000000, 'nweights': 26, 'entries': 3143054}"
    },
    {
     "check": "grid: case 10 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 17252705632061/345600000000000; 53.0 s"
    },
    {
     "check": "grid: case 11 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 489996, 'D': 7776000000, 'nweights': 26, 'entries': 3142540}"
    },
    {
     "check": "grid: case 11 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 10772990232191/216000000000000; 50.6 s"
    },
    {
     "check": "grid: case 12 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 489116, 'D': 7776000000, 'nweights': 26, 'entries': 3141946}"
    },
    {
     "check": "grid: case 12 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 2152839209597/43200000000000; 50.6 s"
    },
    {
     "check": "grid: case 13 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 488241, 'D': 31104000000, 'nweights': 26, 'entries': 3141666}"
    },
    {
     "check": "grid: case 13 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 17210780654981/345600000000000; 53.0 s"
    },
    {
     "check": "grid: case 14 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 487371, 'D': 31104000000, 'nweights': 26, 'entries': 3141194}"
    },
    {
     "check": "grid: case 14 stops at iteration 101 (printed 101)",
     "pass": true,
     "detail": "mu at stop = 86001194747477/1728000000000000; 53.1 s"
    },
    {
     "check": "grid: case 15 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 486505, 'D': 6220800000, 'nweights': 26, 'entries': 3140841}"
    },
    {
     "check": "grid: case 15 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 2398932382399/48000000000000; 49.6 s"
    },
    {
     "check": "grid: case 16 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 485643, 'D': 31104000000, 'nweights': 26, 'entries': 3140706}"
    },
    {
     "check": "grid: case 16 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 4794120632879/96000000000000; 52.6 s"
    },
    {
     "check": "grid: case 17 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 484787, 'D': 31104000000, 'nweights': 26, 'entries': 3140012}"
    },
    {
     "check": "grid: case 17 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 43122643995047/864000000000000; 52.5 s"
    },
    {
     "check": "grid: case 18 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 483934, 'D': 15552000000, 'nweights': 26, 'entries': 3139526}"
    },
    {
     "check": "grid: case 18 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 9575771338643/192000000000000; 51.9 s"
    },
    {
     "check": "grid: case 19 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 483087, 'D': 31104000000, 'nweights': 26, 'entries': 3139234}"
    },
    {
     "check": "grid: case 19 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 10767179790271/216000000000000; 52.5 s"
    },
    {
     "check": "grid: case 20 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 482243, 'D': 31104000000, 'nweights': 26, 'entries': 3138617}"
    },
    {
     "check": "grid: case 20 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 43038733814183/864000000000000; 52.6 s"
    },
    {
     "check": "grid: case 21 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 481404, 'D': 7776000000, 'nweights': 26, 'entries': 3138236}"
    },
    {
     "check": "grid: case 21 stops at iteration 100 (printed 100)",
     "pass": true,
     "detail": "mu at stop = 21506282439629/432000000000000; 50.3 s"
    },
    {
     "check": "grid: case 22 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 480570, 'D': 3110400000, 'nweights': 27, 'entries': 3137595}"
    },
    {
     "check": "grid: case 22 stops at iteration 99 (printed 99)",
     "pass": true,
     "detail": "mu at stop = 21595026416717/432000000000000; 48.9 s"
    },
    {
     "check": "grid: case 23 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 479740, 'D': 1555200000, 'nweights': 27, 'entries': 3137198}"
    },
    {
     "check": "grid: case 23 stops at iteration 99 (printed 99)",
     "pass": true,
     "detail": "mu at stop = 86329331083919/1728000000000000; 48.6 s"
    },
    {
     "check": "grid: case 24 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 478914, 'D': 15552000000, 'nweights': 27, 'entries': 3136772}"
    },
    {
     "check": "grid: case 24 stops at iteration 99 (printed 99)",
     "pass": true,
     "detail": "mu at stop = 319550381317/6400000000000; 51.8 s"
    },
    {
     "check": "grid: case 25 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 478092, 'D': 7776000000, 'nweights': 27, 'entries': 3136184}"
    },
    {
     "check": "grid: case 25 stops at iteration 99 (printed 99)",
     "pass": true,
     "detail": "mu at stop = 3193615722601/64000000000000; 49.8 s"
    },
    {
     "check": "grid: case 26 tables have sum_h w = D_tab for every ordered pair",
     "pass": true,
     "detail": "{'A': 477274, 'D': 15552000000, 'nweights': 27, 'entries': 3135810}"
    },
    {
     "check": "grid: case 26 stops at iteration 99 (printed 99)",
     "pass": true,
     "detail": "mu at stop = 43088860781851/864000000000000; 52.1 s"
    },
    {
     "check": "interval: the parameter table eq. wei:parameters parsed (3 bands)",
     "pass": true,
     "detail": "[(Fraction(0, 1), Fraction(19, 100), -268, -60, -55, 130, Fraction(1, 16), Fraction(15, 32)), (Fraction(19, 100), Fraction(56, 125), -269, 67, 47, 55, Fraction(1, 16), Fraction(15, 32)), (Fraction(56, 125), Fraction(477, 1000), -271, 58, 49, 50, Fraction(1, 12), Fraction(1, 2))]"
    },
    {
     "check": "interval: the xi domain [.03, 1] and the 12 xi intervals as printed (int:xi-grid)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "interval: closed forms for W, Y, That, What, Yhat agree with their definitions at (q, r, a) = (3/4, 1/2, 2/5)",
     "pass": true,
     "detail": "{'W': True, 'Y': True, 'That': True, 'What': True, 'Yhat': True}"
    },
    {
     "check": "interval: P > 0 at the pure, uniform and one mixed message, both xi ends, each band top (point enclosures)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "interval: band 0 [0, 19/100]: P > 0 on {0 <= r <= q <= 4/5} x band x [3/100, 1] (all boxes accepted)",
     "pass": true,
     "detail": "392 boxes, 1.8 s, 0 open"
    },
    {
     "check": "interval: band 1 [19/100, 56/125]: P > 0 on {0 <= r <= q <= 4/5} x band x [3/100, 1] (all boxes accepted)",
     "pass": true,
     "detail": "3504 boxes, 13.0 s, 0 open"
    },
    {
     "check": "interval: band 2 [56/125, 477/1000]: P > 0 on {0 <= r <= q <= 4/5} x band x [3/100, 1] (all boxes accepted)",
     "pass": true,
     "detail": "20324 boxes, 74.1 s, 0 open"
    }
   ],
   "sources": {
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/appendix-certificates.tex": "febb78570d48362eac64a836f35b20541d7727dc8cd7eb13a03ab5fa7c5ad080",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/04-local.tex": "faa55078a370600f7cb8b9c4f7ac76905028948526317d00766eec129278de4f",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/appendix-grid.tex": "fc74a91b5ec1d5dcab8bb23c0d91bdd3ca3c792f3cdc2d582c5d2bc0f0132e24",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/07-conclusion.tex": "3dc6ed24c788e85aa61f2f7206104bc54f9885845b8e23958b11d3a39a23039d",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/06-grid.tex": "a2989f955efcd0891437eb77ea6f7f4305b22343f8aa6f374a667a6abe577aa4",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/05-weighted.tex": "e896690d5cd0642260c12de5a0ec7e2b082635a78a4ecc36d3ff98235fc31bae",
    "preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/build/sections/appendix-interval.tex": "c1bd6082f83444242bb6999c52b2fc50995d14896e44f8623e39d2dc150c9039"
   },
   "forges": [
    {
     "forge": "pair row 1 with 12Xz in place of 13Xz (d_n <= 6Xz)",
     "verdict": "REFUTED"
    },
    {
     "forge": "pair row 5 without the reflection correction I",
     "verdict": "REFUTED"
    },
    {
     "forge": "interval with xi in [0, 1] instead of [3/100, 1]",
     "verdict": "REFUTED"
    },
    {
     "forge": "grid regular case printed as 161 instead of 162",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1688.7
  },
  {
   "id": "F-467",
   "index": 467,
   "family": "238",
   "title": "Routing densities and representation contraction for Thorp sweeps",
   "tier": "A",
   "dir": "Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026",
   "leanChallenges": [
    "BinarySweep",
    "CoordinateSweeps",
    "CoordinateTrace",
    "FourRowPermanent",
    "OccupiedOverlap",
    "PartialPermutation",
    "SignedSweepMoment",
    "SpinAngle",
    "ThorpFirstReciprocal",
    "ThorpRemaining",
    "ThorpRouting",
    "ThorpWeightedCompatibility",
    "WeightedSweepMoments"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the five rational inequalities of Section 4 (H(1/8), H(13/50), H(1) and the two entropy brackets at 7/25 and 1) with every supporting finite fact the text states (closed form of h_j from its definition, h_j <= sqrt(2/M), increasing ratios, the 169/70 tail cap, slopes 7/3 and 3/2, the 25/24 slope bound) — not the probabilistic/asymptotic arguments, and not the headline, which rests on the factorial proof of Section 3; this section is the paper's independent second route",
   "decider": "instruments/openaimath/finite/f467_thorp_sweeps.py",
   "decides": "a finite component: the five rational inequalities of Section 4 (H(1/8), H(13/50), H(1) and the two entropy brackets at 7/25 and 1) with every supporting finite fact the text states (closed form of h_j from its definition, h_j <= sqrt(2/M), increasing ratios, the 169/70 tail cap, slopes 7/3 and 3/2, the 25/24 slope bound) — not the probabilistic/asymptotic arguments, and not the headline, which rests on the factorial proof of Section 3; this section is the paper's independent second route",
   "value": {
    "H_recipe_bounds": {
     "H(1/8)": "0.7914019315",
     "H(13/50)": "1.0980685981",
     "H(1)": "1.9580685981"
    },
    "H_true_enclosures": {
     "H(1/8)": [
      "0.791401384190",
      "0.791401384190"
     ],
     "H(13/50)": [
      "1.098068050856",
      "1.098068050856"
     ],
     "H(1)": [
      "1.958068050856",
      "1.958068050856"
     ]
    },
    "brackets_paper_series": {
     "bracket(7/25)": "0.9040537558",
     "bracket(1)": "0.5153735573"
    },
    "brackets_independent": {
     "bracket(7/25)": "0.9040532085",
     "bracket(1)": "0.5153730100"
    },
    "facts": {
     "cited_checker": "support/network_bounds.py (04-certificate.tex:294)",
     "cited_checker_in_release": false,
     "release_files": [
      "README.md",
      "build/figures/network-node.tex",
      "build/main.tex",
      "build/references.bib",
      "build/sections/00-introduction.tex",
      "build/sections/01-preliminaries.tex",
      "build/sections/02-routing.tex",
      "build/sections/03-factorial.tex",
      "build/sections/04-certificate.tex",
      "build/sections/05-strong-providers.tex",
      "build/sections/06-cycle-windows.tex",
      "build/sections/06b-contact-cancellation.tex",
      "build/sections/07-casimir.tex",
      "build/sections/08-harmonic-cycle.tex",
      "build/sections/09-adaptive-alphabets.tex",
      "build/sections/10-core-cutoff.tex",
      "paper.pdf"
     ]
    },
    "seconds": 0.17
   },
   "checks": [
    {
     "check": "the headline as printed (abstract; Theorem fac:main)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the five printed inequalities and the cap, read from 04-certificate.tex:270-287",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. h_j from its definition (the LP over laws of L, greedy) equals the printed closed form (net:hj), j = 1..24",
     "pass": true,
     "detail": "h_1..h_4 = ['1', '3/4', '7/12', '7/16']"
    },
    {
     "check": "1. q(q+1) <= 2M < (q+1)(q+2) for the q used, and the simplification q/(2M) + 1/(q+1), j = 1..200",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. h_j <= sqrt(2/M) (as h_j^2 <= 2/M), j = 1..200; the endpoint values -4q and -4(q+1) of (q + x/(q+1))^2 - 4x are identities (degree 2 in q, checked at four points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the allocation ratios 2^(j-1) h_j strictly increase, j = 1..200 (general j: the LP value increases with M — theory)",
     "pass": true,
     "detail": "ratios 1..5: ['1', '3/2', '7/3', '7/2', '31/6']"
    },
    {
     "check": "4. (1/2) sum_{j>20} 2^(1-j/2) = (1+sqrt2)/1024 (even j: 2^-9; odd j: sqrt2 2^-9); (99/70)^2 > 2 so it is < 169/(70*1024)",
     "pass": true,
     "detail": "cap 169/71680 = 0.0023577009"
    },
    {
     "check": "4. independently: (1/2) sum_{j>20} h_j <= exact levels 21..200 + geometric tail < the printed cap",
     "pass": true,
     "detail": "(1/2) sum_{j>20} h_j <= 2.3571536012e-03 vs cap 2.3577008929e-03"
    },
    {
     "check": "5. H(1/8) < .791402 by the paper's recipe (levels 1..20 greedy with budget rho - 2^-20, plus the cap)",
     "pass": true,
     "detail": "recipe bound 0.791401931; margin 6.852e-08"
    },
    {
     "check": "5. H(1/8) < .791402 by the independent enclosure (200 exact levels + geometric tail)",
     "pass": true,
     "detail": "H in [0.791401384190, 0.791401384190]"
    },
    {
     "check": "5. H(13/50) < 1.098069 by the paper's recipe (levels 1..20 greedy with budget rho - 2^-20, plus the cap)",
     "pass": true,
     "detail": "recipe bound 1.098068598; margin 4.019e-07"
    },
    {
     "check": "5. H(13/50) < 1.098069 by the independent enclosure (200 exact levels + geometric tail)",
     "pass": true,
     "detail": "H in [1.098068050856, 1.098068050856]"
    },
    {
     "check": "5. H(1) < 1.958069 by the paper's recipe (levels 1..20 greedy with budget rho - 2^-20, plus the cap)",
     "pass": true,
     "detail": "recipe bound 1.958068598; margin 4.019e-07"
    },
    {
     "check": "5. H(1) < 1.958069 by the independent enclosure (200 exact levels + geometric tail)",
     "pass": true,
     "detail": "H in [1.958068050856, 1.958068050856]"
    },
    {
     "check": "6. the two ln 2 enclosures (paper's 40-term atanh(1/3) series with remainder; sum 1/(k 2^k) with geometric tail) overlap",
     "pass": true,
     "detail": "paper [0.693147180559945, 0.693147180559945]"
    },
    {
     "check": "6. -ln(1 - 7/25): the paper's 80-term upper bound lies above the independent atanh enclosure's lower end",
     "pass": true,
     "detail": "-ln(18/25) in [0.328504066972036, 0.328504066972036]"
    },
    {
     "check": "6a. bracket(7/25) < .904054 with the paper's series (numerator < 0, divided by the upper ln 2)",
     "pass": true,
     "detail": "bound 0.9040537558; margin 2.442e-07"
    },
    {
     "check": "6b. bracket(7/25) < .904054 independently (true-H enclosure, atanh -ln(1-rho), ln 2 = sum 1/(k 2^k))",
     "pass": true,
     "detail": "bound 0.9040532085; margin 7.915e-07"
    },
    {
     "check": "6a. bracket(1) < .515374 with the paper's series (numerator < 0, divided by the upper ln 2)",
     "pass": true,
     "detail": "bound 0.5153735573; margin 4.427e-07"
    },
    {
     "check": "6b. bracket(1) < .515374 independently (true-H enclosure, atanh -ln(1-rho), ln 2 = sum 1/(k 2^k))",
     "pass": true,
     "detail": "bound 0.5153730100; margin 9.900e-07"
    },
    {
     "check": "7. the slopes of H: 4 h_3 = 7/3 on [1/8,1/4] and 2 h_2 = 3/2 on [1/4,7/25] (also as exact differences of the greedy H)",
     "pass": true,
     "detail": "difference slopes 7/3, 3/2"
    },
    {
     "check": "7. ln 2 > 2/3 (rigorous) and 1/(2(1-7/25)(2/3)) = 25/24, so |d/drho entropy correction| < 25/24 < 3/2 < 7/3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "7. the useful bounds .80, 1.11, 1.97 (line 279) and margins .96, .54 (line 287) follow",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/build/sections/04-certificate.tex": "31a12b5ea1dbc4b99574cc81b4de87259c704af960df74438b1cc8c4834a8661",
    "preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/build/main.tex": "757e54eb9c9f6191c8bb72ff424054892c2f5e823fab6d2e8e85ea17847c69be",
    "preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/build/sections/00-introduction.tex": "2af424fe1cdfb7cc5c61d2a6379627eb04389f8e57db31bf50e3fc9bb13f9e06"
   },
   "forges": [
    {
     "forge": "printed H(1/8) < .791402 -> .791401",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed bracket at rho = 1 below .515374 -> .515373",
     "verdict": "REFUTED"
    },
    {
     "forge": "the tail cap 169/(70*1024) -> 168/(70*1024) (i.e. sqrt2 < 1.4)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the closed form with q -> q+1 in (net:hj)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.2
  },
  {
   "id": "F-532",
   "index": 532,
   "family": "266",
   "title": "The maximum number of mutually unbiased bases in dimension six",
   "tier": "C",
   "dir": "The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026",
   "leanChallenges": [
    "HadamardCubeFiber",
    "MUBSix"
   ],
   "word": "NEEDS DATA",
   "why": "Algorithm and error analysis in build/sections/{seed,completion,phase-cover,refinement,exclusion,rounding,execution}.tex (5,886 tex lines). The code and data are NOT PUBLISHED in the clone: build/sections/execution.tex:16-20 names verification/code/ (seed.cpp, covout.cpp, ub.cpp, patchx.cpp, atlas.cpp, stageB.cpp, stageC.cpp, extraction.json) and execution.tex:380-387 an execution archive (verification/computation/README.md, full-run-01/...); neither exists anywhere in the clone at fd4aeeb2.",
   "decider": null
  },
  {
   "id": "F-533",
   "index": 533,
   "family": "266",
   "title": "Exact Fourier certificates for complex Hadamard matrices of order six",
   "tier": "A",
   "dir": "Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026",
   "leanChallenges": [
    "HadamardCubeFiber",
    "MUBSix"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the exact finite core of both theorems — the single-matrix certificate (Prop. pair-certificate, both q), the finite arithmetic of the cubic-pair obstruction (Prop. cubic-exclusion), and the mixed-moment certificate (Prop. mixed-certificate: Z_lambda positive definite and a strictly negative exact bound). Theory (the lemmas behind the identities, the cubic classification, the 2-design identity, Weiner completion) is not decided.",
   "decider": "instruments/openaimath/finite/f533_hadamard_six.py",
   "decides": "a finite component: the exact finite core of both theorems — the single-matrix certificate (Prop. pair-certificate, both q), the finite arithmetic of the cubic-pair obstruction (Prop. cubic-exclusion), and the mixed-moment certificate (Prop. mixed-certificate: Z_lambda positive definite and a strictly negative exact bound). Theory (the lemmas behind the identities, the cubic classification, the 2-design identity, Weiner completion) is not decided.",
   "value": {
    "pair_residuals": {
     "q_cross": 0,
     "q_same": 0
    },
    "cubic": {
     "S0": "-4/5",
     "S3": "1/10",
     "S7": "1/10"
    },
    "mixed_c0": "-153535649133541/1620",
    "mixed_sum_abs": "5174469826045/162",
    "mixed_bound": "-33930316957697/540",
    "mixed_bound_decimal": "-6.283392e+10",
    "mixed_residual_terms": 374,
    "not_reproduced": "printed c0 = -613302797399911/6480 and sum|c_K| = 179721388988719/6480 depend on the program's coordinate priority lists I (05-verification.tex:31-34), which the text does not give; not decided here",
    "runtime_s": 160.3
   },
   "checks": [
    {
     "check": "the theorems as stated (paper.tex:65-71 thm:fourier, 142-149 thm:main, abstract 11-19)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the printed rank tables (paper.tex:244-245; 05-verification.tex:316-317)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "sha256 of verification/verify.py equals the checksum printed at paper.tex:260-262",
     "pass": true,
     "detail": "104a7306205b8544c8f913f6afdb59603f2e04a207c9ac592c383b80eb0e2aa9"
    },
    {
     "check": "A. pair table: 8871 nonzero seven-row equations (printed 8871)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pair table: 909 moment coordinates (printed 909)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. pair table: 897 pivots mod 1000000007 (printed 897 witness rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. exact rank over Q of all 8871 pair equations = 897 (so the modular count is the rational rank)",
     "pass": true,
     "detail": "59s"
    },
    {
     "check": "A. Prop. pair-certificate for q_cross = (2, 0, 0, -2, 0, 0): every coefficient of 3<|L|^2> + <|g(alpha)g(q)|^2> vanishes after exact substitution",
     "pass": true,
     "detail": "28 terms before, 0 nonzero after"
    },
    {
     "check": "A. Prop. pair-certificate for q_same = (2, -2, 0, 0, 0, 0): every coefficient of 3<|L|^2> + <|g(alpha)g(q)|^2> vanishes after exact substitution",
     "pass": true,
     "detail": "28 terms before, 0 nonzero after"
    },
    {
     "check": "A. consistency instance: Tao T is Hadamard (exact) and the identity evaluated directly on it, exactly in Q(zeta12), is 0 for both q",
     "pass": true,
     "detail": "['0', '0']"
    },
    {
     "check": "A. consistency instance: Fourier F6 is Hadamard (exact) and the identity evaluated directly on it, exactly in Q(zeta12), is 0 for both q",
     "pass": true,
     "detail": "['0', '0']"
    },
    {
     "check": "A. consistency instance: the Fourier-family member F6(z, z^4) [z = exp(2 pi i/12)] is Hadamard (exact) and the identity evaluated directly on it, exactly in Q(zeta12), is 0 for both q",
     "pass": true,
     "detail": "['0', '0']"
    },
    {
     "check": "B. T is Hadamard: row inner products exactly 6 delta_ij in Z[omega]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. T^{o2} (entrywise square) is Hadamard: row inner products exactly 6 delta_ij in Z[omega]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Lean MUBSix.taoExponent equals the paper's t_ij mod 3 (eq. cubic-matrix)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. |O| = 180 and the loop parametrization (A = e_p+e_q-e_h-e_i, C = e_q-e_p) is exactly O",
     "pass": true,
     "detail": "|O| = 180"
    },
    {
     "check": "B. Table tab:cubic-counts parsed: 11 rows j = 0..10",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. every (N0,N1,N2) and (U0,U1,U2) of the table, by the printed loops and by the definitions (cubic-R)",
     "pass": true,
     "detail": "mismatches []"
    },
    {
     "check": "B. the 11 representatives are exactly the signed-permutation classes of balanced charges of order <= 3",
     "pass": true,
     "detail": "11 classes, 1281 charges"
    },
    {
     "check": "B. R(b) = lambda_j G(b) for EVERY balanced b of order <= 3 (lambda_3,6,7,8 = 1/2,3,9,4; 0 elsewhere)",
     "pass": true,
     "detail": "1281 charges checked, 0 fail"
    },
    {
     "check": "B. the support of R has order <= 3 (eps b = A + d with ord A = 2, d in D)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. orbit sizes of w3, w6, w7, w8 are 360, 120, 20, 90 (printed 03-complete-family.tex:357)",
     "pass": true,
     "detail": "[360, 120, 20, 90]"
    },
    {
     "check": "B. coefficients lambda_j |O_j| = 180, 360, 180, 360 (printed :359)",
     "pass": true,
     "detail": "[Fraction(180, 1), Fraction(360, 1), Fraction(180, 1), Fraction(360, 1)]"
    },
    {
     "check": "B. G = 0 on every charge of orbits 1, 4, 5, 9 (so S1 = S4 = S5 = S9 = 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. |G|^2 = 1/4 on every charge of orbits 6 and 8 (so S6 = S8 = -1/20 via f = -G/5)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. G(3d) = 1 for all d in D, and the signed orbit of 3(e5 - e0) is exactly {3d : d in D}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. orbit counts of w5 + D and w8 + D equal the printed array (03-complete-family.tex:379-383)",
     "pass": true,
     "detail": "w5: {6: 8, 9: 1, 1: 8, 3: 12, 0: 1}; w8: {6: 4, 8: 8, 9: 4, 4: 8, 7: 2, 3: 4}"
    },
    {
     "check": "B. the three relations solve to S3 = S7 = 1/10, S0 = -4/5 (printed :397)",
     "pass": true,
     "detail": "S0=-4/5 S3=1/10 S7=1/10"
    },
    {
     "check": "B. E|sum z_i^3|^2 = 6 + 30 S0 = -18 < 0 (printed :403)",
     "pass": true,
     "detail": "-18"
    },
    {
     "check": "C. the six records match the printed sizes and scales (04-mixed.tex:238-245)",
     "pass": true,
     "detail": "[((1, 1, 1, 2), 5, 14), ((1, 1, 3), 21, 13), ((1, 2, 2), 11, 13), ((1, 4), 27, 14), ((2, 3), 22, 14), ((5,), 12, 15)]"
    },
    {
     "check": "C. every Z_lambda is well formed and positive definite (exact Schur pivots, 98 pivots)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. all 98 pivots exceed 40000, last coordinate first (05-verification.tex:329-344)",
     "pass": true,
     "detail": "min pivot 41331.2"
    },
    {
