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AI and open problems: claimed vs checked

Since July, AI systems have been credited with results on problems that stood for decades. Most claims still await review. Here is each one, how it was checked, and by whom.

The AI-assisted mathematics results are also a dataset on GitHub (CSV and JSON, CC BY 4.0), updated twice a day. OpenAI's 721-manuscript release, family by family.

194claims since July 2026

29confirmed

164awaiting review

1disputed

Median 52 years open. 8,338 problem-years in total, from the 151 claims with a known start year.

Every claim, and how it was checked

One row per claim. The line runs from the year a problem was posed to 2026; the pink dot marks the AI claim. Hover over or tap a row for the result and who did what.

All 194 claims, oldest problem first.

  1. Gauss class number problem 225 years OpenAI internal model and 1 more Awaiting review Lean-verified
  2. Quasi-Riemann hypothesis for Dedekind zeta functions of arbitrary number fields 167 years AxiomProver Awaiting review Lean-verified
  3. Quasi-Riemann hypothesis 167 years OpenAI internal model Awaiting review Lean proof, not yet reviewed
  4. Sylvester’s conjecture 147 years ChatGPT and 2 more Awaiting review Preprint only
  5. Hilbert’s sixteenth problem, second part 126 years OpenAI internal model Awaiting review Lean-verified
  6. Carathéodory’s conjecture 102 years Claude and 1 more Awaiting review Preprint only
  7. Artin’s primitive root conjecture 99 years OpenAI internal model Awaiting review Preprint only
  8. Schiffer’s conjecture 97 years GPT-5.6 and 4 more Awaiting review Lean-verified
  9. Tammes problem for 15 points 96 years unnamed AI system Awaiting review Lean-verified
  10. Köthe conjecture 96 years GPT-6 Astra Result confirmed Lean-verified
  11. Hopf’s conjecture 95 years GPT-6 Astra Awaiting review Preprint only
  12. Hopf’s sign conjecture 95 years Odin Automatic AI Research Agent Awaiting review Preprint only
  13. Erdős problems #1, #74, #126 and #571 95 years GPT-6 Astra Result confirmed Lean-verified
  14. Banach’s isometric subspace conjecture 94 years ChatGPT 5.5 Pro and 2 more Awaiting review Preprint only
  15. Borsuk’s conjecture 93 years OpenAI internal model Awaiting review Peer-reviewed
  16. Uniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equations 92 years Claude Opus and 1 more Awaiting review Lean proof, not yet reviewed
  17. Landau–Siegel zeros 91 years OpenAI internal model and 1 more Awaiting review Lean-verified
  18. Bloch constant conjecture 89 years ChatGPT and 1 more Awaiting review Lean proof, not yet reviewed
  19. Collatz conjecture 89 years OpenAI GPT models via Codex and 3 more Result confirmed Lean-verified
  20. Mahler’s conjecture on the minimal volume product |K||K°| of convex bodies, symmetric and general, with equality cases 87 years OpenAI internal model Awaiting review Lean-verified
  21. Jacobian conjecture 87 years Claude Fable 5 Result confirmed Unverified claim
  22. Kaplansky’s zero-divisor conjecture 86 years OpenAI internal model Awaiting review Peer-reviewed
  23. Hilbert–Smith conjecture 85 years OpenAI internal model and 2 more Awaiting review Lean-verified
  24. Maximum number of Hamiltonian cycles in an n-vertex tournament 83 years GPT-5.6 Awaiting review Preprint only
  25. Hadwiger’s conjecture 83 years OpenAI internal model Awaiting review Peer-reviewed
  26. Linear Hadwiger conjecture 83 years GPT-6 Astra and 1 more Awaiting review Lean proof, not yet reviewed
  27. Linnik’s constant 82 years Claude Opus 5.5 and 2 more Awaiting review Lean-verified
  28. Erdős unit distance problem 80 years OpenAI internal model Awaiting review Peer-reviewed
  29. Hopf problem 79 years Claude Awaiting review Lean proof, not yet reviewed
  30. Weil conjectures 77 years Axiom agents Awaiting review Lean-verified
  31. Hodge conjecture (CM abelian varieties) 76 years OpenAI internal model Result confirmed Lean-verified
  32. Hadwiger–Nelson problem 76 years OpenAI internal model Awaiting review Peer-reviewed
  33. Kaplansky’s quasitrace conjecture 75 years Unreleased OpenAI model and 1 more Awaiting review Preprint only
  34. Irrationality exponent of π 73 years OpenAI internal model Awaiting review Lean-verified
  35. Is every group good 73 years OpenAI internal model Awaiting review Preprint only
  36. Existence of four-dimensional interacting quantum field theories 72 years OpenAI internal model Awaiting review Lean-verified
  37. Harary–Hill conjecture on cr(K_n); Zarankiewicz’s crossing-number conjecture for K_m,n 72 years OpenAI internal model and 1 more Awaiting review Peer-reviewed
  38. Linear MDS conjecture 71 years ChatGPT and 2 more Awaiting review Lean-verified
  39. Kadison’s similarity problem 71 years OpenAI internal model Awaiting review Peer-reviewed
  40. Optimal constant in the i.i.d. Berry–Esseen inequality 70 years ChatGPT and 3 more Awaiting review Expert-checked
  41. Pierce–Birkhoff conjecture 70 years GPT-5.6 and 3 more Awaiting review Lean proof, not yet reviewed
  42. Simon’s Problem 2 68 years OpenAI internal model Awaiting review Lean proof, not yet reviewed
  43. Sendov’s conjecture 68 years GPT-5.6 Pro Result confirmed Lean-verified
  44. Nagata’s conjecture on plane curves 67 years OpenAI internal model Result confirmed Preprint only
  45. Serre’s 1960 conjecture 66 years GPT-6 Astra Awaiting review Preprint only
  46. Ulam’s problem 66 years GPT-6 Astra Awaiting review Preprint only
  47. Complexity of optimal prefix-free coding with unequal letter costs 65 years Codex Awaiting review Preprint only
  48. Vaught’s conjecture for L_ω1,ω 65 years Claude Code and 5 more Awaiting review Lean-verified
  49. Erdős–Sós conjecture 64 years GPT-6 Astra Result confirmed Lean-verified
  50. Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞ 63 years GPT-6.1 Sol and 5 more Awaiting review Lean-verified
  51. Formalization of Hironaka’s resolution of singularities in characteristic zero 62 years Claude Fable 5.1 and 1 more Result confirmed Lean-verified
  52. Erdős Matching Conjecture 61 years unnamed generative AI Awaiting review Preprint only
  53. Birch and Swinnerton-Dyer conjecture in rank ≤ 1 61 years OpenAI internal model Awaiting review Preprint only
