Results

145 statements

1. ●Proved
   A cited upper bound Λ ≤ 0.2

   BoundRiemann hypothesisPlatt & Trudgian 2020

2. ●Proved
   A constant determinant with three inputs sharing an output

   CounterexampleJacobian conjectureGao 2026Formalized

3. ●Proved
   A critical point within distance one

   Main statementSendov’s conjectureMazur 2026 · per Tao 2026Formalized

4. ●Proved
   A cubic witness in four real variables

   Supporting resultGaussian moments conjectureLong 2026

5. ●Proved
   A distinct symmetric 112-vertex counterexample

   CounterexamplePetersen coloring conjecturePutman 2026

6. ●Proved
   A double cover has total length twice the edge count

   Supporting resultCycle double cover conjectureOpenAI 2026

7. ●Proved
   A family attaining the parameter-dependent bound

   Related resultČerný synchronization conjectureZhu 2026

8. ●Proved
   A fixed power gain for infinitely many sizes

   CounterexampleErdős unit-distance problemOpenAI 2026Formalized

9. ●Proved
   A formal non-disk failing the Pompeiu property

   CounterexampleSchiffer and Pompeiu conjecturesJaume de Dios Pont & Cao-LaboraFormalized

10. ●Proved
    A fourth-moment lower bound: 1/8

    Related resultFeige’s probability conjectureHe, Zhang & Zhang 2010 · per Fu et al. 2026

11. ●Proved
    A gap bound of 186 from explicit input assumptions

    Conditional resultHow small can prime gaps be?OpenAI & PrimeGaps186 contributors 2026Formalized

12. ●Proved
    A logarithmic bound

    Related resultKannan–Lovász–Simonovits conjectureKlartag 2023 · per Bizeul et al. 2026

13. ●Proved
    A moment and Berry–Esseen lower bound: 0.1798

    Related resultFeige’s probability conjectureGuo, He, Ling & Liu 2020 · per Fu et al. 2026

14. ●Proved
    A poly-logarithmic bound

    Related resultKannan–Lovász–Simonovits conjectureKlartag & Lehec 2022 · per Bizeul et al. 2026

15. ●Proved
    A prime at or above every bound

    Equivalent formulationWhy are there infinitely many primes?mathlib contributorsFormalized

16. ●Proved
    A prime strictly above every bound

    Main statementWhy are there infinitely many primes?OpenMath staff

17. ●Proved
    A proved 0.3465 triangle threshold

    BoundCaccetta–Häggkvist cycle boundHladký et al. 2017

18. ●Proved
    A quartic witness in three real variables

    CounterexampleGaussian moments conjectureLong 2026

19. ●Proved
    A square-root bound

    Related resultKannan–Lovász–Simonovits conjectureLee & Vempala 2024 · per Bizeul et al. 2026

20. ●Proved
    A sub-polynomial bound

    Related resultKannan–Lovász–Simonovits conjectureChen 2021 · per Bizeul et al. 2026

21. ●Proved
    A truly subcubic deterministic APSP algorithm

    Related resultP versus NPAlman & Williams 2026

22. ●Proved
    A truly subquadratic deterministic 3SUM algorithm

    Related resultP versus NPAlman & Williams 2026

23. ●Proved
    Abelian fourfolds · Markman (2025)

    Special caseHodge conjectureMarkman 2025

24. ●Proved
    Airy factorization of the density and curvature

    Supporting resultStrong log-concavity of the Chernoff densityZhang & Zhou 2026

25. ●Proved
    Algebrizing techniques cannot settle P versus NP

    Proof barrierP versus NPAaronson & Wigderson 2008

26. ●Proved
    All codimensions in dimension ≤ 3

    Special caseHodge conjectureClay Mathematics Institute

27. ●Proved
    All powers of Weil fourfolds with discriminant 1

    Special caseHodge conjectureFloccari & Fu 2025

28. ●Proved
    An additive bound for rainbow cycles

    Related resultCaccetta–Häggkvist cycle boundHompe & Huynh 2025

29. ●Proved
    An analytic tenfold counterexample

    CounterexampleSchiffer and Pompeiu conjecturesColbrook & Stepaniants 2026Formalized