     "check": "C. mixed table: 46041 nonzero equations (45996 shifted-sum + 45 single-label zero) (printed 46041)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. mixed table: 3429 registered coordinates (printed 3429)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. mixed table: 2767 pivots mod 1000000007 (printed 2767 witness rows)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the 2767 kept rows are independent over Q (exact echelon rank 2767)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. Prop. mixed-certificate: after exact substitution Q = c0 + sum c_K mu_K with c0 + sum|c_K| < 0 (my substitution)",
     "pass": true,
     "detail": "c0 = -153535649133541/1620, sum|c_K| = 5174469826045/162, bound = -33930316957697/540 (-6.283392e+10)"
    },
    {
     "check": "C. printed arithmetic: (-613302797399911 + 179721388988719)/6480 = -2007321335237/30 < 0",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/build/paper.tex": "ee580236688b01d573c6da67babcc1e96cbe86fcaa17fcce2d631dc232afdc3c",
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/build/sections/02-pair-moments.tex": "74120b901392dd2d31d72df024d7e384a6ac5b7e885fb88251a694d1fc76dfd2",
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/build/sections/04-mixed.tex": "a7aba4ea164f0105d82893183c6fb072ab0d5ab6ef63d9074524732d2f33e88e",
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/build/sections/05-verification.tex": "56253d91efe54b7d3737e65bcdba15e65edb296284176eae29a9ddef6b4983f9",
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/verification/verify.py": "104a7306205b8544c8f913f6afdb59603f2e04a207c9ac592c383b80eb0e2aa9",
    "lean/ComparatorChallenges/MUBSix.lean": "7a634ee492b54abe357901a69272c42aa8dfc85b7a3401eef723468bb0c6c355",
    "preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/build/sections/03-complete-family.tex": "73c8b506e22f7dcda236d0b684d25798b9163492a243f8e138f52d5b6e5036fd"
   },
   "forges": [
    {
     "forge": "pair certificate with c_1 = 3 instead of 2 (also false on the Fourier-family member)",
     "verdict": "REFUTED"
    },
    {
     "forge": "pair certificate with weight 2 instead of 3 on <|L|^2> (a true consequence by positivity, but not a finite identity of the table)",
     "verdict": "REFUTED"
    },
    {
     "forge": "cubic table: U_0 of row 3 printed 61 instead of 60",
     "verdict": "REFUTED"
    },
    {
     "forge": "Z_(5): first diagonal entry negated",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed mixed witness count 2768 instead of 2767",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 160.3
  },
  {
   "id": "F-539",
   "index": 539,
   "family": "268",
   "title": "The periodic spin-one Haldane gap",
   "tier": "A",
   "dir": "The-periodic-spin-one-Haldane-gap-September-24-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: the finite inputs of the two initializations — the trial MPS contractions at N = 72, 120 (exact U_N, V_N), the thermal recurrences (all nine t_{n,g} at b = 21/2 and W_{n,g} at b = 49/4 for n <= 12 recomputed exactly, all 27 rational enclosures), and every scalar comparison of prop:seed60, prop:seed2304 and the bootstrap parameters. NOT the gap theorem: the transfer operator, sector decomposition, purity bootstrap, filter/capped-mass lemmas, Fox-Glynn and Horner error arguments and the variational principle are analytic and outside this check",
   "decider": "instruments/openaimath/finite/f539_haldane_periodic.py",
   "decides": "a finite component: the finite inputs of the two initializations — the trial MPS contractions at N = 72, 120 (exact U_N, V_N), the thermal recurrences (all nine t_{n,g} at b = 21/2 and W_{n,g} at b = 49/4 for n <= 12 recomputed exactly, all 27 rational enclosures), and every scalar comparison of prop:seed60, prop:seed2304 and the bootstrap parameters. NOT the gap theorem: the transfer operator, sector decomposition, purity bootstrap, filter/capped-mass lemmas, Fox-Glynn and Horner error arguments and the variational principle are analytic and outside this check",
   "value": {
    "U72/V72": "-1.401482112600802",
    "U120/V120": "-1.401482105173254",
    "trial_seconds": 42.5,
    "thermal_seconds": 4841.7,
    "thermal_cpu_seconds": {
     "120 n=6 1": 0.1,
     "120 n=6 P": 0.1,
     "120 n=6 C": 0.2,
     "120 n=8 1": 2.3,
     "120 n=8 P": 2.4,
     "120 n=8 C": 4.6,
     "120 n=10 1": 98.2,
     "120 n=10 P": 99.3,
     "120 n=10 C": 183.9,
     "72 n=4 1": 0.0,
     "72 n=4 P": 0.0,
     "72 n=5 1": 0.0,
     "72 n=5 P": 0.0,
     "72 n=6 1": 0.1,
     "72 n=6 P": 0.1,
     "72 n=7 1": 0.4,
     "72 n=7 P": 0.4,
     "72 n=8 1": 2.4,
     "72 n=8 P": 2.4,
     "72 n=9 1": 13.6,
     "72 n=9 P": 13.7,
     "72 n=10 1": 94.0,
     "72 n=10 P": 95.4,
     "72 n=11 1": 707.7,
     "72 n=11 P": 714.6,
     "72 n=12 1": 5995.2,
     "72 n=12 P": 6487.7
    },
    "enclosures_b49_4": {
     "n=4 1": "[124.88537982, 124.88537982]",
     "n=4 P": "[0.06575703, 0.06575703]",
     "n=5 1": "[0.00292518, 0.00292518]",
     "n=5 P": "[2.96074299, 2.96074299]",
     "n=6 1": "[12.87061404, 12.87061405]",
     "n=6 P": "[0.28659964, 0.28659964]",
     "n=7 1": "[0.05406952, 0.05406952]",
     "n=7 P": "[1.92041479, 1.92041479]",
     "n=8 1": "[4.63928341, 4.63928347]",
     "n=8 P": "[0.51014072, 0.51014074]",
     "n=9 1": "[0.19545503, 0.19545509]",
     "n=9 P": "[1.51484690, 1.51484707]",
     "n=10 1": "[2.65482244, 2.65482371]",
     "n=10 P": "[0.67541938, 0.67542002]",
     "n=11 1": "[0.37762804, 0.37763064]",
     "n=11 P": "[1.31454175, 1.31454661]",
     "n=12 1": "[1.90053154, 1.90056357]",
     "n=12 P": "[0.78739505, 0.78741566]"
    },
    "enclosures_b21_2": {
     "n=6 1": "[8.94530284, 8.94530284]",
     "n=6 P": "[0.34673411, 0.34673411]",
     "n=6 C": "[0.93998742, 0.93998742]",
     "n=8 1": "[3.74128142, 3.74128142]",
     "n=8 P": "[0.56716683, 0.56716683]",
     "n=8 C": "[0.98410787, 0.98410787]",
     "n=10 1": "[2.32754704, 2.32754704]",
     "n=10 P": "[0.72109034, 0.72109034]",
     "n=10 C": "[1.00261268, 1.00261268]"
    },
    "gap_L>=60": "0.000479191703118",
    "gap_L>=2304": "0.003821087083614",
    "seed2304_p0_q0": [
     "0.047915771",
     "0.181520168"
    ],
    "seed60_S_T_lower": [
     "0.875600",
     "0.876056"
    ],
    "runtime_s": 4884.8
   },
   "checks": [
    {
     "check": "the theorem as stated: unique ground state for even L >= 60, gamma_L > (4/105) log(80/79) (L >= 60), > log(20)/784 (L >= 2304)",
     "pass": true,
     "detail": "introduction.tex:37-53; a = 700741/500000 (model.tex:86)"
    },
    {
     "check": "the printed tables were read: 9 + 3 x 3 thermal centres, 18 W totals, 9 t totals, 8 D_alpha, 15 P_ab, 2 c_N, 11 block dims, 6 update rows",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lem:energy input: the bond h has minimal polynomial dividing (h-1)(h+1)(h+2), so spec h = {-2,-1,1} and h <= I",
     "pass": true,
     "detail": ""
    },
    {
     "check": "lem:energy table: -h = sum G_a (x) G_a in Cartesian coordinates, T = -h12 - h23 has no diagonal, and every weighted absolute row sum is <= 3q (so h12 + h23 >= -3I)",
     "pass": true,
     "detail": "pattern (q, row sum): [(2, 4), (2, 6), (3, 9), (4, 12)] as printed (aaa 4/12, aab 3/9, aba 2/6, abc 2/4)"
    },
    {
     "check": "the symmetries behind the representative reduction, exactly for n = 4..8 and theta = 0, pi, 2pi/3: the phased shift commutes with H_n(g); reversal and negation map H_n(g) to its complex conjugate (so exact exponential columns have equal norms on orbits)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the 26 x 26 matrices A_s built from Tables tab:trial-diagonal / tab:trial-pairs and eq:trial-entry equal matrix-data.json entry for entry",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:trial-blocks: the d-e blocks have dimensions 1,8,32,80,136,162,136,80,32,8,1 (sum 676) and no A_s entry leaves its block",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N = 72: U_72, V_72 recomputed equal certificate72.json (one_bond_energy_numerator, norm) and its positive margin",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N = 120: U_120, V_120 recomputed equal certificate120.json (U, V, margin, and all eleven per-block U and V)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:trial-integer-results N = 72: V_N > 0 and -c_N V_N <= 10^12 U_N < (-c_N + 1) V_N with c_N = 1401482112601",
     "pass": true,
     "detail": "floor(1e12 U/V) = -1401482112601"
    },
    {
     "check": "lem:trial-inputs N = 72: 500000 U_N + 700741 V_N < 0, i.e. E_0(72) <= N U_N/V_N < -72 a",
     "pass": true,
     "detail": "U/V + a = -0.000000112600"
    },
    {
     "check": "eq:trial-integer-results N = 120: V_N > 0 and -c_N V_N <= 10^12 U_N < (-c_N + 1) V_N with c_N = 1401482105174",
     "pass": true,
     "detail": "floor(1e12 U/V) = -1401482105174"
    },
    {
     "check": "lem:trial-inputs N = 120: 500000 U_N + 700741 V_N < 0, i.e. E_0(120) <= N U_N/V_N < -120 a",
     "pass": true,
     "detail": "U/V + a = -0.000000105173"
    },
    {
     "check": "lem:trial-contraction re-derived for N = 4,5,6 by explicit amplitudes and the explicit Hamiltonian: ||psi||^2 = V_N, <psi,H psi> = N U_N (N = 4 also equals exact-checks.json)",
     "pass": true,
     "detail": "N=4: norm ok energy ok; N=5: norm ok energy ok; N=6: norm ok energy ok"
    },
    {
     "check": "thermal72: c_j = floor(Q p_j) vanishes exactly for j > 191; thermal120: q_j vanishes exactly for j > 172",
     "pass": true,
     "detail": "last nonzero: 191, 172"
    },
    {
     "check": "thermal72 error constants: tail 2^-280 e^98 < 2^-84 < 1/Q (e < 4), d_n <= 730 and d_n > sqrt(3^n), Q + 281(d_n+1) < 2Q, row sums <= 76, 76*2Q = 5476377146882523136 < 2^63 - 1, 76*2Q/32 + Q = 207165582859042816, 0 <= x_n < 15, tail ratio 15/86",
     "pass": true,
     "detail": ""
    },
    {
     "check": "thermal120 error constants: e^84/2^257 < 27^28/2^257 < 2^-117 (e < 3, 3^84 < 2^140), 257(2 sqrt(3^n)+1) <= 125159, 32(Q+125159) = 2305843009217699040 < 2^63, 3^6 243 (125159/2^56 + 2^-116) < 4e-7, x_n < 12, 2e-6 + (8+4e-7)4e-7 < 1e-5",
     "pass": true,
     "detail": "3^6 243(...) = 0.000000307691"
    },
    {
     "check": "thermal120 totals t_{n,g} recomputed by the integer recurrence equal all nine printed integers (thermal120.tex:186-194)",
     "pass": true,
     "detail": "all nine equal"
    },
    {
     "check": "source2 summary.json totals equal the printed t_{n,g} (data file vs paper)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:thermal120-rational-enclosure: c - 2e-6 < L_n t/Q^2 < U_n t/Q^2 < c + 2e-6 for all nine (exact), hence |Z_n(21/2,g) - c| < 1e-5",
     "pass": true,
     "detail": "min margin 0.000001571"
    },
    {
     "check": "thermal72 totals W_{n,g} recomputed by the integer recurrence equal the printed integers for n = 4..12, g = 1, P (thermal72.tex:186-194)",
     "pass": true,
     "detail": "18 of 18 recomputed and equal"
    },
    {
     "check": "thermal72: every recomputed weight-block contribution equals the released sectors.jsonl entry",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the packed floors are exact: EVERY lane of EVERY stage was checked at run time to lie in [-2^56, 2^56) (b = 49/4; row sums <= 76) or [-2^57, 2^57) (b = 21/2; row sums < 64), so every lane of DX or BX lies in (-2^63, 2^63) at the next floor",
     "pass": true,
     "detail": "max row sums [19, 20, 21, 22, 23, 36, 37, 38, 39, 40, 41, 42, 45, 52]; out-of-window rows 0"
    },
    {
     "check": "eq:thermal72-enclosure: L_{n,g} > (C-30)/1e6 and U_{n,g} < (C+30)/1e6 with margins > 13/1e6, all eighteen (exact)",
     "pass": true,
     "detail": "min margin 0.000013420"
    },
    {
     "check": "seed60: m_I, m_O, m_N exceed the sector centres + 1e-5 by exactly 19/(12e6), 7/(12e6), 3/(4e6) (initialization.tex:134-140)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed60 filters: F_I(U) - B_I > 0.0301, F_N(V) - B_N > 0.3463, and both filters increase beyond (c1 > 0, c0 + c1 x^2 > 0)",
     "pass": true,
     "detail": "margins 0.030165, 0.346312"
    },
    {
     "check": "seed60: U^10 < m_I < 2U^10 and V^10 < m_N < 2V^10",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed60 eq:seed60-scalars: m^5 < D^2 (margin > 0.001068), p^5 < E^2 (> 0.000433), q >= 1, (q-1)^5 < R_q^2 (> 0.0000472), (D+11E)/12 <= tau, (D+3E)/4 <= tau0",
     "pass": true,
     "detail": "margins 0.0010684, 0.0004334, 0.00004727"
    },
    {
     "check": "seed60: D (0.88)^3 < 1, 2 (0.88) > D and D > tau0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed60 R(t) table: R(0.88) = 0.1359, R(0.99) = 0.0721, R(0.998) = 0.068404 exactly",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed60: 1 - 0.99^4 - R(0.88)^2 = 0.02093518 and 1 - 0.998^4 - R(0.99)^2 = 0.002777621984 exactly",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed60: S(60,105/4) >= (1 + R(0.998)/0.998^2)^-2 > 0.8756 > 7/8 and T(120,105/4) >= (1+R_q)^-2 > 0.8760 > 7/8",
     "pass": true,
     "detail": "S >= 0.875600, T >= 0.876056"
    },
    {
     "check": "seed2304: the filter bounds are exactly B_J = 318604.54848175 and B_N = 48169.880699 (initialization.tex:291)",
     "pass": true,
     "detail": "B_J = 318604.54848175, B_N = 48169.880699"
    },
    {
     "check": "seed2304: F_J(v_J) - B_J > 180, F_J(-v_J) - B_J > 496888, F_N(+-v_N) - B_N > 654",
     "pass": true,
     "detail": "margins 180.038, 496888.975, 654.285"
    },
    {
     "check": "seed2304: Q(1+y) = 559+2902y+5165y^2+3815y^3+1000y^4 and Q(-1-y) = 895+3978y+6275y^2+4185y^3+1000y^4 (all coefficients > 0)",
     "pass": true,
     "detail": "[559, 2902, 5165, 3815, 1000], [895, 3978, 6275, 4185, 1000]"
    },
    {
     "check": "seed2304: 12 - 394 v_N^2 + 1000 v_N^4 > 0 and 1000 v_N^2 - 197 > 0 (F_N increases beyond v_N)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed2304: m_J = J_12 centre + 30e-6 = 1.0657205 and m_N = N_12 centre + 30e-6 = 0.2783155 (exactly)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "seed2304: m_J > ell^12 and C(m_J, ell^12, 6) + B(72) < 1 with margin > 0.0693",
     "pass": true,
     "detail": "margin 0.069344"
    },
    {
     "check": "seed2304: p_0 < 0.047916 and q_0 < 0.181521 (exact rational p_0, q_0)",
     "pass": true,
     "detail": "p0 = 0.0479157, q0 = 0.1815201"
    },
    {
     "check": "seed2304: the six rounded coupled updates reproduce the printed (p_j, q_j) table exactly, all in [0,1)",
     "pass": true,
     "detail": "1:(151249/2500000,433453/2500000) 2:(343303/5000000,1632381/10000000) 3:(44601/625000,361773/2500000) 4:(632939/10000000,1055161/10000000) 5:(375793/10000000,445941/10000000) 6:(84513/10000000,27493/5000000)"
    },
    {
     "check": "seed2304: 1 - S(2304,784) <= 84513/10^7 < 1/100 and 1 - T(4608,784) <= 27493/(5 10^6) < 1/100 (and 36 * 64 = 2304, 49/4 * 64 = 784)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "cor:boundary-inputs: 1 - S(72, 49/2) <= 151249/2500000 is the first update row",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eq:bootstrap-factorization f(1-(1-u)^3) = G(u)u^2 (polynomial identity, checked at 8 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "bootstrap row 1: G(1/8) = 913952/117649 < 79/10, 3/(C(1-3u0)) = 48/79 < 1, 0 < 1/8 < 1/6, r = 79/80 < 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "bootstrap row 2: G(1/100) <= 9/(2(1-3u)^2) = 45000/9409 < 5, 3/(C(1-3u0)) = 60/97 < 1, r = 5/100 = 1/20",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the two gap constants enclosed (log by atanh series, proved tail): (4/105) log(80/79) and log(20)/784 are positive",
     "pass": true,
     "detail": "(4/105) log(80/79) in [0.000479191703118, 0.000479191703118]; log(20)/784 in [0.003821087083614, 0.003821087083614]"
    }
   ],
   "sources": {
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/introduction.tex": "c120d799db4e691a66ad1990e92331fdd16581e55aad2107c5b662fd28cab8dc",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/model.tex": "234933822c1af019a721b5d0df8b7afec47ad4a8ba7ea39287d4e7119756a8b8",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/bootstrap.tex": "e740ac5a29ea2041d788b648ef1b705138a1ff09c86b689514781511315421b4",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/initialization.tex": "f38e13e1ee99e2201ce01535b2e968ab519dc41f0ae67a29a4fc3fbebed34d95",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/assembly.tex": "d7c1415b78c2d2b6d1cf00f38fa578e1e1c1075b50f3342186380855afbd0eb3",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/thermal72.tex": "58098e7f04a139db12716dc7a97c852d7777abc451326686e37ee714a253523e",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/thermal120.tex": "fe8fc5f980138e1e91164b0653afa07bb957a5885a26f5b5b925058ce07fcd5b",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/trials.tex": "0b10ec33da7816be727062f00e7bf03fb00c59b0beb78622ce03a1014c6ccfcc",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/trials/matrix-data.json": "51c4b94117621aa44556368caa0d2cc7f7261c85e10e068d5590f33919fc71b1",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/trials/certificate72.json": "a24c13abd7278143f5e0dc8ac0b83b19503e8a5fa9a78b76d6bd0fbc3a8cfd12",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/trials/certificate120.json": "5456ef9cc81055f8a5e8c51c181f19af4f5d2dcfead8d137c2993f7934abf9b1",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/trials/exact-checks.json": "1f0bf73a53067a838a76b00d775722a11464987f1e2f188f54c1b5945a088e61",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/source1/thermal-run/sectors.jsonl": "2f6e5e9112ebcf3723f493cdf232a0150e6e9500e6926ac34772761804c1c4c6",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/computations/evidence/source2/runs/summary.json": "2f32748182c664d371be58a561dd90752feaea97135ed4e783de0bd17f97c6ab"
   },
   "forges": [
    {
     "forge": "printed W_{8,P} + 1 -> fails: thermal72 totals W_{n,g} recomputed by the integer recurrence equal the printed integers f",
     "verdict": "REFUTED"
    },
    {
     "forge": "b=49/4 centre C_{12,P} 787405 -> 787436 (moved by 31e-6) -> fails: eq:thermal72-enclosure: L_{n,g} > (C-30)/1e6 and U_{n,g} < (C+30)/1e6 with margins > 13/1e; seed2304: the filter bounds are exactly B_J = 318604.54848175 and B_N = 48169.880699 (init; seed2304: F_J(v_J) - B_J > 180, F_J(-v_J) - B_J > 496888, F_N(+-v_N) - B_N > 654; seed2304: m_J = J_12 centre + 30e-6 = 1.0657205 and m_N = N_12 centre + 30e-6 = 0.2783155 ; seed2304: the six rounded coupled updates reproduce the printed (p_j, q_j) table exactly, ; seed2304: 1 - S(2304,784) <= 84513/10^7 < 1/100 and 1 - T(4608,784) <= 27493/(5 10^6) < 1/; cor:boundary-inputs: 1 - S(72, 49/2) <= 151249/2500000 is the first update row",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed t_{10,C} - 10^9 -> fails: thermal120 totals t_{n,g} recomputed by the integer recurrence equal all nine printed inte; source2 summary.json totals equal the printed t_{n,g} (data file vs paper)",
     "verdict": "REFUTED"
    },
    {
     "forge": "trial table D_1 = -289 -> -290 -> fails: the 26 x 26 matrices A_s built from Tables tab:trial-diagonal / tab:trial-pairs and eq:tri; N = 72: U_72, V_72 recomputed equal certificate72.json (one_bond_energy_numerator, norm) a; N = 120: U_120, V_120 recomputed equal certificate120.json (U, V, margin, and all eleven p; eq:trial-integer-results N = 72: V_N > 0 and -c_N V_N <= 10^12 U_N < (-c_N + 1) V_N with c; lem:trial-inputs N = 72: 500000 U_N + 700741 V_N < 0, i.e. E_0(72) <= N U_N/V_N < -72 a; eq:trial-integer-results N = 120: V_N > 0 and -c_N V_N <= 10^12 U_N < (-c_N + 1) V_N with ; lem:trial-inputs N = 120: 500000 U_N + 700741 V_N < 0, i.e. E_0(120) <= N U_N/V_N < -120 a; lem:trial-contraction re-derived for N = 4,5,6 by explicit amplitudes and the explicit Ham",
     "verdict": "REFUTED"
    },
    {
     "forge": "update row p_3 lowered by 1e-7 -> fails: seed2304: the six rounded coupled updates reproduce the printed (p_j, q_j) table exactly, ",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 4884.8
  },
  {
   "id": "F-540",
   "index": 540,
   "family": "268",
   "title": "A boundary-field gap for the spin-one Heisenberg chain",
   "tier": "A",
   "dir": "A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "a finite component: the fifteen thermal integers k_{n,g} recomputed by the printed recurrence and every enclosure built on them (Y_6 < .027, c_0 P_*(lambda_0)^2 < .003745), the four trial-vector certificates (m = 34, 78, 142, 240) recomputed exactly, and every printed scalar comparison of the initialization, periodic extraction and conclusion. NOT the gap theorem: the boundary transfer identity, the averaging identity, the boundary recurrence lemmas, the purity lemma and the variational principle are analytic; the periodic inputs are the companion's (F-539), used here as printed",
   "decider": "instruments/openaimath/finite/f540_haldane_boundary.py",
   "decides": "a finite component: the fifteen thermal integers k_{n,g} recomputed by the printed recurrence and every enclosure built on them (Y_6 < .027, c_0 P_*(lambda_0)^2 < .003745), the four trial-vector certificates (m = 34, 78, 142, 240) recomputed exactly, and every printed scalar comparison of the initialization, periodic extraction and conclusion. NOT the gap theorem: the boundary transfer identity, the averaging identity, the boundary recurrence lemmas, the purity lemma and the variational principle are analytic; the periodic inputs are the companion's (F-539), used here as printed",
   "value": {
    "thermal_seconds": 18.0,
    "Y6_interval": [
     "0.026631944058",
     "0.026631960663"
    ],
    "polynomial_upper": "0.003744031842",
    "H/(500000V) m=34": "0.4031974157",
    "H/(500000V) m=78": "0.4041175606",
    "H/(500000V) m=142": "0.4041110066",
    "H/(500000V) m=240": "0.4041006997",
    "trial_seconds": 3.5,
    "gap_L>=960": "0.005873941563760",
    "runtime_s": 21.6
   },
   "checks": [
    {
     "check": "the theorem as stated: E_1(L,3/5) - E_0(L,3/5) > log(10)/392 for every integer L >= 960 (main.tex:77-83)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the printed tables were read: 15 k_{n,g}, 5 centres, 8 endpoint rows, 15 couplings, 4 certificate rows, 4 in:table rows",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A is a contraction (finite part): a bond has spectrum in {-2,-1,1}; S_1.S_2 + S_2.S_3 has spectrum in {-3,..,2} and attains -3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "exact floor by 80 in 128-bit lanes: MAGIC = ceil(2^68/80) with 0 <= 80 MAGIC - 2^68 < 80 and 2^61 * 80 <= 2^68 (so floor(y M / 2^68) = floor(y/80) for 0 <= y < 2^61), (2^61) MAGIC < 2^128, quotient < 2^60, bias 80 * 2^53 > 256 * 2^51",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the packed floors are exact: every |D| row sum <= 256 and EVERY lane of EVERY stage lies in [-2^51, 2^51) (checked at run time), so |D x| < 256 * 2^51 < 80 * 2^53 = bias at the next stage",
     "pass": true,
     "detail": "max row sum 120, final max |x| 199365047214242, out-of-window rows 0"
    },
    {
     "check": "appth:integer-table: all fifteen k_{n,g} recomputed by the integer recurrence equal the printed integers (thermal.tex:199-206)",
     "pass": true,
     "detail": "all fifteen equal"
    },
    {
     "check": "thermal_certificate_results.json: its fifteen quotients equal the recomputed ones, its 193 coefficients equal t_i, and its recorded max |x| (over all stages) is below 2^51 and at least our final-stage max",
     "pass": true,
     "detail": ""
    },
    {
     "check": "appth:tail and appth:frobenius-error: R = 49^193/193!/(1-49/194) < 3.152e-33, 193(27 + (3/2)729)/2^50 = 432513/2^51, 54R + 432513/2^51 < 2e-10 (sqrt(3^n) <= 27, sqrt2 < 3/2)",
     "pass": true,
     "detail": "54R + ... = 0.000000000192074"
    },
    {
     "check": "C_n and the printed upper bounds n - 1 + 6/5 + C_n = 5.4, 7.4, 10.4, 12.4, 15.4 (all <= 16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "appth:interval-data: L - eps > 0 in every entry; the exponents b_0(C_n - a n) lie in [-.0544635, 9.691073] (exp enclosed by T_80 + 2y^81/81!)",
     "pass": true,
     "detail": "exponents 4.8636910, -0.0544635, 7.2773820, 2.3592275, 9.6910730"
    },
    {
     "check": "appth:Y6-interval: .0266319440 < Z_{6,1} < .0266319607 < .027",
     "pass": true,
     "detail": "Z_{6,1} in [0.026631944058, 0.026631960663]"
    },
    {
     "check": "appth:moment-centers: each I(n) = (Z_{n,1} + Z_{n,-1} + 4 Z_{n,0})/6 lies within 1e-7 of its printed centre (n = 2..6)",
     "pass": true,
     "detail": "I(2) in [0.0270307424, 0.0270307435]; I(3) in [0.0120356158, 0.0120356158]; I(4) in [0.0119351085, 0.0119351112]; I(5) in [0.0093155999, 0.0093156001]; I(6) in [0.0089327109, 0.0089327196]"
    },
    {
     "check": "appth:polynomial: P_*(x)^2 has coefficients 1, -.28, -.3404, .0504, .0324 (sum of absolute values 1.7032); at the centres it equals .003744031780; .003744031780 + .00000017032 = .0037442021 < .003745",
     "pass": true,
     "detail": ""
    },
    {
     "check": "appth:bounds (second claim) re-derived from the rational enclosures directly: I(6) - .28I(5) - .3404I(4) + .0504I(3) + .0324I(2) < .003745",