  54. Lehmer’s permutation conjecture 61 years Claude Opus 5.5 and 1 more Awaiting review Lean-verified
  55. Kinetic Krylov–Safonov theorem and Pucci’s conjecture 60 years GPT-6 Astra Ultra and 2 more Awaiting review Preprint only
  56. Lonely Runner Conjecture 59 years ChatGPT 6 Pro and 1 more Awaiting review Preprint only
  57. Stein’s Fourier restriction conjecture in ℝ³ 59 years OpenAI internal model Awaiting review Preprint only
  58. Grad’s conjecture 59 years GPT-6 Astra Pro and 5 more Result confirmed Lean-verified
  59. Barnette’s conjecture 57 years OpenAI internal model Result confirmed Lean proof, not yet reviewed
  60. Exponent ω of matrix multiplication 57 years OpenAI internal model Result confirmed Lean-verified
  61. Erdős problem #108 57 years ChatGPT and 2 more Result confirmed Lean-verified
  62. Matrix mortality problem in dimension 2 56 years OpenAI Codex and 2 more Awaiting review Lean-verified
  63. Golomb–Welch conjecture 56 years ChatGPT Awaiting review Preprint only
  64. Erdős–Szemerédi sunflower conjecture, all k 56 years ChatGPT 6 Astra Pro Awaiting review Lean-verified
  65. Open Erdős problems 56 years AlphaProof Nexus and 2 more Result confirmed Peer-reviewed
  66. Hilbert’s tenth problem over ℚ 56 years OpenAI internal model Awaiting review Preprint only
  67. Singmaster’s conjecture on multiplicities of binomial coefficients 55 years Claude Opus 5.5 Awaiting review Lean-verified
  68. Chvátal’s conjecture 54 years ChatGPT and 2 more Result confirmed Lean-verified
  69. Hochster’s small Cohen–Macaulay module conjecture 53 years OpenAI internal model and 2 more Awaiting review Lean-verified
  70. Penrose conjecture 53 years Claude Awaiting review Preprint only
  71. Amenability of Thompson’s group F 53 years OpenAI internal model Result confirmed Lean-verified
  72. Cycle double cover conjecture 53 years GPT-5.6 Sol Ultra Result confirmed Expert-checked
  73. Double-cap conjecture / Witsenhausen’s problem on spherical sets without orthogonal pairs 52 years Codex and 1 more Awaiting review Lean-verified
  74. Rauch’s hot spots conjecture, simply connected planar case 52 years OpenAI internal model and 1 more Awaiting review Lean-verified
  75. Strong cosmic censorship near Kerr, in Christodoulou’s formulation 52 years OpenAI internal model Awaiting review Preprint only
  76. Kung–Traub conjecture on the maximal order of multipoint iterations without memory 52 years ChatGPT 6 and 1 more Awaiting review Lean-verified
  77. Murty–Simon conjecture 51 years GPT-5.6 Sol and 1 more Awaiting review Preprint only
  78. Erdős–Lovász problem 51 years unnamed generative AI tools Awaiting review Preprint only
  79. Polyakov’s conjecture 51 years OpenAI internal model Awaiting review Peer-reviewed
  80. Erdős’s conjecture on arithmetic progressions 50 years OpenAI internal model Awaiting review Peer-reviewed
  81. Serre’s cyclicity problem 49 years GPT-6 and 1 more Awaiting review Unverified claim
  82. Erdős–Hajnal for all paths 49 years ChatGPT-6 Astra and 2 more Awaiting review Lean-verified
  83. Asymptotic value of Delsarte’s linear programming bound for binary codes and spherical codes 49 years Claude Opus 5.5 and 3 more Awaiting review Preprint only
  84. Brennan’s conjecture on integrability of |g′|^p for conformal maps; Kraetzer’s t²/4 conjecture for the universal integral-means spectrum 48 years OpenAI internal model Result confirmed Lean-verified
  85. De Giorgi’s conjecture 48 years OpenAI internal model Awaiting review Preprint only
  86. Irrationality of ζ(5) 48 years Claude Opus 5 and 3 more Result confirmed Lean-verified
  87. Abundance conjecture 46 years OpenAI internal model and 1 more Awaiting review Preprint only
  88. Smale’s mean value problem 45 years Claude and 1 more Awaiting review Lean-verified
  89. Smale’s mean value conjecture 45 years GPT-6 Astra and 1 more Result confirmed Lean-verified
  90. Erdős’s question 44 years GPT-6 Pro and 2 more Awaiting review Preprint only
  91. Baum–Connes conjecture without coefficients 44 years OpenAI internal model Awaiting review Preprint only
  92. Yau’s conjecture 44 years OpenAI internal model Awaiting review Peer-reviewed
  93. Classification of Finite Simple Groups 43 years AxiomProver and 1 more Awaiting review Lean-verified
  94. Haldane conjecture / Haldane gap for the spin-one antiferromagnetic Heisenberg chain 43 years OpenAI internal model and 1 more Awaiting review Preprint only
  95. Rigorous critical storage capacity of the Hopfield associative-memory model at zero temperature 41 years Claude Opus 5.5 and 1 more Awaiting review Unverified claim
  96. Falconer’s distance conjecture 41 years OpenAI internal model Awaiting review Lean-verified
  97. Large-data global classical existence for the three-dimensional relativistic Vlasov–Maxwell system 40 years OpenAI internal model Awaiting review Peer-reviewed
  98. Sidorenko’s conjecture 40 years OpenAI internal model Awaiting review Peer-reviewed
  99. Erdős #993: forest independence sequences are unimodal 39 years Claude Opus 5.5 and 5 more Awaiting review Lean-verified
  100. Aggarwal–Vitter 38 years ChatGPT and 1 more Awaiting review Lean-verified
  101. k-server conjecture 38 years GPT-6 Astra and 2 more Awaiting review Preprint only
  102. Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h) 37 years ChatGPT and 1 more Awaiting review Preprint only
  103. Planar Mumford–Shah conjecture 37 years GPT-6 Astra Awaiting review Preprint only
  104. Alon–Seymour–Thomas conjecture 36 years GPT-6 Astra and 3 more Awaiting review Lean-verified
  105. Deligne–Drinfeld conjecture 36 years OpenAI internal model Result confirmed Lean-verified
  106. Delocalization 35 years AxiomProver and 1 more Awaiting review Lean-verified
  107. Laczkovich’s Lipschitz mapping problem 35 years ChatGPT Awaiting review Preprint only
  108. Sogge’s local smoothing conjecture in all dimensions; consequences 35 years OpenAI internal model Awaiting review Lean-verified
  109. Is ℤ a Diophantine subset of ℚ? 34 years GPT-6 Astra and 6 more Awaiting review Lean-verified
  110. Satisfiability conjecture for random k-SAT 34 years OpenAI internal model Awaiting review Peer-reviewed