30. ●Proved
    An average bound for simplicity and line membership

    Supporting resultRiemann hypothesisLamzouri 2026

31. ●Proved
    An explicit zero-free region near Re(s) = 1

    BoundRiemann hypothesisBellotti et al. 2026

32. ●Proved
    At least 83.625% distinct zeros, asymptotically

    Supporting resultRiemann hypothesisAlpöge & Furman 2026

33. ●Proved
    At least 88.762% simple OR on-line zeros, asymptotically

    Supporting resultRiemann hypothesisLamzouri 2026

34. ●Proved
    Blowup from C^{1,α} data in ℝ³

    Related resultThree-dimensional Euler regularityElgindi 2021 · per OpenAI 2026

35. ●Proved
    Blowup with external forcing

    Related resultThree-dimensional Euler regularityCórdoba & Martínez-Zoroa 2023 · per OpenAI 2026

36. ●Proved
    Bombieri–Vinogradov implies recurring gaps at most 246

    Conditional resultHow small can prime gaps be?Math & PrimeGapsLib contributors 2026Formalized

37. ●Proved
    BSD would complete the odd squarefree congruent-number criterion

    Conditional implicationBirch and Swinnerton-Dyer conjectureper Wiles (official description)

38. ●Proved
    Černý’s examples meet the bound

    Related resultČerný synchronization conjectureČerný 1964 · per Zhu 2026

39. ●Proved
    Characterizing a natural-number prime

    Supporting resultWhy are there infinitely many primes?mathlib contributorsFormalized

40. ●Proved
    Computer-assisted full formula · 16,714 curves

    Special caseBirch and Swinnerton-Dyer conjectureMiller 2010

41. ●Proved
    Conditional transfer through motivated varieties

    Conditional resultHodge conjectureArapura

42. ●Proved
    Decision NP-completeness for three complete ballots

    Main statementKemeny rank aggregation for three votersPeters 2026

43. ●Proved
    Depth-two threshold circuits require superquadratic size

    Restricted modelP versus NPChen et al. 2026

44. ●Proved
    Dimensions where every smooth sphere is standard

    VariantPoincaré conjectureMilnor 2014

45. ●Proved
    Dirichlet GRH implies classical RH

    Conditional implicationRiemann hypothesisAlpöge & Furman 2026

46. ●Proved
    Divisor classes · Lefschetz (1,1)

    Special caseHodge conjectureper Deligne (official description)

47. ●Proved
    Energy bounds for Leray–Hopf solutions

    Supporting resultNavier–Stokes existence and smoothnessKang & Protas 2022

48. ●Proved
    Erdős’s grid lower bound

    Related resultErdős unit-distance problemErdős · per Alon et al. 2026

49. ●Proved
    ETH would separate P and NP

    Conditional implicationP versus NPImpagliazzo & Paturi 2001

50. ●Proved
    Exotic spheres in every other dimension up to 63

    VariantPoincaré conjectureMilnor 2014

51. ●Proved
    Feige’s first universal lower bound: 1/13

    Related resultFeige’s probability conjectureFeige 2004 · per Fu et al. 2026

52. ●Proved
    Fewer possible zeros to the right of a fixed line

    BoundRiemann hypothesisGuth & Maynard 2024

53. ●Proved
    Finite extinction in the simply connected case

    Supporting resultPoincaré conjecturePerelman 2003

54. ●Proved
    Finite-time smooth breakdown requires divergent integrated vorticity

    Conditional implicationThree-dimensional Euler regularityBeale et al. 1984

55. ●Proved
    Fractional cost 58 versus unsplittable cost 60

    CounterexampleCost-preserving unsplittable flowRamos et al. 2026Formalized

56. ●Proved
    From boundary data to vanishing domain averages

    Conditional implicationSchiffer and Pompeiu conjecturesJaume de Dios Pont & Cao-Labora