     "pass": true,
     "detail": "upper bound 0.003744031842"
    },
    {
     "check": "the tables (apptr:entry-table, apptr:endpoint-table, apptr:coupling-table) give A_s, b, c and the restricted T_*, G_*, O_* equal to trial-exact-matrices.json; the bulk triple equals the periodic paper's (F-539 tables)",
     "pass": true,
     "detail": "A_s ok, b ok, c ok, labels ok, pairs ok, T_* ok, G_* ok, O_* ok, L_0 ok, u ok; equals F-539 triple: True"
    },
    {
     "check": "apptr:restricted-dimension: the paired labels ((i,d),(j,d)) number 1+16+64+64+16+1 = 162",
     "pass": true,
     "detail": ""
    },
    {
     "check": "apptr:certificate-table: V_m, H_m^* recomputed by the row recurrence satisfy every printed p_m, q_m, k_m window, 202100 V_m - H_m^* > 0, and equal trial-exact-results.json in full (m = 34, 78, 142, 240)",
     "pass": true,
     "detail": "m=34: p=202 q=715867 k=4031974 ok; m=78: p=440 q=747265 k=4041175 ok; m=142: p=787 q=121288 k=4041110 ok; m=240: p=1317 q=165352 k=4041006 ok"
    },
    {
     "check": "apptr:bound: H_m^*/(500000 V_m) < 202100/500000 = alpha = 2021/5000 for all four lengths",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Lemma apptr:contractions and the recurrence re-derived for m = 2..7 by explicit amplitudes and the explicit opposite-field Hamiltonian (m = 2..6 also equal trial-direct-physical-crosscheck.json)",
     "pass": true,
     "detail": "m=2 ok m=3 ok m=4 ok m=5 ok m=6 ok m=7 ok"
    },
    {
     "check": "per:native (cited): l = .999, U = 1.00139, V = .872, m_J = 1.0657205, m_N = .2783155 and 1 - S(72,b_1) <= 151249/2500000 = .0604996 are the companion's seed2304 values (re-derived by F-539's scalar code from its printed b = 49/4 table)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "per:initial-ratios: m_J - l^12 < l^12, V < l, m_J - l^12 < (.874 l)^12, V < .874 l (so rho_0 < .874)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "cube roots: (m_J - l^12)^(1/3) < .426635 and (m_N - V^12)^(1/3) < .439736 (by cubing); V^12 < m_N < 2V^12",
     "pass": true,
     "detail": ""
    },
    {
     "check": "per:scalar-checks: D(32) < .0442, D(120) < 2.5e-7, .9856^9 < .968^4, U^64(1.0442)^2/.9856 - 1 < .2092 < .9525^32, s > .9002, t > .9372, f(1-.9002^2 .9372) < .0502, f(1-.9372^2(1-.0502)) < .0198 (< .052), .0604996 < 2(.032)(.968)",
     "pass": true,
     "detail": "D32 < 0.044150, D120 < 0.0000002462, tail32 0.209134, s 0.900263, t 0.937311, outputs 0.050152, 0.019736"
    },
    {
     "check": "periodic induction: G(.052) < 5.583 < 6, 6(.052)^2 <= .052, d(.052) < .971^120 (rho_2 < .971)",
     "pass": true,
     "detail": "G(.052) = 5.582189"
    },
    {
     "check": "in:seedbounds: P_* = x(x-1/2)(x+9/25) > 0 on [l, U], 4P_* - xP_*' = x(x^2-.28x-.54) > 0 there; .003745 e^{b_0 alpha}/P_*(U)^2 < 1.133226 and e^{.132} > 1.141108 (b_0 alpha = 4.95145; exp enclosed rigorously)",
     "pass": true,
     "detail": "left coefficient 1.1332253, e^.132 >= 1.1411083"
    },
    {
     "check": "Y_6(b_0) < .027 is the thermal interval above, and eps(34) < .0928 < 1 (c_0 > 0)",
     "pass": true,
     "detail": "eps(34) < 0.092723"
    },
    {
     "check": "in:puritynumbers: (1+R(78))^-2 > .6180, (1+R(142)^2)^-2 > .7533, (1+R(240)^4)^-2 > .7923 (and R(m) > 0 from its lower enclosure, so t = (1+R)^-2 is in (0,1])",
     "pass": true,
     "detail": "0.618101, 0.753475, 0.792482"
    },
    {
     "check": "in:78 / in:142: F(.953^44, .093, (1+R(78))^-2) < .132129 < .133 and F(.971^64, .133, (1+R(142)^2)^-2) < .137301 < .14; 1 - A(240,2) bound < .207518 < .21; rho_1 < .9525 <= .953, .0928 <= .093",
     "pass": true,
     "detail": "outputs 0.1321282, 0.1373000, 0.2075175"
    },
    {
     "check": "in:table: q bounds d(u_3)^(49/120) < .142, d(u_4)^(1/2) < .0282, d(u_5)^(1/2) < .0028 (exponents 98/240, 1/2, 1/2); E-outputs < .117754, .015328, .000170 and eta-outputs < .147078, .025188, .000432, each below the next printed row",
     "pass": true,
     "detail": "E: 0.1177534, 0.0153272, 0.0001694; eta: 0.1470770, 0.0251872, 0.0004310"
    },
    {
     "check": "in:quadratic: u_5 < .000015 < .01, E_5, eta_5 <= .01, sqrt(6x^2/(2-18x^2)) <= 2x, F(2x,x,1-x)/x^2 and f(2x+16x^2)/x^2 identities (at 8 points) with endpoint values 42430000/6792093 < 10 and 3645000/1495729 < 10, x_j = 10^(-1-2^(j-5))",
     "pass": true,
     "detail": "u_5 = 0.000014965302"
    },
    {
     "check": "conclusion: 2x + 26x^2 <= 4x, 4/[10(1-.04)] < 1, b_5 = 392, M_5 = 1920 <= 2L+1 for L >= 960; log(10)/392 enclosed (atanh series)",
     "pass": true,
     "detail": "log(10)/392 in [0.005873941563760, 0.005873941563760]"
    }
   ],
   "sources": {
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/sections/thermal.tex": "35fec44e6c5c0ffdb154ebae857790d1568d0fd648d216e635903d3230b7816d",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/sections/trials.tex": "3a4879391107c9588ee68de8851ecbdf9a8f00d3885169661daa306508fc0172",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/sections/initialization.tex": "92f540c255afae1118e637d36f91cc61a10ad96da7cf8e6244e332dcdbec3f29",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/main.tex": "c2f4a7cdf6d155fbe9083d9118defef16f5b7917d900267f431b76279844ac4f",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/verification/thermal_certificate_results.json": "468f5908f7515547b2a3dad108e01827c8e76571a80ad8a3167c73f6774d1c96",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/verification/trial-exact-matrices.json": "93b1dc3995b82279e359cc8a8aa594f0c2bfa82b01a5a7a781f914ee4a635c1e",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/initialization.tex": "f38e13e1ee99e2201ce01535b2e968ab519dc41f0ae67a29a4fc3fbebed34d95",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/thermal72.tex": "58098e7f04a139db12716dc7a97c852d7777abc451326686e37ee714a253523e",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/thermal120.tex": "fe8fc5f980138e1e91164b0653afa07bb957a5885a26f5b5b925058ce07fcd5b",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/trials.tex": "0b10ec33da7816be727062f00e7bf03fb00c59b0beb78622ce03a1014c6ccfcc",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/verification/trial-exact-results.json": "50c7b0e8653c556bf11aff39eebd5941a1ae4551ec11a8b2e8d5d4ce2d429670",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/verification/trial-direct-physical-crosscheck.json": "54581c0ba9755037b31c153207bf006a3573cc647ca2afc0ac8c9050be4375ae",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/sections/periodic.tex": "712efaab7e75c1faf264cb0ba593cd7ee7604e5b657dc973a8235763a384a2f8",
    "preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/build/sections/conclusion.tex": "1d63e76f7827c31c489345c6deb3a2040ef2acf2c64aafebd512c6c48e93daa4",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/introduction.tex": "c120d799db4e691a66ad1990e92331fdd16581e55aad2107c5b662fd28cab8dc",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/model.tex": "234933822c1af019a721b5d0df8b7afec47ad4a8ba7ea39287d4e7119756a8b8",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/bootstrap.tex": "e740ac5a29ea2041d788b648ef1b705138a1ff09c86b689514781511315421b4",
    "preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/build/assembly.tex": "d7c1415b78c2d2b6d1cf00f38fa578e1e1c1075b50f3342186380855afbd0eb3"
   },
   "forges": [
    {
     "forge": "printed k_{6,1} 16467340956 -> 16467340957 -> fails: appth:integer-table: all fifteen k_{n,g} recomputed by the integer recurrence equal the pr",
     "verdict": "REFUTED"
    },
    {
     "forge": "endpoint beta_7 12475 -> 12476 -> fails: the tables (apptr:entry-table, apptr:endpoint-table, apptr:coupling-table) give A_s, b, c ; apptr:certificate-table: V_m, H_m^* recomputed by the row recurrence satisfy every printed; apptr:bound: H_m^*/(500000 V_m) < 202100/500000 = alpha = 2021/5000 for all four lengths; Lemma apptr:contractions and the recurrence re-derived for m = 2..7 by explicit amplitudes",
     "verdict": "REFUTED"
    },
    {
     "forge": "centre of I(6) .00893272 -> .00893292 -> fails: appth:moment-centers: each I(n) = (Z_{n,1} + Z_{n,-1} + 4 Z_{n,0})/6 lies within 1e-7 of i; appth:polynomial: P_*(x)^2 has coefficients 1, -.28, -.3404, .0504, .0324 (sum of absolute",
     "verdict": "REFUTED"
    },
    {
     "forge": "in:table E_4 = .016 -> .015 -> fails: in:table: q bounds d(u_3)^(49/120) < .142, d(u_4)^(1/2) < .0282, d(u_5)^(1/2) < .0028 (exp",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 21.6
  },
  {
   "id": "F-541",
   "index": 541,
   "family": "269",
   "title": "Uniform Stability of the Spherical Laughlin Gap",
   "tier": "A",
   "dir": "Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026",
   "leanChallenges": [
    "Laughlin",
    "LaughlinFock",
    "LaughlinGap",
    "LaughlinPlanar"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: this paper's restated seven-row certificate (Appendix A) — 13 three-body margins > 0 with the printed floors, M >= 0 exactly for D = 1..23 with the printed sign table, P^2 eta, 61/4096, 1222 and the final margin 4616733319001/10^14 > 1/25 used for 0 < c_0 < 1/25. NOT the stability theorem: its constants lambda_*, Delta_*, N_* are existential and every analytic step is outside this check.",
   "decider": "instruments/openaimath/finite/f541_laughlin_stability.py",
   "decides": "a finite component: this paper's restated seven-row certificate (Appendix A) — 13 three-body margins > 0 with the printed floors, M >= 0 exactly for D = 1..23 with the printed sign table, P^2 eta, 61/4096, 1222 and the final margin 4616733319001/10^14 > 1/25 used for 0 < c_0 < 1/25. NOT the stability theorem: its constants lambda_*, Delta_*, N_* are existential and every analytic step is outside this check.",
   "value": {
    "eta": "93527408868499/10^14",
    "margins": {
     "2": "30910970603109972096383/23808556800000000000000",
     "4": "7546741008522095438231/35712835200000000000000",
     "5": "353876907460004129/1417176000000000000000",
     "6": "26650275529068337/106288200000000000000",
     "7": "46496571968306777/186004350000000000000",
     "8": "124419886955963507/496011600000000000000",
     "9": "2790351684791537/11160261000000000000",
     "10": "17442867967879651/69751631250000000000",
     "11": "11625103382873/46501087500000000",
     "12": "19930497521112107/79716150000000000000",
     "13": "19929758036126629/79716150000000000000",
     "14": "25472378721696121/3936600000000000000",
     "15": "295245254154499/1180980000000000000"
    },
    "floors": {
     "2": 1298313,
     "4": 211317,
     "5": 249,
     "6": 250,
     "7": 249,
     "8": 250,
     "9": 250,
     "10": 250,
     "11": 249,
     "12": 250,
     "13": 250,
     "14": 6470,
     "15": 250
    },
    "min_margin": [
     5,
     "2.497e-04"
    ],
    "e_omitted_blocks": {
     "0": "0.848622",
     "1": "0.355889",
     "3": "0.397674"
    },
    "four_body_ranks": {
     "1": 0,
     "2": 0,
     "3": 0,
     "4": 0,
     "5": 1,
     "6": 0,
     "7": 1,
     "8": 1,
     "9": 2,
     "10": 1,
     "11": 3,
     "12": 2,
     "13": 4,
     "14": 3,
     "15": 5,
     "16": 4,
     "17": 6,
     "18": 5,
     "19": 7,
     "20": 6,
     "21": 8,
     "22": 7,
     "23": 9
    },
    "four_body_signs": {
     "1": "0",
     "2": "0",
     "3": "00",
     "4": "00",
     "5": "+00",
     "6": "000",
     "7": "+000",
     "8": "+000",
     "9": "++000",
     "10": "+0000",
     "11": "+++000",
     "12": "++0000",
     "13": "++++000",
     "14": "+++0000",
     "15": "+++++000",
     "16": "++++0000",
     "17": "++++++000",
     "18": "+++++0000",
     "19": "+++++++000",
     "20": "++++++0000",
     "21": "++++++++000",
     "22": "+++++++0000",
     "23": "+++++++++000"
    },
    "final_margin": "4616733319001/100000000000000",
    "runtime_s": 11.85,
    "three_body_s": 0.08,
    "four_body_s": 11.76
   },
   "checks": [
    {
     "check": "verification/rows.txt is byte-identical to F-542's (sha256)",
     "pass": true,
     "detail": "a058d7dc2ae8407f vs a058d7dc2ae8407f"
    },
    {
     "check": "the paper states the stability theorem, takes the unperturbed gap as input, and restates both certificates on the same rows",
     "pass": true,
     "detail": "thm:main, thm:unperturbed, rb:eq:3certificate, rb:eq:4certificate, lstinputlisting, eq:retained-energy"
    },
    {
     "check": "eps = 3/10^6 in rb:eq:4certificate and in the arithmetic",
     "pass": true,
     "detail": ""
    },
    {
     "check": "rows.txt parses into seven rows",
     "pass": true,
     "detail": "7 rows, 125 entries"
    },
    {
     "check": "every t_rho in {0..7}; every entry in S_rho (0<=p<=7, 0<=j<=8, 0<=p+j-t<=8); every listed entry nonzero",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the two printed forms of eq:alphadata agree on every entry",
     "pass": true,
     "detail": ""
    },
    {
     "check": "three-body levels T = p+j <= 15; four-body levels T = p+j+k <= 23; only modes 0..8",
     "pass": true,
     "detail": "max T3 = 15, max T4 = 23, modes 0..8"
    },
    {
     "check": "P^2 eta = sum l^2 equals the printed 93527408868499",
     "pass": true,
     "detail": "93527408868499"
    },
    {
     "check": "e_z from the definitions (eq:coupling, eq:alphadata, eq:ez) has no irrational part, z = 0..15",
     "pass": true,
     "detail": "radicals cancel exactly"
    },
    {
     "check": "e_z from the definitions equals the paper's rational formula eq:erational, z = 0..15",
     "pass": true,
     "detail": ""
    },
    {
     "check": "m_2 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 1.298313496"
    },
    {
     "check": "floor(10^6 m_2) = printed 1298313",
     "pass": true,
     "detail": "1298313"
    },
    {
     "check": "m_4 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.211317331"
    },
    {
     "check": "floor(10^6 m_4) = printed 211317",
     "pass": true,
     "detail": "211317"
    },
    {
     "check": "m_5 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249706"
    },
    {
     "check": "floor(10^6 m_5) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_6 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250736"
    },
    {
     "check": "floor(10^6 m_6) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_7 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249976"
    },
    {
     "check": "floor(10^6 m_7) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_8 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250841"
    },
    {
     "check": "floor(10^6 m_8) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_9 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250026"
    },
    {
     "check": "floor(10^6 m_9) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_10 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250071"
    },
    {
     "check": "floor(10^6 m_10) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_11 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249996"
    },
    {
     "check": "floor(10^6 m_11) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_12 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250018"
    },
    {
     "check": "floor(10^6 m_12) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_13 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250009"
    },
    {
     "check": "floor(10^6 m_13) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_14 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.006470655"
    },
    {
     "check": "floor(10^6 m_14) = printed 6470",
     "pass": true,
     "detail": "6470"
    },
    {
     "check": "m_15 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250000"
    },
    {
     "check": "floor(10^6 m_15) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "the printed floor table lists exactly z in {2,4,5,...,15}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D = 1..23: Y = E_D/sqrt(u_r u_s) and Z = G_D/sqrt(u_r u_s) from the definitions (eq:S, eq:E, eq:pairstar, eq:wstar) are rational, Y symmetric, and equal eq:Yrational / eq:Zrational",
     "pass": true,
     "detail": "all 23"
    },
    {
     "check": "D = 1..23: G_D (2chi - E_D + eps I) G_D equals D_sqrt(u) M D_sqrt(u) entrywise in the field (eq:M)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fraction)",
     "pass": true,
     "detail": "ranks 1:0 2:0 3:0 4:0 5:1 6:0 7:1 8:1 9:2 10:1 11:3 12:2 13:4 14:3 15:5 16:4 17:6 18:5 19:7 20:6 21:8 22:7 23:9"
    },
    {
     "check": "D = 1: det(xI+M) coefficient signs (exact interpolation) = printed 0; #+ = LDL rank",
     "pass": true,
     "detail": "0"
    },
    {
     "check": "D = 2: det(xI+M) coefficient signs (exact interpolation) = printed 0; #+ = LDL rank",
     "pass": true,
     "detail": "0"
    },
    {
     "check": "D = 3: det(xI+M) coefficient signs (exact interpolation) = printed 00; #+ = LDL rank",
     "pass": true,
     "detail": "00"
    },
    {
     "check": "D = 4: det(xI+M) coefficient signs (exact interpolation) = printed 00; #+ = LDL rank",
     "pass": true,
     "detail": "00"
    },
    {
     "check": "D = 5: det(xI+M) coefficient signs (exact interpolation) = printed +00; #+ = LDL rank",
     "pass": true,
     "detail": "+00"
    },
    {
     "check": "D = 6: det(xI+M) coefficient signs (exact interpolation) = printed 000; #+ = LDL rank",
     "pass": true,
     "detail": "000"
    },
    {
     "check": "D = 7: det(xI+M) coefficient signs (exact interpolation) = printed +000; #+ = LDL rank",
     "pass": true,
     "detail": "+000"
    },
    {
     "check": "D = 8: det(xI+M) coefficient signs (exact interpolation) = printed +000; #+ = LDL rank",
     "pass": true,
     "detail": "+000"
    },
    {
     "check": "D = 9: det(xI+M) coefficient signs (exact interpolation) = printed ++000; #+ = LDL rank",
     "pass": true,
     "detail": "++000"
    },
    {
     "check": "D = 10: det(xI+M) coefficient signs (exact interpolation) = printed +0000; #+ = LDL rank",
     "pass": true,
     "detail": "+0000"
    },
    {
     "check": "D = 11: det(xI+M) coefficient signs (exact interpolation) = printed +++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++000"
    },
    {
     "check": "D = 12: det(xI+M) coefficient signs (exact interpolation) = printed ++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++0000"
    },
    {
     "check": "D = 13: det(xI+M) coefficient signs (exact interpolation) = printed ++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++000"
    },
    {
     "check": "D = 14: det(xI+M) coefficient signs (exact interpolation) = printed +++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++0000"
    },
    {
     "check": "D = 15: det(xI+M) coefficient signs (exact interpolation) = printed +++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++000"
    },
    {
     "check": "D = 16: det(xI+M) coefficient signs (exact interpolation) = printed ++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++++0000"
    },
    {
     "check": "D = 17: det(xI+M) coefficient signs (exact interpolation) = printed ++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++000"
    },
    {
     "check": "D = 18: det(xI+M) coefficient signs (exact interpolation) = printed +++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++0000"
    },
    {
     "check": "D = 19: det(xI+M) coefficient signs (exact interpolation) = printed +++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++000"
    },
    {
     "check": "D = 20: det(xI+M) coefficient signs (exact interpolation) = printed ++++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++0000"
    },
    {
     "check": "D = 21: det(xI+M) coefficient signs (exact interpolation) = printed ++++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++++000"
    },
    {
     "check": "D = 22: det(xI+M) coefficient signs (exact interpolation) = printed +++++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++0000"
    },
    {
     "check": "D = 23: det(xI+M) coefficient signs (exact interpolation) = printed +++++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++++000"
    },
    {
     "check": "the printed sign table lists exactly D = 1..23",
     "pass": true,
     "detail": ""
    },
    {
     "check": "eta printed in rb:eq:Adecomp equals sum l^2 / 10^14",
     "pass": true,
     "detail": "93527408868499/100000000000000"
    },
    {
     "check": "series sums of 4^-m, m 4^-m, m^2 4^-m = 4/3, 4/9, 20/27 (closed form, inside an exact partial sum + geometric tail enclosure)",
     "pass": true,
     "detail": "['4/3', '4/9', '20/27']"
    },
    {
     "check": "delta = (3/2) sum_{z odd >= 17} (3z-1)(z+1) 2^-z = printed 61/4096 (exact; and enclosed directly)",
     "pass": true,
     "detail": "61/4096 in [0.014892578125, 0.014892578125]"
    },
    {
     "check": "sum_{D=1}^{23} n_D (triples p+j+k=D, j<k, counted) = floor((D+1)^2/4) summed = printed 1222 = printed sum_{m<=12} m^2 + sum_{m<=11} m(m+1)",
     "pass": true,
     "detail": "1222"
    },
    {
     "check": "1 - eta - 61/4096 - 1222 eps = the printed 4616733319001/10^14 (rb:eq:finalmargin) = gamma_* of thm:unperturbed",
     "pass": true,
     "detail": "4616733319001/100000000000000"
    },
    {
     "check": "the final margin exceeds 1/25 (so every 0 < c_0 < 1/25 is admissible)",
     "pass": true,
     "detail": "margin - 1/25 = 616733319001/100000000000000"
    }
   ],
   "sources": {
    "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/verification/rows.txt": "a058d7dc2ae8407f85cf69dad22c6d0e622b4b03a2f4ef14945905cb46479d08",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/verification/rows.txt": "a058d7dc2ae8407f85cf69dad22c6d0e622b4b03a2f4ef14945905cb46479d08",
    "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/build/appendices/retained-blocks.tex": "6a7c20318969e233b5cb26ff72df1027e7eadb6901664a074342a76411e24b27",
    "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/build/sections/02-preliminaries.tex": "62c2f79f4b82525110457026ff82ea249a65d058c3bc08f167f94c3182cd734b",
    "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/build/sections/01-introduction.tex": "fcc0b6b26ca3ab45d0e455bb4b84d8beb08e5f16662834c513205e61adf9089a",
    "preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/build/sections/07-energy.tex": "fec0459d72046a5649c02e2e2ee140253006d604f95acfc98198c8bf8efbdb02"
   },
   "forges": [
    {
     "forge": "row 1 (t=0): a_16 = -43084 -> -53084 (one digit) -> fails: floor(10^6 m_2) = printed 1298313 [1299141]; floor(10^6 m_4) = printed 211317 [211395]; floor(10^6 m_5) = printed 249 [122]; m_6 = (3/2)(3z-1)(-1/2)^z - e_z > 0 [m_z ~ -0.001648755]; floor(10^6 m_6) = printed 250 [-1649]; floor(10^6 m_7) = printed 249 [661]; D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fracti [D=13: M not PSD (negative pivot -514535456432129063079664375]; D = 13",
     "verdict": "REFUTED"
    },
    {
     "forge": "row 1 (t=0): a_05 = -849651 -> -749651 (one digit; all 13 three-body margins stay positive) -> fails: floor(10^6 m_2) = printed 1298313 [1298585]; floor(10^6 m_4) = printed 211317 [203317]; floor(10^6 m_5) = printed 249 [467]; D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fracti [D=13: M not PSD (negative pivot -576208731027910235714879040]; D = 13: det(xI+M) coefficient signs (exact interpolation) = printed ++ [+++-000]",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed floor at z = 5: 249 -> 248 -> fails: floor(10^6 m_5) = printed 248 [249]",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed signs at D = 23: +++++++++000 -> ++++++++++00 -> fails: D = 23: det(xI+M) coefficient signs (exact interpolation) = printed ++ [+++++++++000]",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 11.8
  },
  {
   "id": "F-542",
   "index": 542,
   "family": "269",
   "title": "A Fock-space inequality and the Laughlin spectral gap",
   "tier": "A",
   "dir": "A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026",
   "leanChallenges": [
    "Laughlin",
    "LaughlinFock",
    "LaughlinGap",
    "LaughlinPlanar"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the seven published rows give the 13 three-body margins of eq:3certificate (all > 0, floors as printed) and the 23 four-body certificates of eq:4certificate (M >= 0 exactly, sign table as printed), each re-derived from the definitions and from the paper's rational formulas; plus P^2 eta, 61/4096, 1222 and gamma_* = 4616733319001/10^14 > 1/25. NOT the analytic reduction from these certificates to H_Q^2 >= gamma H_Q or to the 1/25 gap (lemmas, limits, transfer, tail, existential Q_gamma and N_0).",
   "decider": "instruments/openaimath/finite/f542_laughlin_gap.py",
   "decides": "a finite component: the seven published rows give the 13 three-body margins of eq:3certificate (all > 0, floors as printed) and the 23 four-body certificates of eq:4certificate (M >= 0 exactly, sign table as printed), each re-derived from the definitions and from the paper's rational formulas; plus P^2 eta, 61/4096, 1222 and gamma_* = 4616733319001/10^14 > 1/25. NOT the analytic reduction from these certificates to H_Q^2 >= gamma H_Q or to the 1/25 gap (lemmas, limits, transfer, tail, existential Q_gamma and N_0).",