  111. Odlyzko–Poonen conjecture 33 years GPT-6 Astra Result confirmed Lean-verified
  112. LeBrun–Salamon conjecture 32 years ChatGPT and 1 more Awaiting review Preprint only
  113. Rapid mixing of Glauber dynamics for proper colourings 31 years GPT-6 Astra Pro Awaiting review Preprint only
  114. 3SUM hypothesis 31 years Claude Awaiting review Lean-verified
  115. Kannan–Lovász–Simonovits conjecture 31 years GPT-6 Astra and 4 more Awaiting review Preprint only
  116. Pach–Shahrokhi–Szegedy conjecture 30 years ChatGPT-6 Awaiting review Expert-checked
  117. Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal conjectured O(log k) 30 years GPT-6 Astra and 1 more Awaiting review Lean-verified
  118. Exact overlaps conjecture (dimension one) 30 years GPT-6 Astra Awaiting review Lean-verified
  119. Periodic tiling conjecture 30 years OpenAI internal model Awaiting review Peer-reviewed
  120. Benjamini–Schramm conjecture 30 years OpenAI internal model and 1 more Awaiting review Lean-verified
  121. List total colouring conjecture 29 years GPT-6 Astra Ultra Awaiting review Preprint only
  122. Alon–Krivelevich conjecture 28 years GPT-6 Astra and 2 more Awaiting review Preprint only
  123. Smale’s 6th problem, five bodies in the plane 28 years unnamed generative AI tools Awaiting review Peer-reviewed
  124. Existence of the q-colourability threshold of G(n, d/n) 27 years Claude Opus 5.5 and 3 more Awaiting review Lean-verified
  125. Exponential Time Hypothesis 27 years internal Claude research model Awaiting review Lean-verified
  126. Exponential Time Hypothesis 27 years OpenAI internal model Awaiting review Lean-verified
  127. Reed–Solomon list decoding up to capacity 27 years GPT-5.6 Sol and 2 more Awaiting review Expert-checked
  128. Barry Simon’s multidimensional L² conjecture 26 years ChatGPT Awaiting review Expert-checked
  129. Woodin’s question 26 years GPT-6 Astra and 1 more Awaiting review Preprint only
  130. Navier–Stokes existence and smoothness 26 years OpenAI internal model Disputed Lean-verified
  131. Is there a binary search tree satisfying Iacono’s unified bound? 25 years Claude Opus 5.5 and 4 more Awaiting review Preprint only
  132. Mézard–Parisi formula for the free energy of diluted spin-glass models, including random K-SAT 25 years ChatGPT Astra and 1 more Awaiting review Preprint only
  133. Period-index conjecture 25 years ChatGPT Awaiting review Preprint only
  134. Spectral isometry conjecture 24 years GPT-6 Astra and 1 more Awaiting review Lean-verified
  135. Finite index over treeable (JKL 2002) 24 years ChatGPT 6 Astra Ultra Awaiting review Preprint only
  136. Zilber–Pink conjecture for abelian varieties 24 years OpenAI internal model Awaiting review Preprint only
  137. Exact polynomial-time approximability of metric k-median 24 years unreleased internal OpenAI model Awaiting review Lean-verified
  138. Unique Games Conjecture 24 years OpenAI internal model Awaiting review Lean proof, not yet reviewed
  139. Multiple-unicast network coding conjecture 22 years GPT-6 Awaiting review Preprint only
  140. Crouzeix’s conjecture 22 years GPT-5.6 Sol Result confirmed Expert-checked
  141. Explicit matrices with the restricted isometry property beyond the square-root bottleneck 19 years GPT-6 Astra and 2 more Awaiting review Lean-verified
  142. Area law for entanglement entropy of ground states of gapped local Hamiltonians in two dimensions 19 years OpenAI internal model and 2 more Awaiting review Lean-verified
  143. Matroid secretary conjecture 19 years GPT-6 Astra and 2 more Awaiting review Preprint only
  144. Aaronson–Ambainis conjecture 18 years non-public Anthropic model and 1 more Awaiting review Lean-verified
  145. Gurvits’s conjecture that mixed volumes admit no FPRAS in the oracle model 17 years GPT-6 Astra Awaiting review Preprint only
  146. Toms–Winter conjecture (C*-algebras) 17 years ChatGPT 6 Astra and 1 more Awaiting review Preprint only
  147. Courtade–Kumar conjecture 13 years Gemini and 2 more Awaiting review Lean-verified
  148. Bounded gaps between primes 12 years GPT-6 Astra Awaiting review Lean proof, not yet reviewed
  149. Dimension of the SO(3) instanton homology J♯ of the dodecahedron web 11 years ChatGPT Astra 6 and 2 more Awaiting review Unverified claim
  150. 3-Uniform Hyperclique Hypothesis 8 years ChatGPT Sol 6.1 Awaiting review Preprint only
  151. Benson–Etingof–Ostrik conjecture on symmetric tensor categories of moderate growth in positive characteristic 3 years GPT-6 Astra Awaiting review Expert-checked
  152. Hyperelliptic Schottky problem Start year not recorded unnamed AI tools Awaiting review Preprint only
  153. Talagrand’s Ultimate Matching Conjecture Start year not recorded GPT-6 Awaiting review Preprint only
  154. Rational points on X_ns+(121); the level-121 case of the Rouse–Sutherland–Zureick-Brown conjecture on ℓ-adic images Start year not recorded Claude Opus 5.5 and 3 more Awaiting review Unverified claim
  155. Order of the weak-coupling Hawking–Page/deconfinement transition in N=4 SYM; optimal large-c spinless modular bootstrap bound Start year not recorded internal Claude research model and 1 more Awaiting review Unverified claim
  156. Tusnády’s problem Start year not recorded Claude and 2 more Awaiting review Preprint only
  157. Formalizing the Stacks Project in Lean Start year not recorded Axiom agents Awaiting review Lean-verified
  158. Torsion groups of elliptic curves over number fields of fixed degree d Start year not recorded GPT-6.1 Sol and 2 more Awaiting review Unverified claim
  159. Best proven running time of deterministic integer factoring Start year not recorded Claude Opus 5.5 Awaiting review Lean-verified
  160. Kervaire’s conjecture Start year not recorded OpenAI internal model Result confirmed Lean-verified
  161. Petty’s equilateral-set conjecture Start year not recorded Tencent Hunyuan hyprover Awaiting review Preprint only
  162. Pure semisimplicity conjecture Start year not recorded GPT-6 Astra and 2 more Awaiting review Lean-verified
  163. Coloring 3-colorable graphs with few colors in polynomial time Start year not recorded GPT-6 Astra and 1 more Awaiting review Preprint only
  164. Rigorous continuum limit of three-dimensional Yang–Mills theory Start year not recorded Claude Opus 5.5 and 2 more Awaiting review Lean-verified