57. ●Proved
    Full BSD in a specified infinite twist family

    Special caseBirch and Swinnerton-Dyer conjectureBurungale et al. 2024

58. ●Proved
    Gaps at most 240, infinitely often

    BoundHow small can prime gaps be?Stadlmann 2026 · per Charton et al. 2026

59. ●Proved
    Gaps at most 246, infinitely often

    BoundHow small can prime gaps be?Polymath 2014

60. ●Proved
    Gaps at most 600, infinitely often

    BoundHow small can prime gaps be?Maynard 2015

61. ●Proved
    Gaps at most 70,000,000, infinitely often

    BoundHow small can prime gaps be?Zhang 2014

62. ●Proved
    General NP circuit lower bounds would separate P and NP

    Conditional implicationP versus NPWigderson 2019

63. ●Proved
    Global regularity for small initial data

    Special caseNavier–Stokes existence and smoothnessper Fefferman (official description)

64. ●Proved
    Global regularity in two dimensions

    VariantNavier–Stokes existence and smoothnessper Fefferman (official description)

65. ●Proved
    Global weak solutions (Leray)

    Supporting resultNavier–Stokes existence and smoothnessper Fefferman (official description)

66. ●Proved
    Hardness for every fixed voter count at least three

    Related resultKemeny rank aggregation for three votersPeters 2026

67. ●Proved
    Hodge would algebraize inverse Lefschetz maps

    Conditional implicationHodge conjectureper Deligne (official description)

68. ●Proved
    Hodge would make cohomological projectors algebraic

    Conditional implicationHodge conjectureper Deligne (official description)

69. ●Proved
    If P ≠ NP, intermediate problems exist

    Conditional implicationP versus NPWigderson 2019

70. ●Proved
    If P = NP, NP witnesses can be found efficiently

    Conditional implicationP versus NPCook

71. ●Proved
    If P ≠ NP, NP-complete problems have no polynomial algorithm

    Conditional implicationP versus NPCook

72. ●Proved
    If P = NP, the polynomial hierarchy collapses to P

    Conditional implicationP versus NPWigderson 2019

73. ●Proved
    Infinitely many Petersen-coloring counterexamples

    Related resultPetersen coloring conjecturePutman 2026

74. ●Proved
    Infinitely many smooth symmetry orders

    CounterexampleSchiffer and Pompeiu conjecturesCao-Labora & Jaume de Dios Pont 2026

75. ●Proved
    Integral Hodge variant · counterexamples

    VariantHodge conjectureper Deligne (official description)

76. ●Proved
    Low-rank formula at supersingular p ≥ 5

    Special caseBirch and Swinnerton-Dyer conjectureBurungale et al. 2024

77. ●Proved
    More than 67.25% simple zeros on the line, asymptotically

    BoundRiemann hypothesisAlpöge & Furman 2026

78. ●Proved
    Near-maximum general circuit lower bound in an oracle class

    Restricted modelP versus NPRen & Williams 2026

79. ●Proved
    NEXP has no polynomial-size ACC circuits

    Restricted modelP versus NPWilliams 2011

80. ●Proved
    No polynomial inverse for the three-variable witness

    Supporting resultJacobian conjecturealerad/alpoge-lean project 2026Formalized

81. ●Proved
    NP versus coNP would separate P and NP

    Conditional implicationP versus NPWigderson 2019

82. ●Proved
    One-way functions would separate P and NP

    Conditional implicationP versus NPWigderson 2019

83. ●Proved
    Optimization NP-hardness for three voters

    Possible answerKemeny rank aggregation for three votersPeters 2026

84. ●Proved
    Partial bounds for a fixed Dirichlet character

    VariantRiemann hypothesisAlpöge & Furman 2026

85. ●Proved
    Petersen and normal-five colorings are equivalent

    Supporting resultPetersen coloring conjecturePutman 2026

86. ●Proved
    Pointwise and tail curvature bounds

    BoundStrong log-concavity of the Chernoff densityZhang & Zhou 2026

87. ●Proved
    Polynomial NP circuits would collapse the hierarchy

    Conditional implicationP versus NPWigderson 2019

88. ●Proved
    Preserving cost with twice the allowance in planar graphs

    VariantCost-preserving unsplittable flowTraub et al. 2023

89. ●Proved
    Proved analogue: curves over finite fields

    VariantRiemann hypothesisper Bombieri (official description)