   "value": {
    "eta": "93527408868499/10^14",
    "margins": {
     "2": "30910970603109972096383/23808556800000000000000",
     "4": "7546741008522095438231/35712835200000000000000",
     "5": "353876907460004129/1417176000000000000000",
     "6": "26650275529068337/106288200000000000000",
     "7": "46496571968306777/186004350000000000000",
     "8": "124419886955963507/496011600000000000000",
     "9": "2790351684791537/11160261000000000000",
     "10": "17442867967879651/69751631250000000000",
     "11": "11625103382873/46501087500000000",
     "12": "19930497521112107/79716150000000000000",
     "13": "19929758036126629/79716150000000000000",
     "14": "25472378721696121/3936600000000000000",
     "15": "295245254154499/1180980000000000000"
    },
    "floors": {
     "2": 1298313,
     "4": 211317,
     "5": 249,
     "6": 250,
     "7": 249,
     "8": 250,
     "9": 250,
     "10": 250,
     "11": 249,
     "12": 250,
     "13": 250,
     "14": 6470,
     "15": 250
    },
    "min_margin": [
     5,
     "2.497e-04"
    ],
    "e_omitted_blocks": {
     "0": "0.848622",
     "1": "0.355889",
     "3": "0.397674"
    },
    "four_body_ranks": {
     "1": 0,
     "2": 0,
     "3": 0,
     "4": 0,
     "5": 1,
     "6": 0,
     "7": 1,
     "8": 1,
     "9": 2,
     "10": 1,
     "11": 3,
     "12": 2,
     "13": 4,
     "14": 3,
     "15": 5,
     "16": 4,
     "17": 6,
     "18": 5,
     "19": 7,
     "20": 6,
     "21": 8,
     "22": 7,
     "23": 9
    },
    "four_body_signs": {
     "1": "0",
     "2": "0",
     "3": "00",
     "4": "00",
     "5": "+00",
     "6": "000",
     "7": "+000",
     "8": "+000",
     "9": "++000",
     "10": "+0000",
     "11": "+++000",
     "12": "++0000",
     "13": "++++000",
     "14": "+++0000",
     "15": "+++++000",
     "16": "++++0000",
     "17": "++++++000",
     "18": "+++++0000",
     "19": "+++++++000",
     "20": "++++++0000",
     "21": "++++++++000",
     "22": "+++++++0000",
     "23": "+++++++++000"
    },
    "gamma_star": "4616733319001/100000000000000",
    "observation_lean_rows_equal_rows_txt": {
     "Certificate.lean": true,
     "RowCompute.lean": true
    },
    "runtime_s": 8.23,
    "three_body_s": 0.05,
    "four_body_s": 8.17
   },
   "checks": [
    {
     "check": "the paper prints the claim and the two certificates as decided here",
     "pass": true,
     "detail": "thm:fock, thm:main, eq:3certificate, eq:4certificate, lstinputlisting"
    },
    {
     "check": "eps = 3/10^6 in the certificate statement and in the arithmetic",
     "pass": true,
     "detail": ""
    },
    {
     "check": "rows.txt parses into seven rows",
     "pass": true,
     "detail": "7 rows, 125 entries"
    },
    {
     "check": "every t_rho in {0..7}; every entry in S_rho (0<=p<=7, 0<=j<=8, 0<=p+j-t<=8); every listed entry nonzero",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the two printed forms of eq:alphadata agree on every entry",
     "pass": true,
     "detail": ""
    },
    {
     "check": "three-body levels T = p+j <= 15; four-body levels T = p+j+k <= 23; only modes 0..8",
     "pass": true,
     "detail": "max T3 = 15, max T4 = 23, modes 0..8"
    },
    {
     "check": "P^2 eta = sum l^2 equals the printed 93527408868499",
     "pass": true,
     "detail": "93527408868499"
    },
    {
     "check": "e_z from the definitions (eq:coupling, eq:alphadata, eq:ez) has no irrational part, z = 0..15",
     "pass": true,
     "detail": "radicals cancel exactly"
    },
    {
     "check": "e_z from the definitions equals the paper's rational formula eq:erational, z = 0..15",
     "pass": true,
     "detail": ""
    },
    {
     "check": "m_2 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 1.298313496"
    },
    {
     "check": "floor(10^6 m_2) = printed 1298313",
     "pass": true,
     "detail": "1298313"
    },
    {
     "check": "m_4 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.211317331"
    },
    {
     "check": "floor(10^6 m_4) = printed 211317",
     "pass": true,
     "detail": "211317"
    },
    {
     "check": "m_5 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249706"
    },
    {
     "check": "floor(10^6 m_5) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_6 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250736"
    },
    {
     "check": "floor(10^6 m_6) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_7 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249976"
    },
    {
     "check": "floor(10^6 m_7) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_8 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250841"
    },
    {
     "check": "floor(10^6 m_8) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_9 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250026"
    },
    {
     "check": "floor(10^6 m_9) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_10 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250071"
    },
    {
     "check": "floor(10^6 m_10) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_11 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000249996"
    },
    {
     "check": "floor(10^6 m_11) = printed 249",
     "pass": true,
     "detail": "249"
    },
    {
     "check": "m_12 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250018"
    },
    {
     "check": "floor(10^6 m_12) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_13 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250009"
    },
    {
     "check": "floor(10^6 m_13) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "m_14 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.006470655"
    },
    {
     "check": "floor(10^6 m_14) = printed 6470",
     "pass": true,
     "detail": "6470"
    },
    {
     "check": "m_15 = (3/2)(3z-1)(-1/2)^z - e_z > 0",
     "pass": true,
     "detail": "m_z ~ 0.000250000"
    },
    {
     "check": "floor(10^6 m_15) = printed 250",
     "pass": true,
     "detail": "250"
    },
    {
     "check": "the printed floor table lists exactly z in {2,4,5,...,15}",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D = 1..23: Y = E_D/sqrt(u_r u_s) and Z = G_D/sqrt(u_r u_s) from the definitions (eq:S, eq:E, eq:pairstar, eq:wstar) are rational, Y symmetric, and equal eq:Yrational / eq:Zrational",
     "pass": true,
     "detail": "all 23"
    },
    {
     "check": "D = 1..23: G_D (2chi - E_D + eps I) G_D equals D_sqrt(u) M D_sqrt(u) entrywise in the field (eq:M)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fraction)",
     "pass": true,
     "detail": "ranks 1:0 2:0 3:0 4:0 5:1 6:0 7:1 8:1 9:2 10:1 11:3 12:2 13:4 14:3 15:5 16:4 17:6 18:5 19:7 20:6 21:8 22:7 23:9"
    },
    {
     "check": "D = 1: det(xI+M) coefficient signs (exact interpolation) = printed 0; #+ = LDL rank",
     "pass": true,
     "detail": "0"
    },
    {
     "check": "D = 2: det(xI+M) coefficient signs (exact interpolation) = printed 0; #+ = LDL rank",
     "pass": true,
     "detail": "0"
    },
    {
     "check": "D = 3: det(xI+M) coefficient signs (exact interpolation) = printed 00; #+ = LDL rank",
     "pass": true,
     "detail": "00"
    },
    {
     "check": "D = 4: det(xI+M) coefficient signs (exact interpolation) = printed 00; #+ = LDL rank",
     "pass": true,
     "detail": "00"
    },
    {
     "check": "D = 5: det(xI+M) coefficient signs (exact interpolation) = printed +00; #+ = LDL rank",
     "pass": true,
     "detail": "+00"
    },
    {
     "check": "D = 6: det(xI+M) coefficient signs (exact interpolation) = printed 000; #+ = LDL rank",
     "pass": true,
     "detail": "000"
    },
    {
     "check": "D = 7: det(xI+M) coefficient signs (exact interpolation) = printed +000; #+ = LDL rank",
     "pass": true,
     "detail": "+000"
    },
    {
     "check": "D = 8: det(xI+M) coefficient signs (exact interpolation) = printed +000; #+ = LDL rank",
     "pass": true,
     "detail": "+000"
    },
    {
     "check": "D = 9: det(xI+M) coefficient signs (exact interpolation) = printed ++000; #+ = LDL rank",
     "pass": true,
     "detail": "++000"
    },
    {
     "check": "D = 10: det(xI+M) coefficient signs (exact interpolation) = printed +0000; #+ = LDL rank",
     "pass": true,
     "detail": "+0000"
    },
    {
     "check": "D = 11: det(xI+M) coefficient signs (exact interpolation) = printed +++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++000"
    },
    {
     "check": "D = 12: det(xI+M) coefficient signs (exact interpolation) = printed ++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++0000"
    },
    {
     "check": "D = 13: det(xI+M) coefficient signs (exact interpolation) = printed ++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++000"
    },
    {
     "check": "D = 14: det(xI+M) coefficient signs (exact interpolation) = printed +++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++0000"
    },
    {
     "check": "D = 15: det(xI+M) coefficient signs (exact interpolation) = printed +++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++000"
    },
    {
     "check": "D = 16: det(xI+M) coefficient signs (exact interpolation) = printed ++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++++0000"
    },
    {
     "check": "D = 17: det(xI+M) coefficient signs (exact interpolation) = printed ++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++000"
    },
    {
     "check": "D = 18: det(xI+M) coefficient signs (exact interpolation) = printed +++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++0000"
    },
    {
     "check": "D = 19: det(xI+M) coefficient signs (exact interpolation) = printed +++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++000"
    },
    {
     "check": "D = 20: det(xI+M) coefficient signs (exact interpolation) = printed ++++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++0000"
    },
    {
     "check": "D = 21: det(xI+M) coefficient signs (exact interpolation) = printed ++++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "++++++++000"
    },
    {
     "check": "D = 22: det(xI+M) coefficient signs (exact interpolation) = printed +++++++0000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++0000"
    },
    {
     "check": "D = 23: det(xI+M) coefficient signs (exact interpolation) = printed +++++++++000; #+ = LDL rank",
     "pass": true,
     "detail": "+++++++++000"
    },
    {
     "check": "the printed sign table lists exactly D = 1..23",
     "pass": true,
     "detail": ""
    },
    {
     "check": "(3z-1)(z+1) = printed 900 + 208m + 12m^2 at z = 17+2m (degree 2: 5 points decide it)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "series sums of 4^-m, m 4^-m, m^2 4^-m = 4/3, 4/9, 20/27 (closed form, inside an exact partial sum + geometric tail enclosure), as printed",
     "pass": true,
     "detail": "['4/3', '4/9', '20/27']"
    },
    {
     "check": "delta = (3/2) sum_{z odd >= 17} (3z-1)(z+1) 2^-z = printed 61/4096 (exact; and enclosed directly)",
     "pass": true,
     "detail": "61/4096 in [0.014892578125, 0.014892578125]"
    },
    {
     "check": "sum_{D=1}^{23} n_D (triples p+j+k=D, j<k, counted) = floor((D+1)^2/4) summed = printed 1222 = printed sum_{m<=12} m^2 + sum_{m<=11} m(m+1)",
     "pass": true,
     "detail": "1222"
    },
    {
     "check": "1 - eta - 61/4096 - 1222 eps = gamma_* exactly, with eta from the rows and gamma_* as printed (intro and eq:finalmargin)",
     "pass": true,
     "detail": "4616733319001/100000000000000 = 4616733319001/10^14"
    },
    {
     "check": "gamma_* > 1/25",
     "pass": true,
     "detail": "gamma_* - 1/25 = 616733319001/100000000000000"
    }
   ],
   "sources": {
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/verification/rows.txt": "a058d7dc2ae8407f85cf69dad22c6d0e622b4b03a2f4ef14945905cb46479d08",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/08-arithmetic.tex": "0c15dc1a8eaf213b503584920028290880b238be4e569a9df8c8b0e3f6a69a66",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/01-introduction.tex": "24a96abec7534a93d049495f5322d145b76f1b556e72d7613518da522d492a0a",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/06-gap.tex": "2cb86b7a8136afc22639dbe2be903d372d935b9e7a25b805704ebbab6a1b8b03",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/05-transfer.tex": "9f4b50a2a0369b58423359020fc4f62c953cc44a0e52a38a520aded1fd2654e6",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/04-comparison.tex": "3ed5f4078a0d551b33d1db4cb35ae3e9abdaf24627cb2c44c75e609b3f71f39a",
    "preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/03-spin.tex": "73734849efe93172b9930899d6b05f54d8f4e18989180ac449e09f5a8c11bdb8",
    "lean/OAI/Analysis/Laughlin/Operators/Certificate.lean": "9afb6d0f748c98154607a3ef3666146636c7b9d76e9cd36eb66647d054d3fb56",
    "lean/OAI/Analysis/Laughlin/FourBody/RowCompute.lean": "3e69482c984b59c22eae05c3d78527b469cf985c029cbcf2818419e83072e36e"
   },
   "forges": [
    {
     "forge": "row 1 (t=0): a_16 = -43084 -> -53084 (one digit) -> fails: floor(10^6 m_2) = printed 1298313 [1299141]; floor(10^6 m_4) = printed 211317 [211395]; floor(10^6 m_5) = printed 249 [122]; m_6 = (3/2)(3z-1)(-1/2)^z - e_z > 0 [m_z ~ -0.001648755]; floor(10^6 m_6) = printed 250 [-1649]; floor(10^6 m_7) = printed 249 [661]; D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fracti [D=13: M not PSD (negative pivot -514535456432129063079664375]; D = 13",
     "verdict": "REFUTED"
    },
    {
     "forge": "row 1 (t=0): a_05 = -849651 -> -749651 (one digit; all 13 three-body margins stay positive) -> fails: floor(10^6 m_2) = printed 1298313 [1298585]; floor(10^6 m_4) = printed 211317 [203317]; floor(10^6 m_5) = printed 249 [467]; D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fracti [D=13: M not PSD (negative pivot -576208731027910235714879040]; D = 13: det(xI+M) coefficient signs (exact interpolation) = printed ++ [+++-000]",
     "verdict": "REFUTED"
    },
    {
     "forge": "the allowance removed, eps = 0 -> fails: D = 1..23: M = Z B Z is positive semidefinite (exact LDL^T over Fracti [D=11: M not PSD (negative pivot -762433910411671227377227275]; D = 11: det(xI+M) coefficient signs (exact interpolation) = printed ++ [++-000]; D = 13: det(xI+M) coefficient signs (exact interpolation) = printed ++ [+++-000]; D = 14: det(xI+M) coefficient signs (exact interpolation) = printed ++ [++-0000]; D = 15: det(xI+M) co",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed floor at z = 14: 6470 -> 6471 -> fails: floor(10^6 m_14) = printed 6471 [6470]",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed signs at D = 9: ++000 -> +++00 -> fails: D = 9: det(xI+M) coefficient signs (exact interpolation) = printed +++ [++000]",
     "verdict": "REFUTED"
    },
    {
     "forge": "gamma_* = 4616733319002/10^14 -> fails: 1 - eta - 61/4096 - 1222 eps = gamma_* exactly, with eta from the rows [4616733319001/100000000000000 = 4616733319001/10^14]",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 8.2
  },
  {
   "id": "F-549",
   "index": 549,
   "family": "272",
   "title": "Entanglement with zero distillable secret key in local dimension ten",
   "tier": "A",
   "dir": "Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026",
   "leanChallenges": [
    "DimensionTenChannel",
    "DimensionTenPair"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the integer pencil and L (PPT), the twenty singular directions with every modular datum the paper prints, Phi_1 and Phi_2 PPT, Z PSD/PPT and nonzero with all twenty xhat x xhat in ker Z, rho = Z/tr Z in the class C, and Theta on M_21 CP, PPT, trace preserving with the corner c1 c2 Z of J(Theta^2). NOT the Galois/Dedekind step to \"every ten span\" (its finite inputs only), and NOT the secret-key bound K_D(rho) = 0 (analytic).",
   "decider": "instruments/openaimath/finite/f549_ppt_ten.py",
   "decides": "a finite component: the integer pencil and L (PPT), the twenty singular directions with every modular datum the paper prints, Phi_1 and Phi_2 PPT, Z PSD/PPT and nonzero with all twenty xhat x xhat in ker Z, rho = Z/tr Z in the class C, and Theta on M_21 CP, PPT, trace preserving with the corner c1 c2 Z of J(Theta^2). NOT the Galois/Dedekind step to \"every ten span\" (its finite inputs only), and NOT the secret-key bound K_D(rho) = 0 (analytic).",
   "value": {
    "detH": "4647838136575323254753500448472406723500212063461504450560",
    "f_height_digits": 68,
    "rank J(Phi_1)": 15,
    "rank (id x T) J(Phi_1)": 15,
    "rank J(Phi_2)": 15,
    "rank (id x T) J(Phi_2)": 15,
    "rank Z": 80,
    "tr Z": "130208350782",
    "kernel vectors independent (rank of the 100 coordinate classes in A)": 20,
    "c1": "1/1485980",
    "c2": "1/1485980",
    "runtime_s": 51.5,
    "timing_s": {
     "A": 0.0,
     "B-minors-charpoly": 1.0,
     "B": 4.2,
     "C-maps-psd": 26.1,
     "C-kernel": 0.4,
     "D": 20.8
    }
   },
   "checks": [
    {
     "check": "the claims as printed (thm:main-pair, thm:main-channel, eq:L, eq:phi-pair, eq:channel)",
     "pass": true,
     "detail": "introduction.tex:80-105, certificate.tex:35, geometry.tex:104-106, applications.tex:78-81"
    },
    {
     "check": "parsed: four 6x4 integer matrices; the table rows for 41, 131, 139; the recorded output for 41, 131, 139",
     "pass": true,
     "detail": "certificate.tex:14-30, :246-248; verification/certificate-output.txt"
    },
    {
     "check": "A1. M_i^T M_j = M_j^T M_i for all 16 (i, j) (eq:block-symmetry)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A2. J(L) is symmetric and PSD (L is CP)",
     "pass": true,
     "detail": "rank 6 "
    },
    {
     "check": "A2. (id x T) J(L) is PSD (T o L is CP: L is PPT)",
     "pass": true,
     "detail": "rank 6 "
    },
    {
     "check": "A3. L(E_ij) = sum_r Q_r E_ij Q_r^T with Q_r the paper's (column i = row r of M_i)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A4. L(P_e0) = M_0^T M_0 = 36 I_4 (as printed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B1. 15 maximal minors of M(1,t), each of degree <= 4, give D (15 x 20) and H (15 x 15) in the order eq:monomial-order",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B2. det H is a nonzero integer",
     "pass": true,
     "detail": "det H = 4647838136575323254753500448472406723500212063461504450560"
    },
    {
     "check": "B2. det H mod 41 = 40 (table) = 40 (recorded output)",
     "pass": true,
     "detail": "computed 40"
    },
    {
     "check": "B2. det H mod 131 = 23 (table) = 23 (recorded output)",
     "pass": true,
     "detail": "computed 23"
    },
    {
     "check": "B2. det H mod 139 = 9 (table) = 9 (recorded output)",
     "pass": true,
     "detail": "computed 9"
    },
    {
     "check": "B3. f = det(zI - N) over Q: Hessenberg reduction = Faddeev-LeVerrier, monic of degree 20",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B4. f is 41-integral and its reduction equals the 21 recorded coefficients",
     "pass": true,
     "detail": "computed (leading first) [1, 6, 22, 32, 32, 4, 37, 27, 19, 9, 26, 6, 32, 4, 4, 4, 33, 2, 12, 24, 36]"
    },
    {
     "check": "B4. f is 131-integral and its reduction equals the 21 recorded coefficients",
     "pass": true,
     "detail": "computed (leading first) [1, 100, 35, 54, 51, 41, 36, 76, 115, 33, 118, 25, 65, 61, 124, 110, 18, 61, 53, 39, 17]"
    },
    {
     "check": "B4. f is 139-integral and its reduction equals the 21 recorded coefficients",
     "pass": true,
     "detail": "computed (leading first) [1, 40, 61, 0, 51, 49, 50, 50, 107, 89, 57, 84, 54, 20, 40, 37, 60, 126, 44, 44, 97]"
    },
    {
     "check": "B5. d_41(k), k = 1..20, equals the recorded list and the table entries",
     "pass": true,
     "detail": "computed [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20]"
    },
    {
     "check": "B6. f mod 41 is squarefree with irreducible factor degrees [20] (table: [20])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B5. d_131(k), k = 1..20, equals the recorded list and the table entries",
     "pass": true,
     "detail": "computed [1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 18, 3, 1, 3]"
    },
    {
     "check": "B6. f mod 131 is squarefree with irreducible factor degrees [1, 2, 17] (table: [1, 2, 17])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B5. d_139(k), k = 1..20, equals the recorded list and the table entries",
     "pass": true,
     "detail": "computed [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 20, 1]"
    },
    {
     "check": "B6. f mod 139 is squarefree with irreducible factor degrees [1, 19] (table: [1, 19])",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B7. 41, 131, 139 are prime (trial division by 2..11 suffices below 144) and exceed 20",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B8. 1, t1, ..., t1^19 are a basis of the span of the low monomials (t1 generates A)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B9. every maximal minor p_rho(z, h2(z), h3(z)) = 0 in Q[z]/(f): a surjective algebra map A -> Q[z]/(f), so dim A = 20",
     "pass": true,
     "detail": "h2, h3 heights: 921, 920 digits"
    },
    {
     "check": "B10. gcd(f, f') = 1 over Q: f has 20 distinct complex roots, hence exactly 20 distinct singular directions (1, l, h2(l), h3(l))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B11. no singular direction has x_0 = 0 (det H != 0 forces t = 0), and the paper says exactly twenty",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C1. K from the sign rule eps_ijkl equals the printed list (Ke_01..Ke_23) = (e23, -e13, e12, e03, -e02, e01); K = K^T = K^-1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C2. J(Phi_1) is symmetric and PSD",
     "pass": true,
     "detail": "rank 15 "
    },
    {
     "check": "C2. (id x T) J(Phi_1) is symmetric and PSD",
     "pass": true,
     "detail": "rank 15 "
    },
    {
     "check": "C2. J(Phi_2) is symmetric and PSD",
     "pass": true,
     "detail": "rank 15 "
    },
    {
     "check": "C2. (id x T) J(Phi_2) is symmetric and PSD",
     "pass": true,
     "detail": "rank 15 "
    },
    {
     "check": "C3. E^T E = I_6 and J(Phi_2 o Phi_1) = J(R^dagger o S o T_10)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C4. Z is symmetric and PSD",
     "pass": true,
     "detail": "rank 80 "
    },
    {
     "check": "C4. Z has positive partial transpose",
     "pass": true,
     "detail": "rank 80 "
    },
    {
     "check": "C5. S(P_e00) = R(P_e00) = 36^2 I_6 (as printed)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C5. (e00 x e00)^* Z (e00 x e00) = tr[R(P_e00) S(P_e00)] = 6 * 36^4 = 10077696 (as printed)",
     "pass": true,
     "detail": "computed 10077696"
    },
    {
     "check": "C6. Z = (Phi_1^# x Phi_2)(Omega Omega^*) (eq:choi-transfer), entry by entry",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C6. Phi_1^#, Phi_1^# o T, Phi_2, Phi_2 o T are CP (Choi matrices PSD), tr Z > 0: rho = Z / tr Z is in the class C",
     "pass": true,
     "detail": "[True, True, True, True]"
    },
    {
     "check": "C7. Z (xhat x xhat) = 0 in Q(sqrt2) (x) A for x = (1, t1, t2, t3): every one of the twenty xhat_j x xhat_j is in ker Z (eq:kernel-vectors)",
     "pass": true,
     "detail": "largest number of nonzero basis coordinates in a row: 0"
    },
    {
     "check": "D1. c_1, c_2 > 0 are rational, and F_i - c_i I is PSD (F_i >= c_i I)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D2. J(Theta) (441 x 441) is symmetric and PSD (Theta is CP)",
     "pass": true,
     "detail": "rank 51 "
    },
    {
     "check": "D2. (id x T) J(Theta) is PSD (Theta is PPT)",
     "pass": true,
     "detail": "rank 51 "
    },
    {
     "check": "D3. tr Theta(E_ab) = delta_ab for all 441 (a, b): Theta is trace preserving",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D4. Theta(P_*) = P_* (the flag is absorbing)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D5. (W0^* x W0^*) J(Theta o Theta) (W0 x W0) = c_1 c_2 Z, entry by entry (eq:choi-corner)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/build/source/sections/introduction.tex": "a2eda7fe4b6cc97df88acdcc56c8a5392b64cc994ff456cd298e0ae6a3f35d1d",
    "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/build/source/sections/certificate.tex": "ea74b4f09a963b92b33ec9d67c0a87c235ccb332b61c53636a6eaf0a53a40506",
    "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/build/source/sections/geometry.tex": "cdc6f0039f436f5948ef76d3748f72b226ecc0bd4d2651a5836a4322491597db",
    "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/build/source/sections/applications.tex": "8ba7427a5bfc4bc3e82d446bb322cfbaa6ce2c51b2984acfdae95d77ed2750c4",
    "preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/verification/certificate-output.txt": "836e98249143b903bb2ce69c2cec085f7252db333f6fd2e415af37385bb4e228"