  165. Constructing a solution of the two-dimensional KPZ equation at criticality Start year not recorded ChatGPT 5.5 and 2 more Awaiting review Expert-checked
  166. Integer programming with exponent linear in d Start year not recorded GPT-6 Astra Ultra Awaiting review Lean-verified
  167. FPRAS for counting perfect matchings in general graphs Start year not recorded OpenAI internal model Result confirmed Lean-verified
  168. A non-sofic hyperbolic group Start year not recorded GPT Astra and 1 more Awaiting review Lean-verified
  169. Gauss circle problem Start year not recorded ChatGPT and 1 more Awaiting review Preprint only
  170. Existence of the Smith–Toda complex V(4); V(3) exists for p ≥ 7 Start year not recorded OpenAI internal model and 1 more Awaiting review Preprint only
  171. Yang–Mills existence (YM4) Start year not recorded Claude and 2 more Awaiting review Preprint only
  172. Erdős–Simonovits rational exponents conjecture Start year not recorded GPT-6 Astra Result confirmed Lean-verified
  173. How mathematics should respond to AI-driven ‘proof abundance’ after OpenAI’s 722-manuscript release Start year not recorded n/a Result confirmed Unverified claim
  174. Dualization lower bound under ETH Start year not recorded GPT-6 Astra Awaiting review Preprint only
  175. Hyperkähler SYZ conjecture Start year not recorded ChatGPT Astra Awaiting review Preprint only
  176. Artin–Davenport conjecture Start year not recorded ChatGPT 6.0 and 2 more Awaiting review Preprint only
  177. Free group factor isomorphism problem Start year not recorded OpenAI internal model Result confirmed Expert-checked
  178. Generalized Lax conjecture Start year not recorded unnamed AI system and 1 more Awaiting review Lean-verified
  179. Bloch’s conjecture Start year not recorded OpenAI internal model Awaiting review Expert-checked
  180. Cannon’s conjecture Start year not recorded OpenAI internal model Awaiting review Peer-reviewed
  181. Is Catalan’s constant G Start year not recorded OpenAI internal model Awaiting review Lean-verified
  182. Does every Riemannian metric on Sⁿ have infinitely many geometrically distinct closed geodesics? Start year not recorded OpenAI internal model and 1 more Awaiting review Preprint only
  183. Finitely presented Burnside problem Start year not recorded OpenAI internal model Awaiting review Peer-reviewed
  184. Gromov–Lawson / Schoen–Yau aspherical conjecture Start year not recorded OpenAI internal model Awaiting review Expert-checked
  185. Long-range order of the quantum Heisenberg ferromagnet in d ≥ 3; Bose–Einstein condensation of an interacting continuum Bose gas at fixed density in the thermodynamic limit Start year not recorded OpenAI internal model Awaiting review Peer-reviewed
  186. Is BPL Start year not recorded OpenAI internal model Awaiting review Peer-reviewed
  187. Modularity of elliptic curves over imaginary quadratic fields Start year not recorded OpenAI internal model Awaiting review Preprint only
  188. Nearby Lagrangian conjecture Start year not recorded OpenAI internal model Awaiting review Preprint only
  189. ε-Dvoretzky conjecture Start year not recorded ChatGPT Awaiting review Preprint only
  190. Dying percolation conjecture Start year not recorded Claude and 2 more Awaiting review Lean proof, not yet reviewed
  191. Matrix Spencer conjecture Start year not recorded GPT-5.6 Sol Pro Awaiting review Preprint only
  192. Large gaps between consecutive primes Start year not recorded GPT-5.6 Pro/Sol and 1 more Awaiting review Lean proof, not yet reviewed
  193. Yau–Tian–Donaldson conjecture (cscK metrics) Start year not recorded Danus and 2 more Awaiting review Preprint only
  194. Zeros of ζ(s) on the critical line Start year not recorded Claude in Claude Code Result confirmed Lean-verified
Show as a table
194 claims, with the problem, the result and the human role
ProblemOpen sinceAI creditedStatusHow it was checkedDateResultHuman role
Gauss class number problemGauss class number problem: list all imaginary quadratic fields of a given class number (effectively, unconditionally)1801 (225 years)OpenAI internal model (Siegel-zero proof and its Lean formalization); Claude Opus 5.5 (Claude Code)Awaiting reviewLean-verifiedUnconditional determination of all imaginary quadratic fields with class number ≤ 2000 (16,453,493 fields), via an explicit, Lean-verified version of OpenAI’s Landau–Siegel zero exclusion.Human-directed: the author chose the problem, checks and trusted base; Claude did constants, Lean, code and most text; the underlying theorem is OpenAI’s
Quasi-Riemann hypothesis for Dedekind zeta functions of arbitrary number fields1859 (167 years)AxiomProverAwaiting reviewLean-verifiedFor every number field K, ζ_K has no zeros with Re s > 31/32; likewise finite-order Hecke L-functions over K.AI-assisted: Cureton and Thorner directed the research and made the mathematical judgments; AxiomProver generated the proofs and the Lean formalization
Quasi-Riemann hypothesisQuasi-Riemann hypothesis (a zero-free half-plane Re s > θ, θ < 1, for ζ and Dirichlet L-functions); Landau–Siegel zeros1859 (167 years)OpenAI internal model (unreleased)Awaiting reviewLean proof, not yet reviewedClaimed: ζ, all Dirichlet L-functions and finite-order Hecke L-functions over ℚ(√−3) have no zeros with Re s > 7/8; real zeros of real primitive Dirichlet L-functions satisfy 1 − β ≥ c/log q.Listed by OpenAI as an exception to its fixed automated procedure (details not given); the 11/12 write-up was human-edited
Sylvester’s conjectureSylvester’s conjecture (1879): every prime p ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes1879 (147 years)ChatGPT; Claude; unnamed AI (Yin, Oct 2026)Awaiting reviewPreprint onlyProved in full: the 4, 7 cases by Yin (arXiv 2605.25917, May 2026), the 8 case by Burungale–Tian (arXiv 2609.14893, Sept 2026), who show E_p: y² = x³ + p²/4 has analytic rank one; a shorter proof of the 8 case followed from Yin (arXiv 2610.08015, Oct 2026).Human-led with AI tools. Burungale–Tian used ChatGPT and Claude for exploratory computations, editing and reorganization. In Yin’s Oct 2026 short proof, the ideas are his and the detailed working-out was done by AI and then rewritten by him.