90. ●Proved
    Randomized SETH would separate P and NP

    Conditional implicationP versus NPCygan et al. 2014

91. ●Proved
    Rank BSD would decide finite versus infinite rational points analytically

    Conditional implicationBirch and Swinnerton-Dyer conjectureper Wiles (official description)

92. ●Proved
    Rank equality at analytic rank 0 or 1

    Special caseBirch and Swinnerton-Dyer conjectureSkinner 2020

93. ●Proved
    Rank-one converse assuming finite Sha

    Conditional resultBirch and Swinnerton-Dyer conjectureSkinner 2020

94. ●Proved
    Rank-one formula · selected p-parts

    Special caseBirch and Swinnerton-Dyer conjectureJetchev et al. 2015

95. ●Proved
    Rational points form a finitely generated group

    Supporting resultBirch and Swinnerton-Dyer conjectureper Wiles (official description)

96. ●Proved
    Regularity criteria (Ladyzhenskaya–Prodi–Serrin)

    Conditional resultNavier–Stokes existence and smoothnessKang & Protas 2022

97. ●Proved
    Relative extending words on the unique cycle

    Supporting resultČerný synchronization conjectureZhu 2026

98. ●Proved
    Relativizing techniques cannot settle P versus NP

    Proof barrierP versus NPBaker et al. 1975

99. ●Proved
    RH is equivalent to the prime-counting error target

    Supporting resultRiemann hypothesisper Bombieri (official description)

100. ●Proved
     RH is equivalent to Λ = 0

     Supporting resultRiemann hypothesisRodgers & Tao 2018

101. ●Proved
     RH verified to height 3,000,175,332,800

     Special caseRiemann hypothesisPlatt & Trudgian 2020

102. ●Proved
     SAT captures P versus NP

     Equivalent formulationP versus NPCook

103. ●Proved
     Sendov for all sufficiently large degree

     Special caseSendov’s conjectureTao 2022

104. ●Proved
     Sendov for degree at most eight

     Special caseSendov’s conjectureBrown & Xiang 1999 · per Tao 2022

105. ●Proved
     Smooth solutions on a short time interval

     Special caseNavier–Stokes existence and smoothnessper Fefferman (official description)

106. ●Proved
     Smooth-data blowup with a boundary

     Related resultThree-dimensional Euler regularityChen & Hou 2022 · per OpenAI 2026

107. ●Proved
     Smoothing a topological three-manifold

     Supporting resultPoincaré conjectureQin et al. 2026

108. ●Proved
     Strict distance and the equality exception

     VariantSendov’s conjectureMazur 2026 · per Tao 2026Formalized

109. ●Proved
     Strong BSD implies its rank and finiteness assertions

     Conditional implicationBirch and Swinnerton-Dyer conjectureper Wiles (official description)

110. ●Proved
     Strong BSD would determine Sha’s order from the other invariants

     Conditional implicationBirch and Swinnerton-Dyer conjectureper Wiles (official description)

111. ●Proved
     Tate is proved for abelian fourfolds over finite fields

     VariantHodge conjectureBroe 2026

112. ●Proved
     The bound for positive-level one-cluster automata

     Special caseČerný synchronization conjectureZhu 2026

113. ●Proved
     The classical upper bound

     Related resultErdős unit-distance problemSpencer, Szemerédi & Trotter 1984 · per Alon et al. 2026

114. ●Proved
     The critical L³ endpoint (Escauriaza–Seregin–Šverák)

     Conditional resultNavier–Stokes existence and smoothnessTao 2021

115. ●Proved
     The dimension-free unit-slack inequality

     Main statementFeige’s probability conjectureFu et al. 2026

116. ●Proved
     The earlier ordinary log-concavity theorem

     Supporting resultStrong log-concavity of the Chernoff densityBalabdaoui & Wellner 2014