   },
   "forges": [
    {
     "forge": "M_3[0][2] changed from -4 to -3 [fails: A1. A2. B2. B4. B5. B6.]",
     "verdict": "REFUTED"
    },
    {
     "forge": "one sign of K flipped (K e_02 = +e_13): Phi_1, Phi_2 stay PPT, Z stays PSD, the kernel vectors are lost [fails: C1. C7.]",
     "verdict": "REFUTED"
    },
    {
     "forge": "recorded d_131 list with the 18 moved from k = 17 to k = 19 [fails: B5.]",
     "verdict": "REFUTED"
    },
    {
     "forge": "Theta built with c_1 = 1 (no rescaling of Phi_1) [fails: D1. D2.]",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 51.5
  },
  {
   "id": "F-561",
   "index": 561,
   "family": "281",
   "title": "Full support of the zero-temperature Sherrington-Kirkpatrick order parameter",
   "tier": "A",
   "dir": "Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026",
   "leanChallenges": [
    "SKFullSupport",
    "SKValue"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the exact rational certificate of Lemma cert:identity-lemma (the identity, every printed constant and the positivity of all 22 remainder coefficients) and the pointwise algebra of Lemma shape:ibp; not the sign lemmas, the integration by parts, nor the curvature and gap arguments that turn the inequality into full support",
   "decider": "instruments/openaimath/finite/f561_sk_full_support.py",
   "decides": "a finite component: the exact rational certificate of Lemma cert:identity-lemma (the identity, every printed constant and the positivity of all 22 remainder coefficients) and the pointwise algebra of Lemma shape:ibp; not the sign lemmas, the integration by parts, nor the curvature and gap arguments that turn the inequality into full support",
   "value": {
    "constants": {
     "A": "1/13",
     "C": "53/24",
     "k": "19/16",
     "T_0": "73/48",
     "g_0": "2331/10000",
     "h": "87/2500",
     "x_0": "4663/5000",
     "y": "6853/10000",
     "m": "-1051/400",
     "n_0": "-1073/500",
     "p_0": "337/10000",
     "p_1": "183/2500",
     "p_2": "411/10000",
     "p_3": "11461/5000",
     "p_4": "647/10000"
    },
    "P_rem_terms": 22,
    "v4_coefficient": "1661507/1521000000",
    "margin_over_1/1000": "140507/1521000000",
    "residual_terms": 22,
    "F2_terms": 10,
    "W_F2_terms": 30
   },
   "checks": [
    {
     "check": "1. parsed from the LaTeX: 15 constants, Q, T, F_2, P_0..P_4, S, the five derivative rules, the identity's two sides, 22 table rows",
     "pass": true,
     "detail": "A=1/13; C=53/24; k=19/16; T_0=73/48; g_0=2331/10000; h=87/2500; x_0=4663/5000; y=6853/10000; m=-1051/400; n_0=-1073/500; p_0=337/10000; p_1=183/2500; p_2=411/10000; p_3=11461/5000; p_4=647/10000"
    },
    {
     "check": "3. d_b F_2 = 2Q + T, and equals the printed expansion",
     "pass": true,
     "detail": "d_b F_2 = 2Avb-4kRb+2Cv^2+2kKv+(g_0+2T_0)Rv+hjv"
    },
    {
     "check": "3. d_j F_2 = x_0 v^2 + hbv and W(F_2) = W_0(F_2) - L(x_0 v^2 + hbv) (eq cert:L-correction)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. W(F_2) is linear in z with z-coefficient -d_b F_2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. z^2 - 12Rvb^2 + 6v^4 - (z+Q)^2 - ST - W(F_2) has no z dependence",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. THE IDENTITY: LHS - (z+Q)^2 - ST - sum p_i P_i - W(F_2) - L(x_0v^2+hbv), after b = B + 2v, equals the table polynomial P_rem exactly",
     "pass": true,
     "detail": "residual has 22 terms (no z, no L: True); every coefficient matches; 0.00s"
    },
    {
     "check": "4. the table: 22 distinct monomials in R,K,v,B,j, every coefficient > 0",
     "pass": true,
     "detail": "min coefficient 16259/23400000"
    },
    {
     "check": "4. p_0..p_4 > 0",
     "pass": true,
     "detail": "337/10000, 183/2500, 411/10000, 11461/5000, 647/10000"
    },
    {
     "check": "4. each P_i is a product of sign-constrained variables (R,K,v,B,j >= 0) times a square",
     "pass": true,
     "detail": "P_0 = Rj * (...)^2; P_1 = RB * (...)^2; P_2 = Kv * (...)^2; P_3 = vB * (...)^2; P_4 = 1 * (...)^2"
    },
    {
     "check": "4. T = v(g_0 R + hj) and L(x_0 v^2 + hbv) have positive coefficients (g_0, h, x_0 > 0)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. the v^4 coefficient 1661507/1521000000 - 1/1000 = 140507/1521000000 > 0, as printed",
     "pass": true,
     "detail": "140507/1521000000"
    },
    {
     "check": "5. the five rules shape:D-rules follow from R = r^2, K = rs, v = r_x, b = -r_xx/r, j = s_x - (s/r)v, L = -r s_xx (rational jet identities)",
     "pass": true,
     "detail": "DR ok, DK ok, Dv ok, Db ok, Dj ok"
    },
    {
     "check": "5. B = b - 2v = -2H and S = z + bv - 2Rb = 2rH_x with H = w/(2r) + v; Rb^2 = w^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. end to end for F_2: d/dx(r F_2 f) = W(F_2) f when f_x/f = r - s (eq cert:divergence, pointwise part)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/build/certificate.tex": "8943edb57f7b7560ce35af962ccdbe072c359bf6c0dd0992ef97862e9cd7b484",
    "preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/build/signs.tex": "48eda6c618cbbd6d95e6e7518c22888572900a951d7a4b4f3de2cbb3668a36b5"
   },
   "forges": [
    {
     "forge": "the table v^4 coefficient printed as 1661508/1521000000 (one more)",
     "verdict": "REFUTED"
    },
    {
     "forge": "p_4 = 648/10000 instead of 647/10000",
     "verdict": "REFUTED"
    },
    {
     "forge": "p_4 raised by 1/100 and the table REPLACED by the exact new remainder (identity holds by construction; min coefficient -3067/40000)",
     "verdict": "REFUTED"
    },
    {
     "forge": "h = 88/2500 instead of 87/2500 (enters T, F_2 and the L term)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.0
  },
  {
   "id": "F-605",
   "index": 605,
   "family": "309",
   "title": "The Kervaire invariant problem at the prime three",
   "tier": "B",
   "dir": "The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026",
   "leanChallenges": [],
   "word": "REFUSED",
   "why": "a finite component: the structure constants, the low-degree Ext groups D_{1,27} = <v^26 a, y, t_7'> and D_{6,10} = <v b^3> by an independent ordinary-cobar computation (matching the paper's printed checkpoints), the named cocycles and Bocksteins, the resolution's cochain dimensions and every rational inequality of Section finite; the weight-60/92 tables that carry the stem-322 argument are REFUSED, so the row is REFUSED",
   "decider": "instruments/openaimath/finite/f605_kervaire_three.py",
   "decides": "a finite component: the structure constants, the low-degree Ext groups D_{1,27} = <v^26 a, y, t_7'> and D_{6,10} = <v b^3> by an independent ordinary-cobar computation (matching the paper's printed checkpoints), the named cocycles and Bocksteins, the resolution's cochain dimensions and every rational inequality of Section finite; the weight-60/92 tables that carry the stem-322 argument are REFUSED, so the row is REFUSED",
   "value": {
    "ordinary cobar": {
     "C^0,27": 12,
     "C^1,27": 470,
     "ranks (1,27)": [
      11,
      456
     ],
     "C^5,10": 363,
     "C^6,10": 253,
     "ranks (6,10)": [
      162,
      90
     ]
    },
    "ordinary cobar filtration-6 cochains at weight 81": 1397584924909,
    "runtime_s": 2.1
   },
   "checks": [
    {
     "check": "the theorem and the finite lemma as printed",
     "pass": true,
     "detail": "introduction.tex:24-30, finite.tex:206-222, recurrence.tex:765-769, 863-879"
    },
    {
     "check": "S. R_1..R_3 and D_1..D_3 from the recursion are 3-integral (denominators prime to 3)",
     "pass": true,
     "detail": "terms: R [2, 6, 56], D [2, 5, 48]"
    },
    {
     "check": "S. eta_R(v_1) = v_1 (mod 3) and v_1 + 3t_1 (mod 9), as printed",
     "pass": true,
     "detail": ""
    },
    {
     "check": "S. eta_R(v_2) = v_2 + v_1 t_1^3 - v_1^3 t_1 (mod 3) and v_2 + 3t_2 + 5v_1^3 t_1 + v_1 t_1^3 (mod 9), as printed",
     "pass": true,
     "detail": "{(0, 0, 0, 0, 1, 0, 0, 0, 0): 3, (1, 0, 0, 3, 0, 0, 0, 0, 0): 1, (0, 1, 0, 0, 0, 0, 0, 0, 0): 1, (3, 0, 0, 1, 0, 0, 0, 0, 0): 5}"
    },
    {
     "check": "C. d o d = 0 on every basis cochain of C^0, C^1 (weight 27), C^2 (weight 14), C^5 (weight 10); every image normalized",
     "pass": true,
     "detail": "basis sizes {'0,27': 12, '1,27': 470, '2,14': 412, '5,10': 363}"
    },
    {
     "check": "K. weight 27: ordinary cobar dimensions 12, 470 in filtrations 0, 1 and ranks 11, 456 (printed 12, 470; 11, 456)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. h, r, s', y, t_4', t_7' are divisible as stated, normalized one-cocycles of weights 3, 9, 7, 27, 15, 27 (bidegrees (1,3), (1,9), (1,7), (1,27), (1,4i-1))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. delta_9(v_2^9) = y + t_7' (finite:corrected-y) and (v_2 + v_1 w)^9 - v_2^9 = v_1^9 w^9 mod 3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. D_{1,27} = <v^26 a, y, t_7'>: dimension 3 = 470 - 11 - 456, and appending the three raises the incoming rank 11 -> 14",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. their t_1^27 coefficients are 0, 1, 0",
     "pass": true,
     "detail": "[0, 1, 0]"
    },
    {
     "check": "K. b = B(h) = t_1^2|t_1 + t_1|t_1^2 - v t_1|t_1 (alg:b-sign-check), a cocycle of bidegree (2,3)",
     "pass": true,
     "detail": "{((0, 0, 0), (1, 0, 0), (2, 0, 0)): 1, ((0, 0, 0), (2, 0, 0), (1, 0, 0)): 1, ((1, 0, 0), (1, 0, 0), (1, 0, 0)): 2}"
    },
    {
     "check": "K. c' = B(r) and P = a B(t_4') are cocycles of bidegrees (2,9) and (3,16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. (6,10): middle dimension 253, ranks 162 and 90 (printed 253; 162, 90)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "K. D_{6,10} = <v b^3> = F_3: dimension 1, v b^3 a cocycle raising the incoming rank by 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R. cochain dimensions of the smaller resolution (n_-, n, n_+) at (1,27), (6,10), (6,81), by enumeration of (Q, g) and by the generating function (alg:dimension-series), equal the printed ((12, 49, 94), (3, 2, 1), (9301, 11419, 12864))",
     "pass": true,
     "detail": "enumerated [(12, 49, 94), (3, 2, 1), (9301, 11419, 12864)], series [(12, 49, 94), (3, 2, 1), (9301, 11419, 12864)]"
    },
    {
     "check": "R. Table finite:low-ranks: n - r_- - r equals the cohomology found in K (3 at (1,27), 1 at (6,10)) and r_C - r_- = |C| (3, 1, and 0 for the boundary B(y)^3)",
     "pass": true,
     "detail": "the (6,81) row gives dim D_{6,81} = 7 (not re-derived)"
    },
    {
     "check": "G. |G_60| = 1213 and |G_92| = 2269 slots (alg:finite-grid; recurrence.tex:778, 788)",
     "pass": true,
     "detail": "1213, 2269"
    },
    {
     "check": "G. the cancellation degree table (u, p, z), z + R = m, |P| + R = 6|b|, and m = (f, (322+f)/4) at f = 14, 18, 22, 26",
     "pass": true,
     "detail": ""
    },
    {
     "check": "G. deficits 9f/2 - d on the additive page: exterior positives exactly (1, 7/2), (3, 3/2), (4, 1/2), sum 11/2; the only positive polynomial generator is c_{1,0} = b with 6; the elementary deficit 3f/2 - d is positive only for e_{1,0} (1/2)",
     "pass": true,
     "detail": "[(1, Fraction(7, 2)), (3, Fraction(3, 2)), (4, Fraction(1, 2))] [(('c', 2, 3), Fraction(6, 1))] [(('e', 1, 1), Fraction(1, 2))]"
    },
    {
     "check": "G. 94 < I_1(24) = 98, 89 < I_6(35) = 235/2, 88 < S_6(30) = 199/2, I_k(f) = S_k(f-1); the four b^6 conditions at m = (f, (322+f)/4), m+J, m+R, m+R-J are exactly 17f > 464, 396, 328, 396 and hold for f >= 30",
     "pass": true,
     "detail": ""
    },
    {
     "check": "THE TABLES CARRYING STEM 322 (finite:affine-ranks, finite:enclosures, the span / cancellation / injection / product ranks, B(y)^3 = 0 in D_{6,81}): NOT RE-DERIVED (REFUSED: the paper's perturbed resolution through weight 93 and filtration 33 is not re-implemented; the ordinary cobar route has 1,397,584,924,909 cochains at (6,81))",
     "pass": false,
     "detail": "the decisive finite part of the headline"
    }
   ],
   "sources": {
    "preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/build/sections/finite.tex": "f5441f0e6b9be324f5df3442d5eb360fc90470e019431e855a640ceab8a4389d",
    "preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/build/sections/recurrence.tex": "c8eb952bf15e06785af84b96256783f30d39035805b9255575e5ed7a01570a8c",
    "preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/build/sections/introduction.tex": "e7a1e4308ded4c600ac735b24fbc79c3e94bfda0cbe768aa9dcc7cb3ef46f809"
   },
   "forges": [
    {
     "forge": "b printed with + v t_1|t_1 instead of - v t_1|t_1",
     "verdict": "REFUTED"
    },
    {
     "forge": "cochain dimension 11419 at (6,81) printed as 11418",
     "verdict": "REFUTED"
    },
    {
     "forge": "the (6,10) incoming rank printed as 161",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 2.1
  },
  {
   "id": "F-664",
   "index": 664,
   "family": "347",
   "title": "Three fixed points on the symplectic quadric threefold",
   "tier": "A",
   "dir": "A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026",
   "leanChallenges": [
    "ArnoldCounterexample"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: f(x,y) = e.x + (x-p).y on S^2 x S^2 has exactly three critical points, q_0 = (p,-e) degenerate (Hessian rank 2); NOT the passage to Hamiltonian fixed points on Q^3 nor Crit(Q^3) = 4",
   "decider": "instruments/openaimath/finite/f664_quadric_fixed_points.py",
   "decides": "a finite component: f(x,y) = e.x + (x-p).y on S^2 x S^2 has exactly three critical points, q_0 = (p,-e) degenerate (Hessian rank 2); NOT the passage to Hamiltonian fixed points on Q^3 nor Crit(Q^3) = 4",
   "value": {
    "quotient_dim": 6,
    "multiplicity_q0": 4,
    "distinct_complex": 3,
    "distinct_real": 3,
    "groebner_basis_size": 13,
    "charpoly_ell_degree": 6,
    "standard_monomials": [
     "1",
     "m",
     "l",
     "y3",
     "y1",
     "lm"
    ]
   },
   "checks": [
    {
     "check": "the paper states the function and the count",
     "pass": true,
     "detail": "04-three-points.tex:14-24, eq:critical-multipliers 34-38"
    },
    {
     "check": "1. the output is a Groebner basis of the ideal: every S-pair and every generator reduces to 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. reduced Groebner basis (grevlex): the ideal is zero-dimensional",
     "pass": true,
     "detail": "13 basis elements; dim_Q Q[x,y,l,m]/I = 6 (complex solutions with multiplicity)"
    },
    {
     "check": "1. Hermite form Tr(M_{b_i b_j}): number of distinct REAL critical points = signature = 3",
     "pass": true,
     "detail": "rank 3 (distinct complex solutions), signature 3"
    },
    {
     "check": "2. a separating linear form: deg squarefree(charpoly M_ell) = number of distinct complex solutions",
     "pass": true,
     "detail": "deg 3 vs rank 3"
    },
    {
     "check": "2. Sturm count of the real roots of that squarefree polynomial = 3",
     "pass": true,
     "detail": "3"
    },
    {
     "check": "3. q_0 = (p,-e) with l = m = 0 and q_s with l = m = s sqrt3 solve the system exactly in Q(sqrt3)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the three printed points are distinct, so they are the whole real critical set",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. f(q_0) = 0",
     "pass": true,
     "detail": "0 + (0)*sqrt3"
    },
    {
     "check": "3. f(q_s) = s 3sqrt3/2 for s = +1, -1",
     "pass": true,
     "detail": "0 + (3/2)*sqrt3, 0 + (-3/2)*sqrt3"
    },
    {
     "check": "4. Hessian of the Lagrangian at q_0 on its tangent space: charpoly t^4 - t^2 (eigenvalues 1, -1, 0, 0)",
     "pass": true,
     "detail": "['1', '0', '-1', '0', '0']"
    },
    {
     "check": "4. that Hessian has rank two: q_0 is degenerate",
     "pass": true,
     "detail": "rank 2"
    },
    {
     "check": "4. in the chart of Remark rem:degenerate-critical-point, the 2-jet of f at q_0 is exactly va",
     "pass": true,
     "detail": "{(0, 1, 1, 0): 1}"
    },
    {
     "check": "5. (1 - t^2/2)^2 (1 + t^2) - 1 = t^4 (t^2 - 3)/4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. the paper prints the involution A[z] = [z_0:-z_1:-z_2:-z_3:-z_4]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. det [[z1+iz2, z3+iz4], [-z3+iz4, z1-iz2]] = z1^2+z2^2+z3^2+z4^2 in Z[i][z] (02-quadric.tex:102-110)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. the +1 eigenline [1:0:0:0:0] of A is off Q^3, so Fix(A) = Q^3 n {z_0 = 0}",
     "pass": true,
     "detail": "z.z = 1"
    },
    {
     "check": "6. T = diag(0,J,2J): charpoly t^5 + 5t^3 + 4t = t(t^2+1)(t^2+4), five distinct eigenvalues; the printed eigenlines [0:1:+-i:0:0], [0:0:0:1:+-i] are eigenvectors and all four lie on Q^3; [1:0:0:0:0] does not (Cor. cor:quadric-critical)",
     "pass": true,
     "detail": "eigen True, on Q^3 4"
    }
   ],
   "sources": {
    "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/04-three-points.tex": "ab731f2b9659d8a418a37d753d92ae98b4a57a22f4845ce9a9fdf2cfb678ad74",
    "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/00-introduction.tex": "819f8d1abf0d5862e147654027e36450e38ed404b0d7de7c3eb5b61cd8e4a707",
    "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/02-quadric.tex": "82faa26b38f97cb27cb0e13dd9495d313e685e9c41980b9e057c0638ec6ea2e1",
    "preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/03-critical-bound.tex": "f26d7312f00d384b04cc3fddd1f48dbb2f8e3b449d83cf3cb3358a1b680f5e21"
   },
   "forges": [
    {
     "forge": "e scaled to (2,0,0) (real critical count 4)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the e.x term dropped (real critical count 4)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the printed degenerate point moved to (p, +e)",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed f(q_s) halved to s 3sqrt3/4",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.3
  },
  {
   "id": "F-665",
   "index": 665,
   "family": "348",
   "title": "Zero-Plane Rigidity for Einstein Four-Manifolds",
   "tier": "A",
   "dir": "Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026",
   "leanChallenges": [
    "EinsteinFour"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4's sine minorant, upper-deficit integrals and moment-obstruction coefficient table; not the classification theorem",
   "decider": "instruments/openaimath/finite/f665_einstein_zero_plane.py",
   "decides": "a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4's sine minorant, upper-deficit integrals and moment-obstruction coefficient table; not the classification theorem",
   "value": {
    "interval_boxes": 35,
    "triangle_rows": 12,
    "eta_plus_lower": "35568414570146708970215164653657279684840568049287583618357776852209994707811168145719343/436079249052527988124474363762562231471002196310491136000000000000000000000000000000000000",
    "runtime_s": 1.2
   },
   "checks": [
    {
     "check": "the paper prints kappa, Theta, c and the constants used here (06-coupling.tex)",
     "pass": true,
     "detail": "06-coupling.tex:62-75, :189-198, :405-416, :455-457"
    },
    {
     "check": "the theorem as stated (01-introduction.tex:14-24)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the paper prints the two tables and the zero list (07-certificates.tex)",
     "pass": true,
     "detail": "07-certificates.tex:106-120, :223-246, :339-345"
    },
    {
     "check": "A. matrix A: eq:21 at sigma = 1 from Section 6 = the appendix instructions (three entries)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. matrix B: (H_v, H_b, S* K J) from Section 6 = the appendix instructions (H*, its exchange, S*{...})",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. scalar rows: L_v and F + 12 c delta^2 L_b from eq:23 = the appendix instructions",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:22 diagonal clearing: v K r_v (2 Theta R_v - Q_v) = H_v (polynomial identity)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. r_v = 8v + (1-v)^2 and S* = (alpha b r_v + v r_b/alpha)/2 (the AM-GM pair)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. P0 = Pk + 30 h v",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. axial: lambda_ax + 32 v delta = 6(a^2+s^2) + 32 a delta",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. 10/9 (h - 67/50 - 27/100 vb) = c",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. -d/dsigma A_sigma = v^2 [ (30hv, -Pl)(30hv, -Pl)^T + diag(0, 6 h v P0) ] (sigma symbolic)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:20 enlarged g_v = 19/3 - 3v - 17/12 b, g_b = 7/2 - 19/12 b",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:20: (left - enlarged right)/(v b P^4) = A/v + A u^2/b + G v + H b - L (cleared by v b)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:19: (2-v)(4 C0 D0^sc - (2A0+L0)^2) is the printed quintic",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. g-majorant: 1 -+ sigma factorizations, the gap identity, bracket(1) = 16, bracket(2) = 4, concave, Bernstein >= 4 on [1,2]",
     "pass": true,
     "detail": "[Fraction(16, 1), Fraction(18, 1), Fraction(18, 1), Fraction(57, 4), Fraction(4, 1)]"
    },
    {
     "check": "D. c >= 1/10 and M >= 1/4 (so m >= 1/16) on D, by Bernstein coefficients",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. OSC, OCW0, SCU tile D = O U W0 (containment, area, disjoint interiors)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. OAS, ASE, ACE, SBE, CBE, SUB tile D+ = OCU; OSC and SCU tile OCU",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. scalar eq19.defect_C0: degree 2, Bernstein 2 on [('0', '1'), ('1', '2')], every coefficient > 1/2",
     "pass": true,
     "detail": "minima 3971/3200, 1063/1600"
    },
    {
     "check": "B. printed minima of eq19.defect_C0 on the two intervals (07-certificates.tex:122-127)",
     "pass": true,
     "detail": "['3971/3200', '1063/1600'] vs printed ['3971/3200', '1063/1600']"
    },
    {
     "check": "B. scalar eq19.defect_determinant: degree 5, Bernstein 5 on [('0', '1'), ('1', '2')], every coefficient > 1/2",
     "pass": true,
     "detail": "minima 101027/20000, 283729/400000"
    },
    {
     "check": "B. printed minima of eq19.defect_determinant on the two intervals (07-certificates.tex:122-127)",
     "pass": true,
     "detail": "['101027/20000', '283729/400000'] vs printed ['101027/20000', '283729/400000']"
    },
    {
     "check": "B. scalar eq20.G: degree 4, Bernstein 4 on [('0', '1')], every coefficient >= 9/10",
     "pass": true,
     "detail": "minima 117/128"
    },
    {
     "check": "B. scalar eq20.H: degree 4, Bernstein 4 on [('0', '1')], every coefficient >= 1/4",
     "pass": true,
     "detail": "minima 3681/12800"
    },
    {
     "check": "B. scalar eq20.minus_Z: degree 8, Bernstein 8 on [('0', '1/16')], every coefficient >= 13/100",
     "pass": true,
     "detail": "minima 23720719937369031/175921860444160000"
    },
    {
     "check": "B. scalar eq20.radical: degree 16, Bernstein 16 on [('1/16', '1')], every coefficient >= 3/25",
     "pass": true,
     "detail": "minima 6168574436959761676638843427263/49517601571415210995964968960000"
    },
    {
     "check": "B. triangle eq21.A.OSC OSC N=8: bounds 16000 / det 53000000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 16855, min determinant 1260265760821661/23592960; zeros []"
    },
    {
     "check": "B. triangle eq21.A.OCW0 OCW0 N=8: bounds 890 / det 3900000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 42875/48, min determinant 9191328125/2304; zeros []"
    },
    {
     "check": "B. triangle eq21.A.SCU SCU N=8: bounds 2400 / det 3500000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 144809/60, min determinant 19494020649752399/5435817984; zeros []"
    },
    {
     "check": "B. triangle eq22.B.OAS OAS N=9: bounds 43 / det 1050, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 159697622959/150994944; zeros []"
    },
    {
     "check": "B. triangle eq22.B.ASE ASE N=9: bounds 43 / det 490, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 32821379939188729/66045188505600; zeros []"
    },
    {
     "check": "B. triangle eq22.B.ACE ACE N=10: bounds 2 / det 21, all coefficients nonnegative/PSD, 1 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 49859/24192, min determinant 161755713049/7464960000; zeros [(0, 10)]"
    },
    {
     "check": "B. triangle eq22.B.SBE SBE N=9: bounds 43 / det 1300, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 43735879/32400; zeros []"
    },
    {
     "check": "B. triangle eq22.B.CBE CBE N=10: bounds 13/100 / det 4, all coefficients nonnegative/PSD, 2 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 101579/725760, min determinant 78479011/16402500; zeros [(9, 1), (10, 0)]"
    },
    {
     "check": "B. triangle eq22.B.SUB SUB N=9: bounds 5 / det 290, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 140/27, min determinant 865893637/2952450; zeros []"
    },
    {
     "check": "B. triangle eq23.Lv.OCU OCU N=6: bounds 9/10, all coefficients nonnegative/PSD, 1 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 101/108; zeros [(0, 6)]"