Hilbert’s sixteenth problem, second partHilbert’s sixteenth problem, second part: is the number of limit cycles of a real planar polynomial vector field bounded by a function of its degree alone (finiteness of the Hilbert number H(d))?1900 (126 years)OpenAI internal model (math release)Awaiting reviewLean-verifiedClaimed existence of a finite, non-effective bound B(d) for every degree d; companion theorem (in Lean): quintic classical Liénard systems have at most two limit cycles, and two are attainedReleased as model output under OpenAI’s fixed procedure, with no human author named; no human exposition or check yet
Carathéodory’s conjectureCarathéodory’s conjecture (every closed convex surface in ℝ³ has ≥ 2 umbilics) and Loewner’s conjecture (umbilic index ≤ 1), in the C^∞ category1924 (102 years)Claude (Anthropic); ChatGPT (GPT-6.1)Awaiting reviewPreprint onlyExplicit C^∞ counterexamples: umbilics of index 1 + k/2 for every positive integer k, and a strictly convex smooth closed surface with exactly one umbilic (index 2); simplified independently by Ando and UmeharaAI-assisted: Alpöge and John-Paul Smith worked on the problem with Claude, which also checked it; Ando and Umehara used GPT-6.1 to search for candidate seed functions
Artin’s primitive root conjectureArtin’s primitive root conjecture (1927): every integer other than −1 and the squares is a primitive root modulo infinitely many primes1927 (99 years)OpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed: for every admissible a, at least c_a·x/(log x)² primes in (x, 2x) have primitive root a (infinitude, not the conjectured density); a positive lower density was added by a follow-upAutonomous AI output released by OpenAI; follow-ups by human mathematicians with AI assistance
Schiffer’s conjectureSchiffer’s conjecture (Yau Problem 80) and the planar Pompeiu problem1929 (97 years)GPT-5.6; GPT-5.5; Claude Opus 4.8; Claude Fable 5; ChatGPT/CodexAwaiting reviewLean-verifiedNon-circular bounded planar domains carrying a nonconstant Neumann eigenfunction that is constant on the boundary, which disproves both Schiffer’s conjecture and the Pompeiu conjecture in the plane (one explicit domain, and an infinite family).Human-led. Colbrook–Stepaniants: AI only helped debug code. Cao-Labora–de Dios Pont: humans devised the bifurcation strategy; LLMs checked estimates numerically and drafted some lemma proofs, and GPT-5.6 wrote the Lean formalisation
Tammes problem for 15 pointsTammes problem for 15 points (open since Tammes 1930; solved for N ≤ 14 and N = 24); Robbins’ problem (is V > 2?, Bruss–Ferguson 1996); Hoffman–Johnson–Junge frog-model conjecture for d = 3, 4 (2017)1930 (96 years)unnamed AI system(s)Awaiting reviewLean-verifiedTammes optimum for N = 15 determined, with exactly two optimal configurations; V > 2 for Robbins’ problem; frog model recurrent on the 3-ary and transient on the 4-ary tree; plus three smaller conjectures settled.Claimed fully AI-generated (‘results, proofs and code were found and written by AI’); the human operator is anonymous. One Tammes theorem came from a third party using Claude
Köthe conjecture1930 (96 years)GPT-6 Astra (pre-release)Result confirmedLean-verifiedExplicit counterexample disproving the Köthe conjecture, formally verified in Lean.Autonomous discovery; humans checked and wrote up
Hopf’s conjectureHopf’s conjecture: even-dimensional manifolds of positive sectional curvature have positive Euler characteristic (symmetric case)1931 (95 years)GPT-6 AstraAwaiting reviewPreprint onlyProved for manifolds and orbifolds admitting an effective isometric T³-action (previously T⁴), with the rational cohomology of fixed-point components classified.AI-assisted: GPT-6 Astra produced the first proof of Theorem A with minimal human input; the stronger structure theorem, orbifold case and the write-up are human
Hopf’s sign conjectureHopf’s sign conjecture: closed even-dimensional manifolds with positive sectional curvature have positive Euler characteristic1931 (95 years)Odin Automatic AI Research AgentAwaiting reviewPreprint onlyClaimed smooth metric of strictly positive sectional curvature on S³×S³ (χ = 0), a counterexample in dimension 6; preceded by a claimed metric on S²×S³Autonomous per the authors: the agent ‘discovered’ the metric and the proof; human authors posted and presumably edited the paper
Erdős problems #1, #74, #126 and #571Erdős problems #1 (distinct subset sums), #74, #126 and #571 (rational exponents conjecture)1931 (95 years)GPT-6 Astra (pre-release)Result confirmedLean-verified#1 disproved (N ≤ ε·2^n possible for arbitrarily large n); #571 proved (every rational α in [1,2) is a Turán exponent of a bipartite graph); #74 answered in the negative direction; #126 proved with |S(A)| ≫ n^(1/2).Autonomous: the model worked in an agent harness without human steering; humans chose the problems, wrote the Lean statements and wrote informal expositions afterwards
Banach’s isometric subspace conjecture1932 (94 years)ChatGPT 5.5 Pro; ChatGPT 5.6 Pro; GPT-5.6 SolAwaiting reviewPreprint onlyReal case completed (all odd n, with Gromov’s even n); complex and quaternionic analogues also proved.AI-assisted: the humans made the reduction; the key approach came out of dialogue with ChatGPT Pro, and GPT-5.6 Sol drafted proofs that the authors checked and rewrote
Borsuk’s conjectureBorsuk’s conjecture (1933): every bounded set in ℝᵈ can be partitioned into d + 1 sets of smaller diameter; smallest dimension in which it fails1933 (93 years)OpenAI internal model (openai/math release)Awaiting reviewPeer-reviewedClaimed: the compact projector image of ℝP³ in the 9-dimensional trace-one space needs more than ten pieces, so the conjecture fails in every dimension d ≥ 9 (previous smallest known dimension 63, for finite sets). Every finite subset needs at most eight pieces (Damásdi), so the finite version below 63 stays openAutonomous output released without named authors; the construction was known (Kahn–Kalai, Kalai 2015), the covering obstruction is the model’s
Uniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equationsUniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equations (Leray 1934; Jia–Šverák programme 2014–15)1934 (92 years)Claude Opus; CodexAwaiting reviewLean proof, not yet reviewedClaimed continuum of distinct suitable Leray–Hopf solutions from one compactly supported axisymmetric datum in L^{3,∞}, without forceAI-generated text and proofs; authors understood the broad steps of the construction
Landau–Siegel zerosLandau–Siegel zeros (exceptional real zeros of real Dirichlet L-functions)1935 (91 years)OpenAI internal model (original proof); ClaudeAwaiting reviewLean-verifiedExplicit version of OpenAI’s claimed exclusion: (1 − β) log q ≥ 3.0 × 10⁻³ for every primitive real character of conductor q ≥ 3 (with Lu–Zaman–Zhao for small q).Expert reconstruction: the proof is OpenAI’s; Claude drafted the report, expansion and constants; Banks checked every line
Bloch constant conjectureBloch constant conjecture: the Bloch constant equals the Ahlfors–Grunsky value 0.4718616…1937 (89 years)ChatGPT; OpenAI CodexAwaiting reviewLean proof, not yet reviewedDisproved: B < A_AG − 1.3 × 10⁻⁹.Human-led: proofs developed and written by the author; ChatGPT for exploring approaches and editing; Codex generated the Lean formalization
Collatz conjectureCollatz conjecture: lower bounds on the number of integers below X whose orbit reaches 11937 (89 years)OpenAI GPT models via Codex; ProofAtlas.ai harness; Claude (manuscript review); AxiomProver (formalization of Kenny Lau’s refinement)Result confirmedLean-verifiedA positive lower natural density of integers reach 1 within 10.46 ln n ordinary Collatz steps (linear count cX, with explicit but astronomically small c), improving earlier sublinear bounds X^0.84 and X^0.90.AI-led: ‘developed primarily by AI agents’ with high-level guidance from Lech Mazur, who directed the work and is responsible for the claims
Mahler’s conjecture on the minimal volume product |K||K°| of convex bodies, symmetric and general, with equality casesMahler’s conjecture (1939) on the minimal volume product |K||K°| of convex bodies, symmetric and general, with equality cases1939 (87 years)OpenAI internal model (openai/math release)Awaiting reviewLean-verifiedClaimed: sharp lower bounds 4ⁿ/n! (symmetric; equality for Hanner polytopes) and (n+1)ⁿ⁺¹/(n!)² (general; equality for simplices) in every dimension; Gromov width of K × K° equals 4Autonomous AI output released by OpenAI under its fixed procedure (README)
Jacobian conjectureJacobian conjecture (Keller 1939): a polynomial map with non-zero constant Jacobian determinant is invertible1939 (87 years)Claude Fable 5Result confirmedUnverified claimExplicit non-injective polynomial map C³→C³ with constant Jacobian determinant, disproving the conjecture for all n≥3.AI-assisted: human chose the problem and verified; Claude found the counterexample
Kaplansky’s zero-divisor conjectureKaplansky’s zero-divisor conjecture (K[G] has no zero divisors for torsion-free G) and direct-finiteness conjecture (ab = 1 implies ba = 1 in K[G]); Gottschalk’s surjunctivity conjecture; the group-ring Determinant Conjecture1940 (86 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed counterexamples: zero divisors in F₂[G] for a finitely presented torsion-free G with a 2-dimensional classifying space; one-sided inverses in K[G] in characteristic 2 (finitely presented G), odd characteristic and over F₂ for torsion-free G; injective non-surjective cellular automata; a Z[G]-matrix with Fuglede–Kadison determinant in (0, 1)Released as model output under OpenAI’s fixed procedure, with no human author named; specialists have commented on presentation and consequences, not on correctness
Hilbert–Smith conjectureHilbert–Smith conjecture: a locally compact group acting faithfully and continuously on a connected manifold is a Lie group (equivalently, ℤ_p has no faithful action on a manifold)1941 (85 years)OpenAI internal model (math release); Claude Opus 5.5; GPT 6.1Awaiting reviewLean-verifiedClaimed proof in every finite dimension (OpenAI manuscript); separate AI-agent proof with a registered Lean 4 formalization under second-countability assumptionsOpenAI: autonomous output released without named authors. Separate proof: AI agents run ‘under human guidance’ by the repository owner (adbrw)
Maximum number of Hamiltonian cycles in an n-vertex tournamentMaximum number of Hamiltonian cycles (and paths) in an n-vertex tournament: Alon’s question (1990) and Wormald’s conjecture on the extremal tournament1943 (83 years)GPT-5.6Awaiting reviewPreprint onlyC(n) = (cosh 1/cos 1 ± o(1))(n−1)!/2ⁿ, attained asymptotically by the cyclic regular tournament; asymptotic maximum number of Hamiltonian paths followsAuthors chose the problem and the strategy hint (permanent estimates); GPT-5.6 found the lower-bound proof and a second instance the upper bound; authors rewrote, simplified and exposited
Hadwiger’s conjectureHadwiger’s conjecture (1943): every graph with no K_t minor is (t−1)-colourable, i.e. χ(G) ≤ h(G); also its fractional weakening χ_f(G) ≤ h(G)1943 (83 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed arbitrarily large counterexamples with independence number at most two, beating even the fractional form (Lean covers the ordinary form; an independent Lean project derives the fractional one); companion claims: Colin de Verdière chromatic conjecture false, linear list-Hadwiger bound trueReleased as model output under OpenAI’s fixed procedure, with no human author named; independent Lean re-packaging by a volunteer; no human exposition yet
Linear Hadwiger conjectureLinear Hadwiger conjecture (K_t-minor-free graphs have chromatic number O(t))1943 (83 years)GPT-6 Astra; OpenAI Codex (6 Sol, Astra)Awaiting reviewLean proof, not yet reviewedProved: a universal C with χ(G) ≤ Ct for all K_t-minor-free GAI-generated proof under human direction: authors set the strategy and sub-goals; GPT-6 Astra wrote essentially all of the proof; Codex produced the Lean formalization