117. ●Proved
     The global implication to polynomial inversion

     Conditional implicationGaussian moments conjectureDerksen et al. 2015

118. ●Proved
     The July bounds in two deviation ranges

     VariantFeige’s probability conjectureFu et al. 2026

119. ●Proved
     The main 112-vertex counterexample

     CounterexamplePetersen coloring conjecturePutman 2026

120. ●Proved
     The natural-proofs barrier is conditional

     Proof barrierP versus NPRazborov & Rudich 1994

121. ●Proved
     The Newman constant satisfies Λ ≥ 0

     BoundRiemann hypothesisRodgers & Tao 2018

122. ●Proved
     The normalized C² formal statement

     Supporting resultSchiffer and Pompeiu conjecturesJaume de Dios Pont & Cao-LaboraFormalized

123. ●Proved
     The older level-zero circular case

     Supporting resultČerný synchronization conjectureZhu 2026

124. ●Proved
     The one-variable case

     Special caseGaussian moments conjectureDerksen et al. 2015

125. ●Proved
     The optimal bound for a fixed number of variables

     BoundFeige’s probability conjectureFu et al. 2026Formalized

126. ●Proved
     The original congestion guarantee

     Supporting resultCost-preserving unsplittable flowTraub et al. 2023

127. ●Proved
     The Poincaré theorem in dimensions five and higher (Smale)

     VariantPoincaré conjectureper Milnor (official description)

128. ●Proved
     The same witness works in every higher dimension

     Conditional implicationGaussian moments conjectureLong 2026

129. ●Proved
     The set of primes is unbounded

     Equivalent formulationWhy are there infinitely many primes?OpenMath staff

130. ●Proved
     The sharp bound for out-degree two through five

     Special caseCaccetta–Häggkvist cycle boundCaccetta–Häggkvist, Hamidoune, Hoàng–Reed · per Blair D. Sullivan

131. ●Proved
     The sharp bound for vertex-transitive graphs

     Special caseCaccetta–Häggkvist cycle boundHamidoune 1981 · per Blair D. Sullivan

132. ●Proved
     The sharp bound when n ≥ 2r² − 3r + 1

     Special caseCaccetta–Häggkvist cycle boundShen 2000 · per Blair D. Sullivan

133. ●Proved
     The singular set is small (Caffarelli–Kohn–Nirenberg)

     BoundNavier–Stokes existence and smoothnessper Fefferman (official description)

134. ●Proved
     The smooth three-dimensional theorem

     Equivalent formulationPoincaré conjectureMorgan & Tian 2007Formalized

135. ●Proved
     The topological four-dimensional theorem

     VariantPoincaré conjectureManolescu 2026

136. ●Proved
     The topological three-dimensional Poincaré theorem

     Main statementPoincaré conjectureClay Mathematics InstituteFormalized

137. ●Proved
     The two-voter case is polynomial-time solvable

     Supporting resultKemeny rank aggregation for three votersPeters 2026

138. ●Proved
     The universal constant is a limit across dimensions

     Conditional implicationFeige’s probability conjectureFu et al. 2026

139. ●Proved
     The unrestricted compact Kähler extension is false

     VariantHodge conjectureper Deligne (official description)

140. ●Proved
     There are infinitely many primes

     Equivalent formulationWhy are there infinitely many primes?OpenMath staff

141. ●Proved
     Transporting the proposed bound beyond isotropic measures

     Conditional implicationKannan–Lovász–Simonovits conjectureBizeul et al. 2026

142. ●Proved
     Unbounded compactly supported velocity prevents smooth continuation

     Conditional implicationNavier–Stokes existence and smoothnessOpenAI 2026

143. ●Proved
     Uniform positive curvature of the Chernoff density

     Main statementStrong log-concavity of the Chernoff densityZhang & Zhou 2026

144. ●Proved
     Why the radius one is sharp

     Supporting resultSendov’s conjectureTao 2026

145. ●Proved
     Why the rainbow conjecture would imply the directed bound

     Conditional implicationCaccetta–Häggkvist cycle boundHompe & Huynh 2025