    },
    {
     "check": "B. triangle eq23.Fplus.OSC OSC N=9: bounds 19/100, all coefficients nonnegative/PSD, 6 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 323/1680; zeros [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)]"
    },
    {
     "check": "B. triangle eq23.Fplus.SCU SCU N=9: bounds 9/1000, all coefficients nonnegative/PSD, 7 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 37/3780; zeros [(0, 6), (0, 7), (0, 8), (0, 9), (1, 7), (1, 8), (2, 7)]"
    },
    {
     "check": "B. the three printed CBE triples at (8,0), (8,1), (8,2) (07-certificates.tex:316-323)",
     "pass": true,
     "detail": "{(8, 0): ['11041/810', '4063/810', '60557/16200'], (8, 1): ['4328/405', '839/405', '3977/2700'], (8, 2): ['191/81', '191/81', '-1189/1350']}"
    },
    {
     "check": "B. F*(n,t) = F(1-n+t, 1-n-t) is even in t with exactly the five printed f_j (07-certificates.tex:366-377)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. h_i = d_i q_i exactly, i = 0..4, with q_i of degrees 4, 6, 7, 7, 7",
     "pass": true,
     "detail": "[4, 6, 7, 7, 7]"
    },
    {
     "check": "B. sum_i h_i C(4,i) z^i (1-z)^(4-i) = F*, z = t^2/(1-n)^2 (cleared by (1-n)^8), and F*(1,0) = f_0(1) = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. q_0 on the four quarters, Bernstein degree 4, every coefficient >= 3",
     "pass": true,
     "detail": "minima 241103/61440, 282419/61440, 135713/3840, 6580667/61440"
    },
    {
     "check": "B. q_1 on the four quarters, Bernstein degree 6, every coefficient >= 31",
     "pass": true,
     "detail": "minima 2641/60, 258239/7680, 14880659/460800, 5041525/98304"
    },
    {
     "check": "B. q_2 on the four quarters, Bernstein degree 7, every coefficient >= 1/5",
     "pass": true,
     "detail": "minima 37/180, 14355523/786432, 483339817/20643840, 345453127/11796480"
    },
    {
     "check": "B. q_3 on the four quarters, Bernstein degree 7, every coefficient >= 12/5",
     "pass": true,
     "detail": "minima 299/120, 33386881/1966080, 15845401/983040, 34464779/1966080"
    },
    {
     "check": "B. q_4 on the four quarters, Bernstein degree 7, every coefficient >= 9/5",
     "pass": true,
     "detail": "minima 62/15, 31029203/3932160, 2423173/786432, 540781/286720"
    },
    {
     "check": "E. a0 = 157/50 < pi < 22/7 = b0 (pi enclosed in [3.1415926535.., 3.1415926535..] by Machin with alternating tails)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. S(z) coefficients 157/50, -5324/1029, 95388992557/37500000000, -155897368/259416045 (04-volume.tex:306-308)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. S(z) >= z (a0 - b0^3/24 - b0^7/322560) >= 9/5 z on [0,1/2]: the floor 19063149197/10376641800 > 9/5, and S(z)/z - 9/5 has Bernstein >= 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. upper-deficit row [0,1/2]: degree 31, (9 a0/48) * integral > 11869/1000000",
     "pass": true,
     "detail": "0.011869855"
    },
    {
     "check": "E. upper-deficit row [1/2,5/8]: degree 31, (9 a0/48) * integral > 1207/40000",
     "pass": true,
     "detail": "0.030175367"
    },
    {
     "check": "E. upper-deficit row [5/8,1]: degree 23, (9 a0/48) * integral > 19759/500000",
     "pass": true,
     "detail": "0.039518885"
    },
    {
     "check": "E. the three rows sum to more than 163/2000 > 2/25 > 3/40; printed floors sum to 81562/10^6 > 163/2000",
     "pass": true,
     "detail": "0.081564107"
    },
    {
     "check": "E. f(5/8) = 7511875/8388608 < 1; beta^2(3-beta) at u = 5/8 is 57500/59319 > 15/16; the comparison reduces to u^6(6 - 13u^2 + 6u^4) >= 0, positive on [0,5/8]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. L(2x^2) = 3/8 + 3/16 x^4 - 35/64 x^8 and V(2x^2) = 24/5 - 64/15 x^4 - 7/3 x^8 + 7/18 x^12 (04-volume.tex:634-636)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. H''' = L(z) z^(-5/2) and Phi''' = V(z), Phi = 16 H^2 + 9z(1-z)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. eq9.L: degree 8, Bernstein 8, every coefficient > 1/100",
     "pass": true,
     "detail": "least 1/64"
    },
    {
     "check": "E. eq9.third_derivative_sign: degree 17, Bernstein 17, every coefficient > 2/5",
     "pass": true,
     "detail": "least 2459668689/5793382400"
    },
    {
     "check": "E. eq9.endpoint.lower: degree 22, Bernstein 22, every coefficient > 1",
     "pass": true,
     "detail": "least 58454914732902747377800095783827783501884053/50000000000000000000000000000000000000000000"
    },
    {
     "check": "E. eq9.endpoint.upper: degree 22, Bernstein 22, every coefficient > 1",
     "pass": true,
     "detail": "least 132428032291164570287168821949356607977047/770000000000000000000000000000000000000"
    },
    {
     "check": "E. the four printed least coefficients 1/64, 2459668689/5793382400, and those of P_-, P_+ (04-volume.tex:638-649)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 1 + w^2/40 - w^4/1152 = R*(x) at w = 2x^2 (so H(2) - H(w) = sqrt2 (R*(1) - x R*(x)))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. two-point J = -16t(1-t)(H(2)-H(w))^2 + 9((1-t)(w^2-w) + 2t); t_+ - t_- numerator = (2-w)(10-37w); B_-, B_+ > 0 on [0,37/100]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 32 (11/20) = 88/5, (4 sqrt2)^2 = 32 < 289/9; (37/100)^2 = 1369/10000 >= 5/37; q(z) = 25 - 346z/25 + 31z^2/45 has q(2) = 17/225, q'(2) = -2494/225 < 0, q'' = 62/45",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the moment lower bound: 1 + z^2/40 - z^4/1152 - (1 + 31z^2/1440) >= 0 on [0,2]; 16z(1+31z^2/1440)^2 + 9z(1-z) - 121z^2/25 - z q(z) >= 0 on [0,2]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the measure 9/32 at 2, rest at 0: constraint expectations -5/2 and -1/8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. G'(c) = 1 + b^2/8 - b^4/128 >= 0 on [0,2], and with z = b^2/4: (1 + z/2 - z^2/8)^2 - (1+z) = -z^3/8 + z^4/64 <= 0 on [0,1]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R. results.json (data) agrees with every polynomial, coefficient, index, determinant, minimum, minimizing index and zero set computed here (15 interval rows, 12 triangle rows, 3 integrals)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/build/sections/07-certificates.tex": "3865f324bdab0b3462313d3511319ae0b08c9f8159cc305a71b01211680c9a57",
    "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/build/sections/06-coupling.tex": "bed07fc9ce1c1472a70090a9900c37a6134e558b2550af45d110a32d521af56e",
    "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/build/sections/04-volume.tex": "a42b63cfc087de16afa8ea2ba7a44f3f2348f5082e6abee904c9cd17634a4d3c",
    "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/build/sections/01-introduction.tex": "a56db14e908f48c1df9c16032c4509c317f766e3a3a64a4ba11773b4dd61c473",
    "preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/verification/results.json": "d84273376cedd109c7cbf46b77cdc9ae84eac99daced9e5cd3dcbb2f20b2e7dc"
   },
   "forges": [
    {
     "forge": "kappa coefficient 4 z_11 = -41 changed to -40",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed minimum 3971/3200 changed to 3972/3200",
     "verdict": "REFUTED"
    },
    {
     "forge": "printed CBE triple (8,2) off-diagonal -1189/1350 changed to -1188/1350",
     "verdict": "REFUTED"
    },
    {
     "forge": "vertex B moved from (3/2,1/2) to (3/2,3/5)",
     "verdict": "REFUTED"
    },
    {
     "forge": "CBE diagonal bound 13/100 printed as 14/100",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1.2
  },
  {
   "id": "F-667",
   "index": 667,
   "family": "348",
   "title": "Positively curved Einstein four-manifolds",
   "tier": "A",
   "dir": "Positively-curved-Einstein-four-manifolds-September-23-2026",
   "leanChallenges": [
    "EinsteinFour"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4's sine minorant, upper-deficit integrals and moment-obstruction coefficient table and the conditional lower-volume rows; not the classification theorem of F-667",
   "decider": "instruments/openaimath/finite/f667_einstein_positive.py",
   "decides": "a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4's sine minorant, upper-deficit integrals and moment-obstruction coefficient table and the conditional lower-volume rows; not the classification theorem of F-667",
   "value": {
    "interval_boxes": 35,
    "triangle_rows": 12,
    "eta_plus_lower": "35568414570146708970215164653657279684840568049287583618357776852209994707811168145719343/436079249052527988124474363762562231471002196310491136000000000000000000000000000000000000",
    "runtime_s": 1.2
   },
   "checks": [
    {
     "check": "the paper prints the lower-volume table and base term (08-classification.tex:205-219)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the theorem as stated (01-introduction.tex:13-20)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the paper prints kappa, Theta, c and the constants used here (06-coupling.tex)",
     "pass": true,
     "detail": "06-coupling.tex:62-75, :189-198, :405-416, :455-457"
    },
    {
     "check": "the paper prints the two tables and the zero list (07-certificates.tex)",
     "pass": true,
     "detail": "07-certificates.tex:106-120, :223-246, :339-345"
    },
    {
     "check": "A. matrix A: eq:21 at sigma = 1 from Section 6 = the appendix instructions (three entries)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. matrix B: (H_v, H_b, S* K J) from Section 6 = the appendix instructions (H*, its exchange, S*{...})",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. scalar rows: L_v and F + 12 c delta^2 L_b from eq:23 = the appendix instructions",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:22 diagonal clearing: v K r_v (2 Theta R_v - Q_v) = H_v (polynomial identity)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. r_v = 8v + (1-v)^2 and S* = (alpha b r_v + v r_b/alpha)/2 (the AM-GM pair)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. P0 = Pk + 30 h v",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. axial: lambda_ax + 32 v delta = 6(a^2+s^2) + 32 a delta",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. 10/9 (h - 67/50 - 27/100 vb) = c",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. -d/dsigma A_sigma = v^2 [ (30hv, -Pl)(30hv, -Pl)^T + diag(0, 6 h v P0) ] (sigma symbolic)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:20 enlarged g_v = 19/3 - 3v - 17/12 b, g_b = 7/2 - 19/12 b",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:20: (left - enlarged right)/(v b P^4) = A/v + A u^2/b + G v + H b - L (cleared by v b)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. eq:19: (2-v)(4 C0 D0^sc - (2A0+L0)^2) is the printed quintic",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. g-majorant: 1 -+ sigma factorizations, the gap identity, bracket(1) = 16, bracket(2) = 4, concave, Bernstein >= 4 on [1,2]",
     "pass": true,
     "detail": "[Fraction(16, 1), Fraction(18, 1), Fraction(18, 1), Fraction(57, 4), Fraction(4, 1)]"
    },
    {
     "check": "D. c >= 1/10 and M >= 1/4 (so m >= 1/16) on D, by Bernstein coefficients",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. OSC, OCW0, SCU tile D = O U W0 (containment, area, disjoint interiors)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. OAS, ASE, ACE, SBE, CBE, SUB tile D+ = OCU; OSC and SCU tile OCU",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. scalar eq19.defect_C0: degree 2, Bernstein 2 on [('0', '1'), ('1', '2')], every coefficient > 1/2",
     "pass": true,
     "detail": "minima 3971/3200, 1063/1600"
    },
    {
     "check": "B. printed minima of eq19.defect_C0 on the two intervals (07-certificates.tex:122-127)",
     "pass": true,
     "detail": "['3971/3200', '1063/1600'] vs printed ['3971/3200', '1063/1600']"
    },
    {
     "check": "B. scalar eq19.defect_determinant: degree 5, Bernstein 5 on [('0', '1'), ('1', '2')], every coefficient > 1/2",
     "pass": true,
     "detail": "minima 101027/20000, 283729/400000"
    },
    {
     "check": "B. printed minima of eq19.defect_determinant on the two intervals (07-certificates.tex:122-127)",
     "pass": true,
     "detail": "['101027/20000', '283729/400000'] vs printed ['101027/20000', '283729/400000']"
    },
    {
     "check": "B. scalar eq20.G: degree 4, Bernstein 4 on [('0', '1')], every coefficient >= 9/10",
     "pass": true,
     "detail": "minima 117/128"
    },
    {
     "check": "B. scalar eq20.H: degree 4, Bernstein 4 on [('0', '1')], every coefficient >= 1/4",
     "pass": true,
     "detail": "minima 3681/12800"
    },
    {
     "check": "B. scalar eq20.minus_Z: degree 8, Bernstein 8 on [('0', '1/16')], every coefficient >= 13/100",
     "pass": true,
     "detail": "minima 23720719937369031/175921860444160000"
    },
    {
     "check": "B. scalar eq20.radical: degree 16, Bernstein 16 on [('1/16', '1')], every coefficient >= 3/25",
     "pass": true,
     "detail": "minima 6168574436959761676638843427263/49517601571415210995964968960000"
    },
    {
     "check": "B. triangle eq21.A.OSC OSC N=8: bounds 16000 / det 53000000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 16855, min determinant 1260265760821661/23592960; zeros []"
    },
    {
     "check": "B. triangle eq21.A.OCW0 OCW0 N=8: bounds 890 / det 3900000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 42875/48, min determinant 9191328125/2304; zeros []"
    },
    {
     "check": "B. triangle eq21.A.SCU SCU N=8: bounds 2400 / det 3500000, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 144809/60, min determinant 19494020649752399/5435817984; zeros []"
    },
    {
     "check": "B. triangle eq22.B.OAS OAS N=9: bounds 43 / det 1050, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 159697622959/150994944; zeros []"
    },
    {
     "check": "B. triangle eq22.B.ASE ASE N=9: bounds 43 / det 490, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 32821379939188729/66045188505600; zeros []"
    },
    {
     "check": "B. triangle eq22.B.ACE ACE N=10: bounds 2 / det 21, all coefficients nonnegative/PSD, 1 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 49859/24192, min determinant 161755713049/7464960000; zeros [(0, 10)]"
    },
    {
     "check": "B. triangle eq22.B.SBE SBE N=9: bounds 43 / det 1300, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 3137/72, min determinant 43735879/32400; zeros []"
    },
    {
     "check": "B. triangle eq22.B.CBE CBE N=10: bounds 13/100 / det 4, all coefficients nonnegative/PSD, 2 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 101579/725760, min determinant 78479011/16402500; zeros [(9, 1), (10, 0)]"
    },
    {
     "check": "B. triangle eq22.B.SUB SUB N=9: bounds 5 / det 290, all coefficients nonnegative/PSD, 0 zero(s) exactly as printed",
     "pass": true,
     "detail": "min diagonal 140/27, min determinant 865893637/2952450; zeros []"
    },
    {
     "check": "B. triangle eq23.Lv.OCU OCU N=6: bounds 9/10, all coefficients nonnegative/PSD, 1 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 101/108; zeros [(0, 6)]"
    },
    {
     "check": "B. triangle eq23.Fplus.OSC OSC N=9: bounds 19/100, all coefficients nonnegative/PSD, 6 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 323/1680; zeros [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)]"
    },
    {
     "check": "B. triangle eq23.Fplus.SCU SCU N=9: bounds 9/1000, all coefficients nonnegative/PSD, 7 zero(s) exactly as printed",
     "pass": true,
     "detail": "min nonzero coefficient 37/3780; zeros [(0, 6), (0, 7), (0, 8), (0, 9), (1, 7), (1, 8), (2, 7)]"
    },
    {
     "check": "B. the three printed CBE triples at (8,0), (8,1), (8,2) (07-certificates.tex:316-323)",
     "pass": true,
     "detail": "{(8, 0): ['11041/810', '4063/810', '60557/16200'], (8, 1): ['4328/405', '839/405', '3977/2700'], (8, 2): ['191/81', '191/81', '-1189/1350']}"
    },
    {
     "check": "B. F*(n,t) = F(1-n+t, 1-n-t) is even in t with exactly the five printed f_j (07-certificates.tex:366-377)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. h_i = d_i q_i exactly, i = 0..4, with q_i of degrees 4, 6, 7, 7, 7",
     "pass": true,
     "detail": "[4, 6, 7, 7, 7]"
    },
    {
     "check": "B. sum_i h_i C(4,i) z^i (1-z)^(4-i) = F*, z = t^2/(1-n)^2 (cleared by (1-n)^8), and F*(1,0) = f_0(1) = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. q_0 on the four quarters, Bernstein degree 4, every coefficient >= 3",
     "pass": true,
     "detail": "minima 241103/61440, 282419/61440, 135713/3840, 6580667/61440"
    },
    {
     "check": "B. q_1 on the four quarters, Bernstein degree 6, every coefficient >= 31",
     "pass": true,
     "detail": "minima 2641/60, 258239/7680, 14880659/460800, 5041525/98304"
    },
    {
     "check": "B. q_2 on the four quarters, Bernstein degree 7, every coefficient >= 1/5",
     "pass": true,
     "detail": "minima 37/180, 14355523/786432, 483339817/20643840, 345453127/11796480"
    },
    {
     "check": "B. q_3 on the four quarters, Bernstein degree 7, every coefficient >= 12/5",
     "pass": true,
     "detail": "minima 299/120, 33386881/1966080, 15845401/983040, 34464779/1966080"
    },
    {
     "check": "B. q_4 on the four quarters, Bernstein degree 7, every coefficient >= 9/5",
     "pass": true,
     "detail": "minima 62/15, 31029203/3932160, 2423173/786432, 540781/286720"
    },
    {
     "check": "E. a0 = 157/50 < pi < 22/7 = b0 (pi enclosed in [3.1415926535.., 3.1415926535..] by Machin with alternating tails)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. S(z) coefficients 157/50, -5324/1029, 95388992557/37500000000, -155897368/259416045 (04-volume.tex:306-308)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. S(z) >= z (a0 - b0^3/24 - b0^7/322560) >= 9/5 z on [0,1/2]: the floor 19063149197/10376641800 > 9/5, and S(z)/z - 9/5 has Bernstein >= 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. upper-deficit row [0,1/2]: degree 31, (9 a0/48) * integral > 11869/1000000",
     "pass": true,
     "detail": "0.011869855"
    },
    {
     "check": "E. upper-deficit row [1/2,5/8]: degree 31, (9 a0/48) * integral > 1207/40000",
     "pass": true,
     "detail": "0.030175367"
    },
    {
     "check": "E. upper-deficit row [5/8,1]: degree 23, (9 a0/48) * integral > 19759/500000",
     "pass": true,
     "detail": "0.039518885"
    },
    {
     "check": "E. the three rows sum to more than 163/2000 > 2/25 > 3/40; printed floors sum to 81562/10^6 > 163/2000",
     "pass": true,
     "detail": "0.081564107"
    },
    {
     "check": "E. f(5/8) = 7511875/8388608 < 1; beta^2(3-beta) at u = 5/8 is 57500/59319 > 15/16; the comparison reduces to u^6(6 - 13u^2 + 6u^4) >= 0, positive on [0,5/8]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. L(2x^2) = 3/8 + 3/16 x^4 - 35/64 x^8 and V(2x^2) = 24/5 - 64/15 x^4 - 7/3 x^8 + 7/18 x^12 (04-volume.tex:634-636)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. H''' = L(z) z^(-5/2) and Phi''' = V(z), Phi = 16 H^2 + 9z(1-z)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. eq9.L: degree 8, Bernstein 8, every coefficient > 1/100",
     "pass": true,
     "detail": "least 1/64"
    },
    {
     "check": "E. eq9.third_derivative_sign: degree 17, Bernstein 17, every coefficient > 2/5",
     "pass": true,
     "detail": "least 2459668689/5793382400"
    },
    {
     "check": "E. eq9.endpoint.lower: degree 22, Bernstein 22, every coefficient > 1",
     "pass": true,
     "detail": "least 58454914732902747377800095783827783501884053/50000000000000000000000000000000000000000000"
    },
    {
     "check": "E. eq9.endpoint.upper: degree 22, Bernstein 22, every coefficient > 1",
     "pass": true,
     "detail": "least 132428032291164570287168821949356607977047/770000000000000000000000000000000000000"
    },
    {
     "check": "E. the four printed least coefficients 1/64, 2459668689/5793382400, and those of P_-, P_+ (04-volume.tex:638-649)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 1 + w^2/40 - w^4/1152 = R*(x) at w = 2x^2 (so H(2) - H(w) = sqrt2 (R*(1) - x R*(x)))",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. two-point J = -16t(1-t)(H(2)-H(w))^2 + 9((1-t)(w^2-w) + 2t); t_+ - t_- numerator = (2-w)(10-37w); B_-, B_+ > 0 on [0,37/100]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. 32 (11/20) = 88/5, (4 sqrt2)^2 = 32 < 289/9; (37/100)^2 = 1369/10000 >= 5/37; q(z) = 25 - 346z/25 + 31z^2/45 has q(2) = 17/225, q'(2) = -2494/225 < 0, q'' = 62/45",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the moment lower bound: 1 + z^2/40 - z^4/1152 - (1 + 31z^2/1440) >= 0 on [0,2]; 16z(1+31z^2/1440)^2 + 9z(1-z) - 121z^2/25 - z q(z) >= 0 on [0,2]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. the measure 9/32 at 2, rest at 0: constraint expectations -5/2 and -1/8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. G'(c) = 1 + b^2/8 - b^4/128 >= 0 on [0,2], and with z = b^2/4: (1 + z/2 - z^2/8)^2 - (1+z) = -z^3/8 + z^4/64 <= 0 on [0,1]",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. lower-volume row [0,1/2] (08-classification.tex:205-214): degree 15, (a0^3/27) * integral > 349/500000",
     "pass": true,
     "detail": "0.000698489"
    },
    {
     "check": "E. lower-volume row [1/2,1] (08-classification.tex:205-214): degree 15, (a0^3/27) * integral > 20277/500000",
     "pass": true,
     "detail": "0.040554741"
    },
    {
     "check": "E. 2(a0^2-4)/9 = 4883/3750 and base + the two rows > 1343/1000 > 4/3 (printed floors too)",
     "pass": true,
     "detail": "1.343386563"
    },
    {
     "check": "R. results.json (data) agrees with every polynomial, coefficient, index, determinant, minimum, minimizing index and zero set computed here (15 interval rows, 12 triangle rows, 5 integrals)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/07-certificates.tex": "1b43bdbdce970bc74bf46a3e2117271f498659e5ad3aff5994128a49ccc7af20",
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/06-coupling.tex": "25a6ab2d62f36f5dfc09a0d2ba1422701ef96b1f64e9a059cc9a177d8aadb280",
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/04-volume.tex": "a7bc1d2445f3d293ed507a30128ce0e5d7528a70b52bc54a643079320d4ddfbf",
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/verification/results.json": "d84273376cedd109c7cbf46b77cdc9ae84eac99daced9e5cd3dcbb2f20b2e7dc",
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/08-classification.tex": "08269d853bbc54bb05233ae334d7b66677df76642a498630b61e3fb53cce8e06",
    "preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/build/sections/01-introduction.tex": "d5620194e8f419a106ee8d95411bd8538d33c87a75bcf9a3d9a0e0abf80d7e5a"
   },
   "forges": [
    {
     "forge": "kappa coefficient 4 z_1 = -43 changed to -42",
     "verdict": "REFUTED"
    },
    {
     "forge": "lower-volume row [1/2,1] printed as 40555/10^6 (the exact value is 0.0405547...)",
     "verdict": "REFUTED"
    },
    {
     "forge": "vertex E moved from (1,1/2) to (1,2/5)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 1.2
  },
  {
   "id": "F-675",
   "index": 675,
   "family": "352",
   "title": "A finite-time singularity of Calabi flow on projective space",
   "tier": "A",
   "dir": "A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026",
   "leanChallenges": [],
   "word": "CERTIFIED",
   "why": "the whole of Proposition cert:shoot (existence on [0,56], 1 + p > 0, |q| <= .25R, ||(Re xi, Im xi) - z*|| <= .13R for every z* in the square), by an independent validated integration; not the tail lemma, the matching or the theorem",
   "decider": "instruments/openaimath/finite/f675_calabi_flow.py",
   "decides": "the whole of Proposition cert:shoot (existence on [0,56], 1 + p > 0, |q| <= .25R, ||(Re xi, Im xi) - z*|| <= .13R for every z* in the square), by an independent validated integration; not the tail lemma, the matching or the theorem",
   "value": {
    "q_over_R": "0.115049",
    "xi_over_R": "0.003137",
    "Delta_over_R": "2.48",
    "min_1_plus_p": "0.246938",
    "max_T": "8189.1",
    "order": 20,
    "steps": 1152,
    "runtime_s": 7.7
   },
   "checks": [
    {
     "check": "the paper prints the proposition, the ODE, the parameters, p_a, D0 and Lambda as used here",
     "pass": true,
     "detail": "certificate.tex:11-25, profile.tex:60-65, :93-102, :173-205"
    },
    {
     "check": "O. prof:ode re-derived from Q(y d/dy)(G-1) = (y^2/2)(1/G - 1/c): with D = (2/3) s d/ds, Q(D)/(8/27) has coefficients (1, 30, 751/4, -1215/4) on (s^3 d^3, s^2 d^2, s d, 1), and the right side becomes -p/(1+p)",
     "pass": true,
     "detail": "['-1215/4', '751/4', '30', '1']"
    },
    {
     "check": "O. majorant lemma: on |k-k0|, |d-d0| <= 1/10 (complex), 3|k| + 3/50 < |1+d|, (9/D_2)(beta + 1) <= 3/50 and 9 beta^2/D_2 = 0.0468 < 1 (D_2 = 891/2): |a_i| <= (3/50) 3^-i",
     "pass": true,
     "detail": ""
    },
    {
     "check": "O. 1 + p > 0 on [0,1] for every parameter of the polydisc: 1 + p >= |1+d| - (3|k| + 3/50)/3 >= 1.0861",
     "pass": true,
     "detail": ""
    },
    {