Linnik’s constantLinnik’s constant: the least L with P(a, q) ≪ q^L for the least prime in every reduced residue class mod q1944 (82 years)Claude Opus 5.5; GPT-6 Astra; Grok 4.7Awaiting reviewLean-verifiedClaimed L ≤ 3.99 (P(a, q) < q^3.99 for all large q), improving Xylouris’s L ≤ 5Naslund prompted and supervised; the models did the research and wrote the paper; Grok helped with formalization
Erdős unit distance problemErdős unit distance problem: the maximum number u(n) of unit distances among n points in the plane; the upper bound O(n^(4/3)) (Spencer–Szemerédi–Trotter, 1984) had not been improved. Also the weak pinned distinct-distance conjecture1946 (80 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed u(n) = O(n^(4/3−δ)) for an absolute, unspecified δ > 0, and the weak pinned distance theorem; both formalized in LeanReleased as model output under OpenAI’s fixed procedure, with no human author named; detailed public reading by an outside mathematician (forum comment), no written check
Hopf problemHopf problem: does S⁶ admit an integrable complex structure?1947 (79 years)Claude (internal research model)Awaiting reviewLean proof, not yet reviewedClaimed explicit construction of integrable complex structures on S⁶ (answer: yes).AI-assisted: human mathematician (Alpöge) directed and wrote up
Weil conjecturesWeil conjectures (Weil 1949; Dwork 1960, Grothendieck 1960s, Deligne 1974): machine-checked proof1949 (77 years)Axiom agentsAwaiting reviewLean-verifiedClaimed sorry-free Lean 4 proof of rationality, functional equation, Riemann hypothesis and Betti-number statement for smooth projective varieties over finite fieldsAI agents wrote the library on top of Axiom’s Stacks formalization; no human code review yet
Hodge conjecture (CM abelian varieties)Hodge conjecture for CM abelian varieties; Tate conjecture for abelian varieties over finite fields1950 (76 years)OpenAI internal model (unreleased)Result confirmedLean-verifiedClaimed: every rational Hodge class on a complex CM abelian variety is a ℚ-linear combination of algebraic cycle classes; consequences for Tate and the Hodge standard conjecture.Listed by OpenAI as an exception to its fixed automated procedure (details not given)
Hadwiger–Nelson problemHadwiger–Nelson problem: the least number of colours needed so that no two points of the plane at distance one share a colour (bounds 5 ≤ χ ≤ 7 before the release)1950 (76 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed proof that 5 colours never suffice, for arbitrary colour classes, so χ(R²) ∈ {6, 7}; Lean formalization of both boundsReleased as model output under OpenAI’s fixed procedure, with no human author named; no human exposition yet; already used as a tool by human authors (with AI help)
Kaplansky’s quasitrace conjectureKaplansky’s quasitrace conjecture (2-quasitraces / AW*-factors) and the monotone completeness conjecture for AW*-algebras1951 (75 years)Unreleased OpenAI model; GPT-6 AstraAwaiting reviewPreprint onlyCounterexample to the quasitrace conjecture (from an unreleased OpenAI model, reproduced with a modified construction) and a finite AW*-factor that is not monotone completeOpenAI model produced the original quasitrace counterexample; Arulseelan and Hanson re-derived it, devised the monotone completeness argument, and reworked GPT 6 Astra’s preliminary write-up
Irrationality exponent of πIrrationality exponent (measure) of π, conjectured to be 2; convergence of the Flint–Hills series1953 (73 years)OpenAI internal model (unreleased)Awaiting reviewLean-verifiedClaimed: μ(π) = 2 (ineffective); hence Σ 1/(n³ sin² n) converges, and Σ 1/(n^a |sin n|^b) converges iff a > max{1, b}.Autonomous per OpenAI’s README: ‘The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model’ (about three hours of ChatGPT Pro thinking compute per result on average); family 017 is not among the listed exceptions; no human author is named
Is every group goodIs every group good (Freedman’s disk embedding for all fundamental groups)? Wall’s realization of PD4 groups; topological Borel conjecture in dimension 41953 (73 years)OpenAI internal model (math release)Awaiting reviewPreprint onlyClaimed: disk embedding fails in general and F₂ is not good; a PD4 group with no closed aspherical manifold model; homotopy-equivalent but non-homeomorphic closed aspherical 4-manifolds with hyperbolic fundamental groupAutonomous AI output released by OpenAI
Existence of four-dimensional interacting quantum field theoriesExistence of four-dimensional interacting quantum field theories (pure Yang–Mills, QCD); existence half of the Yang–Mills Millennium problem1954 (72 years)OpenAI internal modelAwaiting reviewLean-verifiedClaimed rigorous existence proof; details, method and paper not publicUnknown; a physics panel led by David Gross reviews it and will write its own account
Harary–Hill conjecture on cr(K_n); Zarankiewicz’s crossing-number conjecture for K_m,nHarary–Hill (Guy) conjecture on cr(K_n); Zarankiewicz’s crossing-number conjecture for K_m,n (Turán’s brick-factory problem, 1944)1954 (72 years)OpenAI internal model (openai/math release); ChatGPT (Astra) for Balko’s expositionAwaiting reviewPeer-reviewedClaimed exact formulas for all n and all m, n, with Lean proofs of both equalities; independent simplified human proof of the K_n case (Balko)Manuscripts credited to ‘OpenAI’ as model output with no human author; human reading, simplification and exposition by Balko (with ChatGPT’s help in understanding the proof), another exposition in preparation by Salazar
Linear MDS conjectureLinear MDS conjecture (Segre 1955; maximum length of MDS codes / arcs), even characteristic1955 (71 years)ChatGPT; Claude; Claude Opus 5.5Awaiting reviewLean-verifiedFor q even and 2 ≤ k ≤ q, the maximum length of a k-dimensional linear MDS code over F_q is q + 1, or q + 2 when k ∈ {3, q − 1}Human-led with AI tools: Kumar devised the argument and Claude helped formalize it in Lean; Chen and Huang used ChatGPT to explore proof strategies and check calculations