     "check": "O. the initial error e_0 (series tails, rounding, tangent tails, second-order remainder M_m (j+1) t^j) is below 10^-19 in each jet entry",
     "pass": true,
     "detail": "6.182e-21, 2.909e-21, 4.084e-21"
    },
    {
     "check": "I. the schedule: 288 blocks, lengths 1/32 (32), 1/16 (32), 1/8 (32), 1/4 (192), from 1 to 56 (sum 55); h s_c >= 32 at each of the paper's 8 step starts per block",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. every block enclosed: max Lagrange remainder coefficient 6.69e-26, max jump 8.15e-24, max transfer radius 3.21e-22; min of 1 + p_ref over [1,56] = 0.246938",
     "pass": true,
     "detail": ""
    },
    {
     "check": "I. second way: at s = 81/64 (r = 729/512) the origin series (110 terms, majorant tail) and the integrator agree on (p, p', p'') and on both parameter derivatives (transfer x initial tangent)",
     "pass": true,
     "detail": "max |difference of centres| 4.2e-33"
    },
    {
     "check": "E. bootstrap: assuming |delta p| <= Delta = 2.48R on [1,56], every block gives |delta p| <= 2.4293R < Delta (linear part <= 2.4287R, max weighted-Gronwall factor 1.649), and 1 + p >= m_j - Delta > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "E. endpoint error |e_N| <= (5.94e-04, 5.33e-04, 4.97e-04) R (componentwise, all jumps, transfer radii, forcing and e_0 propagated through the 41,616 products)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "P. |q| <= 0.11505R <= 1/4R for every z* in the square (central miss 0.00001R, linear 0.11454R)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "P. ||(Re xi, Im xi) - z*||_inf <= 0.00314R <= 13/100R for every z* in the square",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. origin: .585 + (9/440)(1 + 1/(1.30-.8)) = 711/1100 < .8; 3(|k0| + .1) < .585; |1+d| > 1.30; |d| < .501; 3(|k0|+.1) + 27/440 < .8; 27/440 < .062; 36/440 < 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. origin jets: sup (3i/2)/3^i = 1/2, sup (3i/2)(3i/2-1)/3^i = 2/3 (ratios decrease after); .501 + .8/3 < .768, .8/2 <= .4, .8(2/3) < .534",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. origin tails: E60 = .8 (3*61/2)^2/3^61 < 5.267e-26; 10 * .01169933 <= .12; |dk| <= 4.44605e-13, |dd| <= 5.40506e-12 < .011R; E_par < 1.816e-20; cert:initial-error < 2.316e-20 < 2e-19",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. disks: 30/31.5 < .954, B/31.5^2 < .190, |C|/31.5^3 < .011, ell^3 <= .015625; H_2 <= e^.477(...) solves to < .123602 < .13; .06/2 + .13/8 = .04625 < .1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. disks: Jacobian norm 30/31.5 + (751/4)/31.5^2 + (1215/2)/31.5^3 + (1/64)/.14^2 < 2.1; 6 e^1.05 < 17.146 < 32; 32 sum_{i>=45} 4^-i = (128/3) 4^-45 < 48 4^-45; .06/8 + .13/128 = .008515625 < .01; (floor(Q/100)+1)/Q > .01",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. transfers: 8500[(1 + 4e-15*8500)^288 - 1] < .001; 570000 + 288(.001) < 580000; 246000 + 55(.001)/.23^3 < 250000; 2.7 + .001(.02) < 2.72; e^L_real < 4.3",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. nonlinear: .23/(.23 - 14R) < 1.001; the endpoint error sum = .1055747504R exactly < .11R; the bootstrap 4.3(2.72+.11)R + 4.3(1.001)(14R)^2/.23^3 < 12.169035R < 14R",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. projection: .2315396 and .0901084 exactly, < .25 and < .13",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. roots: alpha^2 - 3beta = -453/25088; D0(-1.08659), D0(-1.08655) as printed; -1.08659 < lambda0 < -1.08655, u_- < Re lambda1 < u_+, .40933 < u_-, u_+ < .40938, .8671 < Im lambda1 < .86721; the three printed rational comparisons; |lambda1|^2 = H0/r0 < .959^2",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. Gamma: every entry inside the printed table, |Gamma - Lambda| < .001 entrywise (max 0.000005), row sums 0.91587, 0.65038, 0.65815 < .9164, .6507, .6587 (< .92, .68, .68); E > 46876434629/15680000000 > 2.9895 > 1.729^2; 1/(2(.40933)(.8671)(1.729)) = 500000000000/613674044347 < .82",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. p_a: the parameter interval lies in [.40068272185, .40068272187]; |d_j| and |d_j'| below the printed table; max |dP_a/dd| <= 0.001733 < .001735 < .01, so ||P_a(d) - P_a(d0)|| <= .01|d - d0|",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. cert:finite-products on this decider's reference (same schedule): m_j > .23 (min 0.2469), ||Z|| < .02 (0.01081), max ||Tt_(i,j)|| <= 8500 (8189.1), max_i sum ||Tt^(3)_(i,j+1)|| <= 570000 (555715), max_i sum ell_j/m_j^3 ||Tt^(3)|| <= 246000 (227681), max_i ||Tt_(i,0) Z|| <= 2.7 (2.6593)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. cert:finite-matching on this decider's reference: ||U_N - P_a(d0)|| < .04R (0.00556R), ||Lambda A*|| < .006R (0.00277R), ||(Lambda Y)_1|| < .116 (0.11455), ||(Lambda Y)_bottom - I_2|| < .001 (0.000007)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "N. cert:step-start on this decider's a-priori boxes (they contain every step start): -.76 < p < .4 (-0.7531, 0.3055), |ell p'| < .06 (0.0480), |ell^2 p''| < .01 (0.00510)",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/certificate.tex": "7bf137209d05a03e9971cf1fae102a9203bba06ee29c0253c9bcbff387ee3785",
    "preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/profile.tex": "ef35cc4db0e61ace45ebc3155c9250dbee7250dfffcfb93bacda09e716cf6f6d",
    "preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/introduction.tex": "f4be291774a887820f282991b554a24870cf6e0de6a7c6787ba6f622092e3310"
   },
   "forges": [
    {
     "forge": "printed bound .13R for the nonreal coordinates replaced by .001R (this decider proves >= .002R at z* = 0)",
     "verdict": "REFUTED"
    },
    {
     "forge": "parameter matrix with its two columns swapped (the trajectory is reused)",
     "verdict": "REFUTED"
    },
    {
     "forge": "central d0 = .400682721856639 changed to .400682721857639 (+1e-12; a new trajectory)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 7.7
  },
  {
   "id": "F-678",
   "index": 678,
   "family": "354",
   "title": "The Isoperimetric Conjecture for the Cubic Flat Three-Torus",
   "tier": "A",
   "dir": "The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026",
   "leanChallenges": [
    "CubicTorus"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: Proposition num:scalar-positive (B > 0 on [.64,1.02] u [1.62,2.25], B + S > 0 on [1.02,1.62]), decided independently, and the paper's knot and gap tables; NOT the geometric reduction to these inequalities",
   "decider": "instruments/openaimath/finite/f678_cubic_torus.py",
   "decides": "a finite component: Proposition num:scalar-positive (B > 0 on [.64,1.02] u [1.62,2.25], B + S > 0 on [1.02,1.62]), decided independently, and the paper's knot and gap tables; NOT the geometric reduction to these inequalities",
   "value": {
    "pieces": [
     35,
     323,
     33
    ],
    "smallest_lower_bound": [
     1.288250714803733e-05,
     2.9449793985515313e-06,
     3.8633188646813266e-05
    ],
    "B_1.46": "[-0.00273169482904084, -0.00273169482904084]",
    "S_1.46": "83349/25000000",
    "Bt_minus_b_range": [
     1.1043481696459334,
     1.9446495348754724
    ],
    "S_minus_s_range": [
     1.0033038553401452,
     1.6502616462317872
    ],
    "gap_min": "9/100000",
    "knots": {
     "0.64": {
      "b_h": 1744,
      "1e5_Bt": "1745.3103",
      "s_h": null,
      "1e5_S": null
     },
     "0.74": {
      "b_h": 1200,
      "1e5_Bt": "1201.4295",
      "s_h": null,
      "1e5_S": null
     },
     "0.84": {
      "b_h": 735,
      "1e5_Bt": "736.5987",
      "s_h": null,
      "1e5_S": null
     },
     "0.92": {
      "b_h": 432,
      "1e5_Bt": "433.5387",
      "s_h": null,
      "1e5_S": null
     },
     "0.98": {
      "b_h": 247,
      "1e5_Bt": "248.7761",
      "s_h": null,
      "1e5_S": null
     },
     "1.02": {
      "b_h": 144,
      "1e5_Bt": "145.7873",
      "s_h": 76,
      "1e5_S": "77.0033"
     },
     "1.06": {
      "b_h": 57,
      "1e5_Bt": "58.4861",
      "s_h": 98,
      "1e5_S": "99.0700"
     },
     "1.1": {
      "b_h": -15,
      "1e5_Bt": "-13.8117",
      "s_h": 123,
      "1e5_S": "124.2385"
     },
     "1.14": {
      "b_h": -74,
      "1e5_Bt": "-72.0554",
      "s_h": 151,
      "1e5_S": "152.3551"
     },
     "1.18": {
      "b_h": -119,
      "1e5_Bt": "-117.4813",
      "s_h": 182,
      "1e5_S": "183.1991"
     },
     "1.22": {
      "b_h": -153,
      "1e5_Bt": "-151.6548",
      "s_h": 215,
      "1e5_S": "216.4994"
     },
     "1.26": {
      "b_h": -178,
      "1e5_Bt": "-176.5427",
      "s_h": 247,
      "1e5_S": "248.2415"
     },
     "1.3": {
      "b_h": -196,
      "1e5_Bt": "-194.6355",
      "s_h": 271,
      "1e5_S": "272.6420"
     },
     "1.34": {
      "b_h": -211,
      "1e5_Bt": "-209.1511",
      "s_h": 297,
      "1e5_S": "298.5913"
     },
     "1.38": {
      "b_h": -226,
      "1e5_Bt": "-224.3640",
      "s_h": 325,
      "1e5_S": "326.1369"
     },
     "1.42": {
      "b_h": -247,
      "1e5_Bt": "-245.8957",
      "s_h": 332,
      "1e5_S": "333.3960"
     },
     "1.46": {
      "b_h": -275,
      "1e5_Bt": "-273.1695",
      "s_h": 332,
      "1e5_S": "333.3960"
     },
     "1.5": {
      "b_h": -260,
      "1e5_Bt": "-258.7586",
      "s_h": 318,
      "1e5_S": "319.2828"
     },
     "1.54": {
      "b_h": -193,
      "1e5_Bt": "-191.3222",
      "s_h": 293,
      "1e5_S": "294.3773"
     },
     "1.58": {
      "b_h": -76,
      "1e5_Bt": "-74.6539",
      "s_h": 271,
      "1e5_S": "272.2961"
     },
     "1.62": {
      "b_h": 86,
      "1e5_Bt": "87.8095",
      "s_h": 251,
      "1e5_S": "252.6503"
     },
     "1.7": {
      "b_h": 536,
      "1e5_Bt": "537.9097",
      "s_h": null,
      "1e5_S": null
     },
     "1.9": {
      "b_h": 2275,
      "1e5_Bt": "2276.3388",
      "s_h": null,
      "1e5_S": null
     },
     "2.25": {
      "b_h": 6853,
      "1e5_Bt": "6854.7643",
      "s_h": null,
      "1e5_S": null
     }
    }
   },
   "checks": [
    {
     "check": "the proposition as stated (numerics.tex:10-16)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the definitions read (nest:weight, nest:baseline, nest:gain-parameters, nest:gain-coefficient, nest:gain-definition)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "Table num:knot-table read: 24 knots from .64 to 2.25 with 16 s-entries; Table num:gap-table: 23 intervals",
     "pass": true,
     "detail": "24 knots, 23 gaps"
    },
    {
     "check": "pi enclosure (Machin, alternating brackets, here): the printed pi_- < pi < pi_+ holds",
     "pass": true,
     "detail": "pi in [3.14159265358979, 3.14159265358979]"
    },
    {
     "check": "the paper's rational comparisons for pi (A_31(1/5), A_9(1/239), e_N) hold exactly (:370-373)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "g(D) = 7/25 = ell and int_0^D W = 1 - ell = .72 (sqrt(1 - 48/49) = 1/7 exactly)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "W = -g' = X(.96 - .98X^2)/(1-X^2)^(3/2) (numerator identity, a cubic checked at 7 points)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. B > 0 on [0.64, 1.02], decided here by monotone bounds on an adaptive partition",
     "pass": true,
     "detail": "35 pieces, smallest certified lower bound 1.288e-05"
    },
    {
     "check": "A. B + S > 0 on [1.02, 1.62], decided here by monotone bounds on an adaptive partition",
     "pass": true,
     "detail": "323 pieces, smallest certified lower bound 2.945e-06"
    },
    {
     "check": "A. B > 0 on [1.62, 2.25], decided here by monotone bounds on an adaptive partition",
     "pass": true,
     "detail": "33 pieces, smallest certified lower bound 3.863e-05"
    },
    {
     "check": "B. no knot is a branch tie (t = b_0 or h b_0 = D): every branch decided by disjoint enclosures (:428-429)",
     "pass": true,
     "detail": "1.46:hb0>=D, 1.5:t<b0, 1.54:t<b0, 1.58:t<b0, 1.62:t<b0, 1.7:t<b0, 1.9:t<b0, 2.25:t<b0"
    },
    {
     "check": "B. 10^5 Bt(h) > b_h at all 24 knots (Table num:knot-table)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. stated slack 1.10 < 10^5 Bt(h) - b_h < 1.95 at all 24 knots (num:table-slack)",
     "pass": true,
     "detail": "range of 10^5 Bt - b_h: [1.1043, 1.9446]"
    },
    {
     "check": "B. 10^5 S(h) > s_h at all 16 interior knots",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. stated slack 1.00 < 10^5 S(h) - s_h < 1.66 at all 16 knots (num:table-slack)",
     "pass": true,
     "detail": "range of 10^5 S - s_h: [1.0033, 1.6503]"
    },
    {
     "check": "B. -273.170 < 10^5 Bt(1.46) < -273.169 (:473), and Bt(1.46) = B(1.46) (h b_0 > D: no Taylor term)",
     "pass": true,
     "detail": "10^5 Bt in [-273.169483, -273.169483]"
    },
    {
     "check": "B. 10^5 S(1.46) = 333.396 exactly, by the cap term (P >= d_m and the uncapped minimum exceeds C_0 d_m^3) (:474-477)",
     "pass": true,
     "detail": "10^5 C_0 d_m^3 = 83349/250"
    },
    {
     "check": "B. at every knot the rigorous B (and B + S inside) is positive",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Lemma num:taylor-error holds at the 24 knots: B >= Bt - .008 (D/h - b_0)_+^4 (B enclosed independently)",
     "pass": true,
     "detail": "smallest slack 4.091e-10; grids refined at 1.42: step 1/8000"
    },
    {
     "check": "B. 49 D_+^2 > 48, so D < D_+ (:498)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. 2/(1.02)^2 + .4 < 2.4 (:490)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Table num:gap-table: the 23 intervals are the consecutive knots, and every printed integer <= 10^6 M_{a,b} from (num:gap-formula)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. the smallest margin is exactly 9/100000, on [1.42,1.46] and [1.46,1.50]; at 1.46 the corrected bound is .00057 and the loss .00048 (:547-550)",
     "pass": true,
     "detail": "min 9/100000 on [1.42,1.46], [1.46,1.5]"
    },
    {
     "check": "C. num:s-second: .64 < t < .8 on [1.02,1.62] and t(1.62) < .65 (t decreasing)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:s-second: 1792/9350 = 2(.64)^2/((22/7)(2-.64)), 160/471 = 2(.8)^2/(3.14(2-.8)), .19 < 1792/9350 < 160/471 < .34",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:s-second: 1053/1570 = 2(1.62)(.65)/3.14 < .671 and b_0 - 1 + 1053/1570 < .36 (b_0 < 15/22)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:s-second: h_c = .96/b_0 > 1.4; w' > .19 - .36/(2.04) > 0; w'' <= .36/(4(1.02)^2) < .8; -.96/1.4^2 + .19 - .36/2.8 > -.50; 1.92/1.4^3 + .36/(4(1.4)^2) < .8",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:s-second: C_0 <= .36, C_0' = .402h^2 <= 1.08, C_0'' = .804h <= 2.16 for h <= 1.62",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:s-second: the product-rule bound equals .26866728 < .4, and 2.16 d_m^3 < .4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:taylor-error: .68 < b_0 < .69, and 15 sqrt(22/7)(11/14)^2(.69)/(1+(3.14/4)(.68)^2)^(7/2) < 4 (squared: rational)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:taylor-error: 6 sigma b_0^2 >= 1 (|rho''''| decreasing beyond b_0) and .72 * 4 (L/2)^4/24 = .0075 L^4 <= .008 L^4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:arithmetic: j_D^2 = (1 - 2/sqrt7)/2 (m(D)^2 = 3/7 exactly) and (1 - 2/sqrt7)/2 < 1/8 < .36^2 (64 > 63)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. num:layer-cake: rho'(b_0) = 2/(pi+1) (enclosures overlap) and rho(b_0) = b_0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:s-formula: 16/27 - 1 + 21/44 = 83/1188 > 0, and 9/16 - pi/4 < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:coefficient-bound: c d_m = .567; .567(32/37)^2 - 3.14(5/37) = -73/342250; .567(13/15)^2 - (22/7)(2/15) = 3587/525000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:coefficient-bound: M(32/37) = 40448/151959 > .2659, q_m(32/37) = 2041/2923 > .697, p_2(13/15) = 2723587/2500000 < 1.09",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:coefficient-bound: (.91*2.7*.2659)^2 - .574^2 1.09^3 = 14121304169/10^14; .574(.617)^3 = .134824054862; .96 - 1.62*.06363 = .8569194; .574(.856)^3 = .360025437184; 1 - 1/3.14 - .06363 > .617",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:derivative-bound: (.49-n)(1-b^2) - .01 = .0000096, .000395, .000476, .000192; D_1(a)^2 = .20772864, .16467759, .10285104 < .46^2, .41^2, .33^2; n - .91 d = .001, .0214, .0269, .0597",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. nest:shape-gain: 16/27 - .21 = 1033/2700 > .38 > 1/3 > a_0; c d_m^2 = .11907 < .14; 35/24 - 25(.21)^2/16 = 26677/19200 >= 1.35; .525^2 bound -3/20 + w/5 + w^2/25 < 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "consistency: the layer-cake formula (num:layer-cake-formula) equals the direct integration here at all 24 knots (overlap)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "consistency: the paper's moments (num:moments-j) equal the moments derived here at x, y for all knots (overlap)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "consistency: numerics.tex Q = (2/sqrt pi)(1+4h^2/pi)^(1/4) equals nesting.tex Q = (2/pi) sqrt(pi + 2h^2 t) at all knots (overlap)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "the paper says its exact calculations are included with the source (paper.tex:157); the release has no code or data file for them",
     "pass": true,
     "detail": "recorded as an observation; the tables are the checkable certificate"
    }
   ],
   "sources": {
    "preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/build/sections/numerics.tex": "2560ee9a179d718d4c6b5d3899c403ec88eb83e8ebe9704995de11c119df7594",
    "preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/build/sections/nesting.tex": "56162a6f0af399a56599bef12f5007b24975cbb74ce620140f27383ff449afd4",
    "preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/build/paper.tex": "ab75095f542adf4073a41a26a3c3ae97be7b55fa19674ec4fb4c288f7145a813"
   },
   "forges": [
    {
     "forge": "b_h at 1.46 raised from -275 to -272 (above 10^5 Bt = -273.17)",
     "verdict": "REFUTED"
    },
    {
     "forge": "gap entry [1.42,1.46] raised from 90 to 91",
     "verdict": "REFUTED"
    },
    {
     "forge": "s_h at 1.02 raised from 76 to 78",
     "verdict": "REFUTED"
    },
    {
     "forge": "the exterior range claimed up to 1.10 instead of 1.02 (B alone, without the gain S)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 7.1
  },
  {
   "id": "F-688",
   "index": 688,
   "family": "361",
   "title": "A counterexample to integer-degree harmonic dimension comparison",
   "tier": "A",
   "dir": "A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026",
   "leanChallenges": [
    "HarmonicGrowth"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the supplementary exact spectral selection at n = 16, k = 50000 (finite-selection.tex) — every parameter inequality, the eigenvalue data, the multiplicities, every square-root bound and the three printed integers; not the analytic construction of the metric nor the lemmas the eigenvalues and multiplicities come from",
   "decider": "instruments/openaimath/finite/f688_harmonic_selection.py",
   "decides": "a finite component: the supplementary exact spectral selection at n = 16, k = 50000 (finite-selection.tex) — every parameter inequality, the eigenvalue data, the multiplicities, every square-root bound and the three printed integers; not the analytic construction of the metric nor the lemmas the eigenvalues and multiplicities come from",
   "value": {
    "n": 16,
    "k": 50000,
    "selected": "46785605044474340387088811257657817010714687366486217934111",
    "h_k": "46779709349146362239266126538609505511855492134986686636251",
    "surplus": "5895695327978147822684719048311498859195231499531297860",
    "ratio_selected_over_h_k": "1.000126031038",
    "L_full": 49963,
    "first_individual": 49964,
    "L": 50212,
    "stop_degree": 50213,
    "partial_cuts": 233,
    "selection_seconds": 56.7
   },
   "checks": [
    {
     "check": "the paper prints eq:hopf-multiplicity, N_l, eq:angular-operator, eq:link-margin and eq:euclidean-comparator as used here",
     "pass": true,
     "detail": "counting.tex:24-29, links.tex:40-44,71-83, introduction.tex:13-17, geometry.tex:118-122, finite-selection.tex:22-24,44-53,70-71"
    },
    {
     "check": "A. m = 2s - 1 odd with s >= 4; n = m + 1 = 16 even >= 8; k >= 2",
     "pass": true,
     "detail": "m=15 s=8 n=16 k=50000"
    },
    {
     "check": "A. alpha_*^2 = 1 - c(q_* - 1) (eq:dwell-parameters) and c > 2/(m-1)",
     "pass": true,
     "detail": "c - 2/(m-1) = 29/7000"
    },
    {
     "check": "A. the printed endpoint margin 1 - 2(q_*-1)/(m-1) - alpha_*^2 = 29/700000 > 0",
     "pass": true,
     "detail": "29/700000"
    },
    {
     "check": "A. F'(q) = (m+1)/(m(m-1)) q^(1/m-1)(1-2q) (coefficients compared), negative for q >= 1: F decreases on [1, q_*]",
     "pass": true,
     "detail": "so a_*^2 = q_*^(1/m) alpha_*^2 < F(q_*) <= F(q) on [1, q_*] by the margin; the min in eq:link-margin is the horizontal term since 1 - 2(q-1)/(m-1) <= 1 <= q"
    },
    {
     "check": "A. 101 * 99853^15 < 100 * 100000^15, i.e. q_*(alpha_*^2)^15 < 1, so a_*^30 < 1 and a_* < 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. 0 < -log a_* <= -(1/2) log alpha_*^2 < (1 - alpha_*^2)/(2 alpha_*^2) = 147/199706 < 1/1000 (rational series enclosures)",
     "pass": true,
     "detail": "-log a_* in [4.038630599e-04, 4.038630599e-04]; width 4.2e-19"
    },
    {
     "check": "A. int_4^oo (1+t)^(-5/4) dt = 4*5^(-1/4) > 2: (4*5^(-1/4))^4 = 256/5 > 2^4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. mu = -log a_* / I_chi < (1/1000)/2 = 1/2000, so b >= -mu > -1/4 and eq:cutoff-smallness holds (sup chi (1+t)^(-5/4) <= 1)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. num/den = alpha_*^-2 [l(l+m-1) - (1 - 1/q_*)(2b-l)^2] as polynomials in (l, b) over Q (eq:angular-operator at a_*^2 = q_*^(1/m) alpha_*^2)",
     "pass": true,
     "detail": "num = 100000(101 l(l+14) - (2b-l)^2), den = 10085153"
    },
    {
     "check": "C. sum_{l=0}^k N_l = C(m+k,k) + C(m+k-1,k-1) = C(n+k-1,k) + C(n+k-2,k-1) = the printed h_k(R^16)",
     "pass": true,
     "detail": "h_k = 46779709349146362239266126538609505511855492134986686636251 (59 digits); direct sum 0.1s"
    },
    {
     "check": "C. eq:hopf-multiplicity: line 1 = line 2 and sum_b h_{l,b} = N_l for every l <= 200 and every b",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. N_l = C(m+l,m) - C(m+l-2,m) = (l+s-1)/(s-1) C(l+2s-3,2s-3) for every l <= 50213",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. full degrees: 100000 l(l+14) < 99853 k(k+14) exactly for 2 <= l <= 49963 and fails at 49964 (all their exponents are < k)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. individual degrees 49964..50212: every S verified (S^2 den > Q^2(h^2 den + 4 num)), eigenvalues increasing and bounds nondecreasing in b, line 1 = line 2 and sum_b = N_l in every degree, every group mean bound < k",
     "pass": true,
     "detail": "2490502 bounds S, 6236143 (l,b) multiplicities in both lines; 233 degrees cut inside a repeated eigenvalue, 16 taken whole; no failures"
    },
    {
     "check": "C. the search stops at degree 50213, whose smallest eigenvalue is >= k(k+14) exactly (smallest exponent >= k); the last selected degree is L = 50212",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the selected dimension sum_{l=2}^L M_l equals the printed integer",
     "pass": true,
     "detail": "46785605044474340387088811257657817010714687366486217934111 vs printed 46785605044474340387088811257657817010714687366486217934111"
    },
    {
     "check": "C. the printed surplus = selected - h_k, and it is > 0 (eq:selection: sum M_l > sum_{l=0}^k N_l)",
     "pass": true,
     "detail": "recomputed surplus 5895695327978147822684719048311498859195231499531297860"
    }
   ],
   "sources": {
    "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/build/sections/finite-selection.tex": "1bf1945991e9398d15e9e77b2012943c3300efb06daac4b893e992d3aad4cb6e",
    "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/build/sections/counting.tex": "f567f723c45fb15b1925e86a7e28e056df8721eb79c61c17c2fdfe6c02d810d9",
    "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/build/sections/links.tex": "3f09cfc35ba1397cb3ecaacaeece0da797ff40df994082bdfb65a61019c41be4",
    "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/build/sections/introduction.tex": "5d49b4aeae4a849dce5872855459bd3bbee4dd3e4ee8e4ae6a56a5f0ba1858d2",
    "preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/build/sections/geometry.tex": "4d8226bfb401df00dce0934496fcf989489ed3cf455dfe0d731562d4501a858d"
   },
   "forges": [
    {
     "forge": "selected dimension and surplus both printed one larger",
     "verdict": "REFUTED"
    },
    {
     "forge": "alpha_*^2 = 99854/100000, printed data kept (the dwell relation, the 29/700000 margin and the eigenvalue identity break; the count itself does not move)",
     "verdict": "REFUTED"
    },
    {
     "forge": "S = isqrt(...) without the +1 (not a strict upper bound for sqrt)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 57.2
  },
  {
   "id": "F-699",
   "index": 699,
   "family": "367",
   "title": "The critical dimension for one-phase Bernoulli minimizers",
   "tier": "A",
   "dir": "The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026",
   "leanChallenges": [
    "BernoulliNonflatInSeven"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: the exact arithmetic of Proposition cert:universal (the universal algebraic certificate behind the flatness half, d <= 6), from Tables I and II: the interior expansion with its 1714 keys and ten printed D_0 sums, D_0 <= Dbar and the slacks giving eps_*, and the boundary expansion with its eleven coefficients giving delta_*; plus the PSD of the four projection operators. Not the graph bounds L_t (Lemmas cert:tensor-bounds, cert:graph-bound), not the geometric reduction, cutoffs or regularity theory, and not the dimension-7 example, which is cited (De Silva-Jerison 2009)",
   "decider": "instruments/openaimath/finite/f699_bernoulli_critical_dimension.py",
   "decides": "a finite component: the exact arithmetic of Proposition cert:universal (the universal algebraic certificate behind the flatness half, d <= 6), from Tables I and II: the interior expansion with its 1714 keys and ten printed D_0 sums, D_0 <= Dbar and the slacks giving eps_*, and the boundary expansion with its eleven coefficients giving delta_*; plus the PSD of the four projection operators. Not the graph bounds L_t (Lemmas cert:tensor-bounds, cert:graph-bound), not the geometric reduction, cutoffs or regularity theory, and not the dimension-7 example, which is cited (De Silva-Jerison 2009)",
   "value": {
    "keys": 1714,
    "D0_sums": {
     "11": 321268243322644,
     "12": 96616799870252,
     "13": 294707669526536,
     "14": 385792585960684,
     "22": 269583307398596,
     "23": 115717471031956,
     "24": 125155603107368,
     "33": 605509501361296,
     "34": 618424270356936,
     "44": 1004504025266456
    },
    "Dbar_margins": {
     "11": 106756677356,
     "12": 8200129748,
     "13": 42330473464,
     "14": 82414039316,
     "22": 41692601404,
     "23": 32528968044,
     "24": 94396892632,
     "33": 115498638704,
     "34": 75729643064,
     "44": 120974733544
    },
    "slacks": [
     "673/18400",
     "438907/12000000",
     "2844869/24900000",
     "3479389/7600000"
    ],
    "boundary_Q2_coefficients": {
     "0004": -13479110,
     "0005": 17712712,
     "0013": -93201124,
     "0014": 151038620,
     "0022": -78543848,
     "0023": 244636976,
     "0112": -139986162,
     "0113": 310091962,
     "0122": 445177656,
     "1111": -4861942,
     "1112": 157995170
    },
    "delta_star": "6984137359/31250000000",
    "seconds": 6.6
   },
   "checks": [
    {
     "check": "the theorem as stated (paper.tex Theorem thm:main)",
     "pass": true,
     "detail": "paper.tex lines 73-79"
    },
    {
     "check": "the dimension-7 half is cited, not constructed: De Silva-Jerison 2009, Theorem 1.1",
     "pass": true,
     "detail": "consequences.tex lines 22-23; history.tex lines 35-37"
    },
    {
     "check": "the printed constants (cert:constants): a = 29/50, Q = 250000, eps_* = 438907/12000000, delta_* = 6984137359/31250000000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "A. Table I: 231 rows, (f, U, K, M, P) = (34, 57, 34, 74, 32)",
     "pass": true,
     "detail": "231 rows, [34, 57, 34, 74, 32]"
    },
    {
     "check": "A. Table I: every free slot once, no X/l/w/z, every P row has g power > 0, U and K rows have an even number of p endpoints (the parity claims)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "A. Table II: 21 blocks; 19 interior with 529 rows; 2 boundary blocks of 14 rows and 5 columns; 557 rows",
     "pass": true,
     "detail": "21 blocks, 529 interior rows"
    },
    {
     "check": "A. every interior Table II row has degree one in (w, z, l, X); each block has one column count",
     "pass": true,
     "detail": ""
    },
    {
     "check": "B. Pi(anti, r=2) is symmetric with Pi^2 = 2 Pi on (R^5)^(x)2, so PSD",
     "pass": true,
     "detail": "image dimension tr(Pi)/c = 10"
    },
    {
     "check": "B. Pi(sym, r=2) is symmetric with Pi^2 = 10 Pi on (R^5)^(x)2, so PSD",
     "pass": true,
     "detail": "image dimension tr(Pi)/c = 14"
    },
    {
     "check": "B. Pi(sym, r=3) is symmetric with Pi^2 = 42 Pi on (R^5)^(x)3, so PSD",
     "pass": true,
     "detail": "image dimension tr(Pi)/c = 30"
    },
    {
     "check": "B. Pi(hook, r=3) is symmetric with Pi^2 = 48 Pi on (R^5)^(x)3, so PSD",
     "pass": true,
     "detail": "image dimension tr(Pi)/c = 35"
    },
    {
     "check": "B. the arithmetic of Lemma cert:tensor-bounds: |5-2j|^2/(5j(5-j)) is 9/20 (j = 1, 4) and 1/30 (j = 2, 3); 1/3 + (10/21) theta = 5/7; alpha^2 <= 4(1 - alpha^2) gives theta = 4/5",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the single row U = A contributes 5z^2 + |X|^2 + (2a-2)|q|^2 to I (certificate-proof.tex line 296)",
     "pass": true,
     "detail": "3 keys"
    },
    {
     "check": "C. every key of E has degree two in (w, z, l, X) and every Q^2 e_t is an integer (Lemma cert:closure; line 473)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the root rounding: n_t is the least integer with n_t^2 >= 10^10 L_t^2, for every key",
     "pass": true,
     "detail": ""
    },
    {
     "check": "C. the number of nonzero keys of E is the printed 1714",
     "pass": true,
     "detail": "1714"
    },
    {
     "check": "C. entry (1, 1): 89 keys and 2 10^5 Q^2 (D_0)_ij = 321268243322644 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 89 keys, 321268243322644"
    },
    {
     "check": "C. entry (1, 2): 53 keys and 2 10^5 Q^2 (D_0)_ij = 96616799870252 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 53 keys, 96616799870252"
    },
    {
     "check": "C. entry (1, 3): 123 keys and 2 10^5 Q^2 (D_0)_ij = 294707669526536 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 123 keys, 294707669526536"
    },
    {
     "check": "C. entry (1, 4): 170 keys and 2 10^5 Q^2 (D_0)_ij = 385792585960684 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 170 keys, 385792585960684"
    },
    {
     "check": "C. entry (2, 2): 64 keys and 2 10^5 Q^2 (D_0)_ij = 269583307398596 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 64 keys, 269583307398596"
    },
    {
     "check": "C. entry (2, 3): 85 keys and 2 10^5 Q^2 (D_0)_ij = 115717471031956 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 85 keys, 115717471031956"
    },
    {
     "check": "C. entry (2, 4): 106 keys and 2 10^5 Q^2 (D_0)_ij = 125155603107368 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 106 keys, 125155603107368"
    },
    {
     "check": "C. entry (3, 3): 238 keys and 2 10^5 Q^2 (D_0)_ij = 605509501361296 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 238 keys, 605509501361296"
    },
    {
     "check": "C. entry (3, 4): 414 keys and 2 10^5 Q^2 (D_0)_ij = 618424270356936 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 414 keys, 618424270356936"
    },
    {
     "check": "C. entry (4, 4): 372 keys and 2 10^5 Q^2 (D_0)_ij = 1004504025266456 as printed (cert:exact-interior-sums)",
     "pass": true,
     "detail": "recomputed 372 keys, 1004504025266456"
    },
    {
     "check": "C. D_0 <= Dbar entrywise (cert:upper-matrix), from the recomputed sums",
     "pass": true,
     "detail": "least margin 8200129748 at (1, 2)"
    },
    {
     "check": "C. slack 1: d_1 - (Dbar nu)_1 / nu_1 = 673/18400 as printed (cert:slacks)",
     "pass": true,
     "detail": "673/18400"
    },
    {
     "check": "C. slack 2: d_2 - (Dbar nu)_2 / nu_2 = 438907/12000000 as printed (cert:slacks)",
     "pass": true,
     "detail": "438907/12000000"
    },
    {
     "check": "C. slack 3: d_3 - (Dbar nu)_3 / nu_3 = 2844869/24900000 as printed (cert:slacks)",
     "pass": true,
     "detail": "2844869/24900000"
    },
    {
     "check": "C. slack 4: d_4 - (Dbar nu)_4 / nu_4 = 3479389/7600000 as printed (cert:slacks)",
     "pass": true,
     "detail": "3479389/7600000"
    },
    {
     "check": "C. all four slacks are positive, the z^2 slack is eps_* and the |l|^2 slack is >= eps_*",
     "pass": true,
     "detail": "['673/18400', '438907/12000000', '2844869/24900000', '3479389/7600000']"
    },
    {
     "check": "C. evidence (mod 2^61-1, random point 1 on the variety): the graph expansions of I and Gamma equal I and Gamma evaluated directly from the formulas (explicit tensors, D_l by dual numbers)",
     "pass": true,
     "detail": "I 408522513233024923 / 408522513233024923, Gamma 1715481412031208960 / 1715481412031208960"
    },
    {
     "check": "C. evidence (mod 2^61-1, random point 2 on the variety): the graph expansions of I and Gamma equal I and Gamma evaluated directly from the formulas (explicit tensors, D_l by dual numbers)",
     "pass": true,
     "detail": "I 717189234043394387 / 717189234043394387, Gamma 809454268870954873 / 809454268870954873"
    },
    {
     "check": "D. P = 0 at g = 0 (every P row has a positive g power)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. S, Lambda and E_b have degree at most five (certificate-proof.tex lines 606-613)",
     "pass": true,
     "detail": "deg S 5, deg Lambda 5"
    },
    {
     "check": "D. S summed over all indices equals the component formula (cert:boundary-components)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the single row U = A gives S = lambda R + lambda - tr A^3 = 2 lambda - tr A^3 on R = 1 (line 298)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "D. the homogenized E_b is symmetric in x1..x4 and has exactly the eleven printed monomial types",
     "pass": true,
     "detail": "11 nonzero m_beta"
    },
    {
     "check": "D. Q^2 e_(0, 0, 0, 4) = -13479110 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "-13479110"
    },
    {
     "check": "D. Q^2 e_(0, 0, 0, 5) = 17712712 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "17712712"
    },
    {
     "check": "D. Q^2 e_(0, 0, 1, 3) = -93201124 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "-93201124"
    },
    {
     "check": "D. Q^2 e_(0, 0, 1, 4) = 151038620 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "151038620"
    },
    {
     "check": "D. Q^2 e_(0, 0, 2, 2) = -78543848 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "-78543848"
    },
    {
     "check": "D. Q^2 e_(0, 0, 2, 3) = 244636976 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "244636976"
    },
    {
     "check": "D. Q^2 e_(0, 1, 1, 2) = -139986162 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "-139986162"
    },
    {
     "check": "D. Q^2 e_(0, 1, 1, 3) = 310091962 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "310091962"
    },
    {
     "check": "D. Q^2 e_(0, 1, 2, 2) = 445177656 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "445177656"
    },
    {
     "check": "D. Q^2 e_(1, 1, 1, 1) = -4861942 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "-4861942"
    },
    {
     "check": "D. Q^2 e_(1, 1, 1, 2) = 157995170 as printed (cert:eleven-coefficients)",
     "pass": true,
     "detail": "157995170"
    },
    {
     "check": "D. Q^2 sum |e_beta| = 1656725282 as printed",
     "pass": true,
     "detail": "1656725282"
    },
    {
     "check": "D. 1/4 - Q^-2 sum|e_beta| = delta_* (printed line 669-670, Q^2 = 62500000000)",
     "pass": true,
     "detail": "6984137359/31250000000"
    }
   ],
   "sources": {
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/paper.tex": "13195acd1719a8a56dc56c5684a9a60e12ab88034696cd6011711f6e42151b81",
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/sections/certificate.tex": "09168b55dea1e04ba7989a6f6b0107011934d9cecc10213e3b7332e6776e797c",
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/sections/certificate-proof.tex": "f8a7473a63bce4567b5a73f957989dd075eaf0ff7fa99a4bde4d8a9781e69217",
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/appendices/tables.tex": "6b47116a8e5e4398b39abe8c311872b94365e518c510340ee8ae2a266bd6c701",
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/sections/consequences.tex": "155294701915742a32f44133f5018948f6792ab7dcb9f75ca17b1af745031e7f",
    "preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/build/sections/history.tex": "35d08b80739cd3535f3a2150fd36fbcb42ad7481fb1e7b6d4625de4379ad9308"
   },
   "forges": [
    {
     "forge": "Table I: the constant term of f moved by one unit of 1/Q (172575762 -> 172575763)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Dbar_44 printed one unit lower (8037 -> 8036)",
     "verdict": "REFUTED"
    },
    {
     "forge": "Table II: one boundary (H block) coefficient moved by one unit (2926888 -> 2926889)",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 6.6
  },
  {
   "id": "F-704",
   "index": 704,
   "family": "371",
   "title": "Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus",
   "tier": "A",
   "dir": "Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026",
   "leanChallenges": [
    "DefocusingNLS"
   ],
   "word": "CERTIFIED",
   "why": "a finite component: every finite comparison of Appendix cert:arithmetic (integer box, boundary-form positivity, winding sign table and margins, high-angular polynomial, profile enclosures and degree comparisons, exterior positivity), and the low-angular counts 2,1,0,0 for P_l by Routh-Hurwitz; NOT Proposition free:main itself nor the blowup theorem, which rest on the analytic lemmas of Section sec:free",
   "decider": "instruments/openaimath/finite/f704_nls_blowup.py",
   "decides": "a finite component: every finite comparison of Appendix cert:arithmetic (integer box, boundary-form positivity, winding sign table and margins, high-angular polynomial, profile enclosures and degree comparisons, exterior positivity), and the low-angular counts 2,1,0,0 for P_l by Routh-Hurwitz; NOT Proposition free:main itself nor the blowup theorem, which rest on the analytic lemmas of Section sec:free",
   "value": {
    "boundary_minima": {
     "0": [
      63,
      60,
      94
     ],
     "1": [
      21,
      178,
      199
     ],
     "2": [
      4,
      13,
      323
     ],
     "3": [
      1115,
      173,
      298
     ]
    },
    "separator_margins": [
     583,
     432,
     80,
     68
    ],
    "routh_right_half_plane": [
     2,
     1,
     0,
     0
    ],
    "gamma": "15489394593775948843693173/25000000000000000000000000",
    "e_jet+e_tail": "6443280219857887/374999849998125000000000000",
    "exterior_minima": [
     288247,
     17080,
     31628
    ]
   },
   "checks": [
    {
     "check": "the numbers read from the LaTeX agree with this decider's transcription of them (a parse check)",
     "pass": true,
     "detail": "[]"
    },
    {
     "check": "the disk as printed (free.tex:42-44) and the integer box (certificates.tex:40-41)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "1. |H0 b_* - B| + H0 delta < 2 and |H0 Z_* - S| + H0 delta < 2 (certificates.tex:44-46)",
     "pass": true,
     "detail": "282191/250000, 646827/500000"
    },
    {
     "check": "2. cert:boundary-table, l = 0: minima of floor(Re X_j/(1+e_j)) over (0,1,1), (1,10,1), (10,1,0)",
     "pass": true,
     "detail": "computed (63, 60, 94), printed (63, 60, 94)"
    },
    {
     "check": "2. cert:boundary-table, l = 1: minima of floor(Re X_j/(1+e_j)) over (0,1,1), (1,10,1), (10,1,0)",
     "pass": true,
     "detail": "computed (21, 178, 199), printed (21, 178, 199)"
    },
    {
     "check": "2. cert:boundary-table, l = 2: minima of floor(Re X_j/(1+e_j)) over (0,1,1), (1,10,1), (10,1,0)",
     "pass": true,
     "detail": "computed (4, 13, 323), printed (4, 13, 323)"
    },
    {
     "check": "2. cert:boundary-table, l = 3: minima of floor(Re X_j/(1+e_j)) over (0,1,1), (1,10,1), (10,1,0)",
     "pass": true,
     "detail": "computed (1115, 173, 298), printed (1115, 173, 298)"
    },
    {
     "check": "2. every transformed coefficient enclosure is strictly positive (Re X_j > e_j), all l and substitutions",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. consistency check: at (B/H0, S/H0) without errors, det S is real, even and of degree <= 30 (certificates.tex:91-92)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "2. the three maps (U + V w)/(1 + J w), w >= 0, cover [0,1), [1,10), [10, inf): w/(1+w), (1+10w)/(1+w), 10+w",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 0: deg P_l = 15 with p_15 = -8 S H0^15 and p_16 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 0: R_l and I_l are nonzero of degree <= 7 with 5 + 2 and 6 + 1 sign-change brackets = 7 each",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 1: deg P_l = 15 with p_15 = -8 S H0^15 and p_16 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 1: R_l and I_l are nonzero of degree <= 7 with 6 + 1 and 6 + 1 sign-change brackets = 7 each",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 2: deg P_l = 15 with p_15 = -8 S H0^15 and p_16 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 2: R_l and I_l are nonzero of degree <= 7 with 7 + 0 and 7 + 0 sign-change brackets = 7 each",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 3: deg P_l = 15 with p_15 = -8 S H0^15 and p_16 = 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. l = 3: R_l and I_l are nonzero of degree <= 7 with 7 + 0 and 7 + 0 sign-change brackets = 7 each",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the printed sign table: the cycle ++, -+, --, +- from the printed starting signs at every listed q (endpoints 0, 7500), and ++, -- at w = -(1/2)^2, -(2)^2 for l = 0, 1",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. cert:separator-margins: the row minima of floor(1e4 e sum d_j / sum |d_j|) equal 583, 432, 80, 68",
     "pass": true,
     "detail": "computed [583, 432, 80, 68]"
    },
    {
     "check": "3. positive brackets of R_l: 5,6,7,7; of I_l: 6,6,7,7 (certificates.tex:184-185)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. the root-count table of Lemma free:winding (free.tex:563-566): (pos R, pos I, neg R, neg I) per l",
     "pass": true,
     "detail": "[(5, 6, 2, 1), (6, 6, 1, 1), (7, 7, 0, 0), (7, 7, 0, 0)]"
    },
    {
     "check": "3. the winding conversion N_right = (15 - (2n+1))/2 with n = 5, 6, 7, 7 gives 2, 1, 0, 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "3. SECOND WAY: Routh-Hurwitz on P_l(X), X = lambda + 1/32: zeros with Re X > 0 are 2, 1, 0, 0 and none lie on the imaginary axis (regular array)",
     "pass": true,
     "detail": "[2, 1, 0, 0]"
    },
    {
     "check": "4. cos^2 + sin^2 = 1 at theta_0; sin 2t0 < .4, cos 2t0 > .92, cos t0 > .98 (certificates.tex:199-200)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. N <= .1(.4) - 2.70(.92) = -2.444; L' > .98 - 4(-2.444) > 10.7; -3 - 1/32 + .3(10.7) = 143/800 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. at the join (u = 10/101, v = Z + 1/101): 1.13 < L < 1.19 on the disk; -1.5 - .3(1.19) + 2.444 = 587/1000",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. on the arc (free.tex:394-397): u <= sin(t0)/2 = 10/101 <= .1, 2.70 < Z <= v <= Z + 1/101 < 2.72 on the disk, and 3 - .3^2 > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. on the ray (free.tex:410-411): .099 < u(eta_0) = 10/101 < .100, so u_0 = .099 + .98t <= u <= .100 + .981t",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. ray slopes 99/101 in (.98, .981), 20/101 < .20; M/2 - Z >= 9/2 - Z_* - delta > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. P(t) of free:high-polynomial has degree 6 and P - Q has the seven printed coefficients (cert:high-difference)",
     "pass": true,
     "detail": "[Fraction(142518401, 500000000), Fraction(325932989, 1000000000), Fraction(15777215591, 200000000000), Fraction(8728703679, 200000000000), Fraction(34209914771, 800000000000), Fraction(16113799, 3125000000), Fraction(60018849, 40000000000)]"
    },
    {
     "check": "4. the seven coefficients of P - Q are strictly positive",
     "pass": true,
     "detail": ""
    },
    {
     "check": "4. discriminants (72.2)^2 - 4(33)(44.8) = -17519/25 and (4.61)^2 - 4(17.9)(.54) = -174119/10000, leading coefficients > 0",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. 9 < |m_0| < 10",
     "pass": true,
     "detail": "|m0|^2 = 86.904025"
    },
    {
     "check": "5. U_00/m_0 in (2.835, 2.837) + i(1.426, 1.428)",
     "pass": true,
     "detail": "2.835733 +1.426910i"
    },
    {
     "check": "5. |U_11/m_0| < 1e-12",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. X_11/m_0 in (-.0811, -.0810) + i(.1361, .1363)",
     "pass": true,
     "detail": "-0.081069 +0.136183i"
    },
    {
     "check": "5. Y_11/m_0 in (-.0001, .0001) + i(.1237, .1240)",
     "pass": true,
     "detail": "-0.000000 +0.123759i"
    },
    {
     "check": "5. max |X_ij/m_0|, max |Y_ij/m_0| < 20",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. delta^2 E < 50000 |m_0| delta^2 entrywise (the second-order remainder)",
     "pass": true,
     "detail": "max E = 3.771e+05"
    },
    {
     "check": "5. SECOND WAY: the exact bivariate product reproduces U_0, X_0, Y_0 term by term",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. SECOND WAY: the exact Taylor remainder of L over the box |b - b_*|, |Z - Z_*| <= delta is < 50000 |m_0| delta^2",
     "pass": true,
     "detail": "remainder / (|m_0| delta^2) <= 99.27 (the majorant E gives 4.045e+04)"
    },
    {
     "check": "5. cert:profile-determinant: |det L|^2 <= prod (n^2 + (b_*+d)^2)((n-5)^2 + (b_*+d)^2)/(34!)^4 < (6.2e-12)^2 |m_0|^4",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. Sylvester: |x|^2 > .048^2 and (|x|^2 - k)(|y|^2 - k) > (Re conj(x) y)^2: minimum stretch > .048",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. |1 - i Z_* u/5| > 2.34, |u| < 3.18, |x| < .159, |y| < .124 (cert:profile-extra)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. negative determinant of the linear map: Re x Im y - Im x Re y < 0 (also from the rectangles alone)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. |L(eta, zeta)| <= delta (|x| + |y|) < (.159 + .124) delta = .283 delta (triangle inequality; also .159^2 + .124^2 < .283^2)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "5. gamma > .6 and chi < 5e-7 (free:denominator-gap)",
     "pass": true,
     "detail": "gamma = 15489394593775948843693173/25000000000000000000000000"
    },
    {
     "check": "5. cert:degree-comparisons: e_jet + e_tail < 2.5e-11, .283 delta + 2.5e-11 < 3e-9, .048 delta > 2.5e-11",
     "pass": true,
     "detail": "e_jet + e_tail = 1.71821e-11"
    },
    {
     "check": "5. boundary margin 4.8e-10 - 2.5e-11 = 4.55e-10 (.048 delta = 4.8e-10)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. cert:exterior-table: minima 288247, 17080, 31628 at truncation degrees 10, 10, 8",
     "pass": true,
     "detail": "computed (288247, 17080, 31628)"
    },
    {
     "check": "6. the three truncated polynomials have every coefficient strictly positive",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. consistency: at b = B/H0 without errors, deg d <= 10, Im d = 0, deg c <= 9 and deg Im c <= 8 (cert:exterior-degrees)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "6. 0 < b < .335 on the disk and prod_{n<5} (n^2 + .335^2)((n-5)^2 + .335^2) < 1130^2 (cert:exterior-determinant)",
     "pass": true,
     "detail": "1270789.9"
    },
    {
     "check": "6. all Z in the disk exceed 2.704 (free.tex:708)",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R. RESULTS.json agrees: boundary minima, exterior minima, separator margins, q lists, brackets, starting signs",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R. RESULTS.json agrees: P(t) coefficients, P - Q, Q, discriminants, 143/800, 587/1000, M/2 - Z lower bound",
     "pass": true,
     "detail": ""
    },
    {
     "check": "R. RESULTS.json agrees: gamma, e_jet + e_tail, 2.5e-11, 4.55e-10; the exterior determinant bound and 1130^2",
     "pass": true,
     "detail": ""
    }
   ],
   "sources": {
    "preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/build/sections/certificates.tex": "ec6ceef384ba15779a71db90524db11a00fce09033cc3dc0e66ef2dff09cd504",
    "preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/build/sections/free.tex": "33ab5da74e8fc68fe9bec4f7f4fc87bc3012d22ba3963c6ac6099783adf175d6",
    "preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/verification/certificates/RESULTS.json": "fc628544db75c85ee848896ebbd8eea765c225c73f2e31bcfaf31d0fa0324756"
   },
   "forges": [
    {
     "forge": "the l = 2, (0,1,1) boundary minimum printed as 5",
     "verdict": "REFUTED"
    },
    {
     "forge": "the third P - Q coefficient changed in its last digit",
     "verdict": "REFUTED"
    },
    {
     "forge": "the disk centre b_* moved by 1e-6 (outside the integer box)",
     "verdict": "REFUTED"
    },
    {
     "forge": "the l = 0 sign table printed as starting at ++",
     "verdict": "REFUTED"
    }
   ],
   "seconds": 0.3
  }
 ]
}