Kadison’s similarity problemKadison’s similarity problem (1955): is every bounded representation of a C*-algebra on a Hilbert space similar to a *-representation?1955 (71 years)OpenAI internal model (openai/math release)Awaiting reviewPeer-reviewedClaimed: yes, for every unital C*-algebra and every Hilbert space, together with a uniform amplified commutator estimate and one hyperreflexivity constant for all von Neumann algebrasAutonomous AI output released by OpenAI under its fixed procedure (README)
Optimal constant in the i.i.d. Berry–Esseen inequalityOptimal constant in the i.i.d. Berry–Esseen inequality (Esseen 1956 lower bound; sharpness conjecture attributed to Zolotarev, 1966–67)1956 (70 years)ChatGPT; Codex; Gemini; AntigravityAwaiting reviewExpert-checkedClaimed: C_n ≤ C_E = (3+√10)/(6√(2π)) ≈ 0.40973 for every n ≥ 1, so the optimal constant equals Esseen’s lower boundAI-assisted: the two authors directed the reductions, certificate schemas and verification runs; AI systems helped with exploration, checking code, distributed verification and drafting; Hexagon text label ‘primarily AI-generated’, understanding ‘some parts’
Pierce–Birkhoff conjecture1956 (70 years)GPT-5.6; GPT-6; Claude Opus 5; Claude Fable 5.1Awaiting reviewLean proof, not yet reviewedExplicit continuous piecewise-quadratic function on R^30 that is not a finite lattice combination of polynomials; further counterexamples in dimensions 5, 6, 7, 22, 24 and 72.Humans gave high-level direction on which targets mattered and wrote the final exposition; a multi-model agent system found and reviewed the construction
Simon’s Problem 2Simon’s Problem 2 (localization at all disorder in 2D) and Problem 1 (absolutely continuous spectrum / extended states at weak disorder in d ≥ 3) for the Anderson model1958 (68 years)OpenAI internal model (math release)Awaiting reviewLean proof, not yet reviewedClaimed for uniform site potentials: pure-point spectrum throughout the spectrum in 2D at every disorder; purely absolutely continuous spectrum on an open energy interval in every d ≥ 3 at small disorderAutonomous AI output released by OpenAI; an expert appendix shortens one step; a third-party AI-assisted Lean formalization of the 3D result
Sendov’s conjecture1958 (68 years)GPT-5.6 ProResult confirmedLean-verifiedProof of Sendov’s conjecture for all degrees, formally verified in Lean.Human orchestrated (Mazur); Tao simplified and formalised with AI agents
Nagata’s conjecture on plane curvesNagata’s conjecture on plane curves (1959)1959 (67 years)OpenAI internal model (unreleased)Result confirmedPreprint onlyFor r ≥ 10 very general points in P², every nonzero curve of degree d with multiplicities m_i satisfies Σ m_i < d√r (claimed by OpenAI; reorganized and re-proved by Tevelev).Autonomous AI proof (OpenAI release, no human author); Tevelev re-derived, simplified and presented it, using ChatGPT Astra only for editing and figures
Serre’s 1960 conjectureSerre’s 1960 conjecture: the Artin function of a finite group acting on a regular local ring with isolated fixed points is the character of an ℓ-adic representation1960 (66 years)GPT-6 AstraAwaiting reviewPreprint onlyProof in mixed characteristic, completing the conjecture (positive characteristic: Abe 2025; residue characteristic zero: classical)Kisin directed the work; GPT Astra was used extensively to find the proofs and write the first draft; Kisin checked, refined and rewrote
Ulam’s problemUlam’s problem: is every Lie group countably representable (isomorphic to a group of permutations of a countable set)?1960 (66 years)GPT-6 AstraAwaiting reviewPreprint onlyYes: every Lie group, and hence every locally compact second countable group, embeds in Sym(N).AI-assisted: GPT-6 Astra produced the key countable-index deformation for SL3(R) after three weeks of discussion; the authors extended it to all simple Lie groups, checked it and rewrote the proof
Complexity of optimal prefix-free coding with unequal letter costs1961 (65 years)CodexAwaiting reviewPreprint onlyNP-complete even for letter costs (M, M+1) given in the inputAI-assisted: authors posed the question and guided; Codex produced the reduction strategy and much of the proof
Vaught’s conjecture for L_ω1,ωVaught’s conjecture (1961) for L_ω1,ω: a countable theory/sentence has either countably many or perfectly many (2^ℵ0) countable models1961 (65 years)Claude Code; Codex; Claude Opus 4.5–5.5; Claude Fable 5/5.1; GPT-5.2–GPT-6; GPT ProAwaiting reviewLean-verifiedDisproved for L_ω1,ω: an explicit sentence with exactly ℵ1 countable models and no perfect set of non-isomorphic models.Human-led: Knight’s framework and draft; Ackerman and Freer directed the work; coding agents did the Lean formalization and helped search for the combinatorial template, with SMT/Horn-clause solvers
Erdős–Sós conjectureErdős–Sós conjecture (Erdős problem #548)1962 (64 years)GPT-6 Astra (pre-release)Result confirmedLean-verifiedFull proof: every n-vertex graph with more than (k−2)n/2 edges contains every tree on k vertices.Autonomous discovery and Lean proof in Epoch’s benchmark harness; humans checked, simplified and generalised it afterwards
Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞ (beyond Yudovich’s uniqueness class; the unforced form of Vishik’s theorem)1963 (63 years)GPT-6.1 Sol; Claude Opus 5.5; ChatGPT 5.6; ChatGPT 6.0; ChatGPT 6.1; GPT-6 AstraAwaiting reviewLean-verifiedThree independent claimed constructions of two distinct solutions with the same initial vorticity: vanishing-viscosity solutions for every p > 2 (Pappalettera–Sorella, fully AI-generated), renormalized weak solutions in L¹ ∩ L^q for every q < ∞ (Coiculescu, Lean-formalized), and finite-energy solutions in every finite L^p (Choi)Ranges from none (Pappalettera–Sorella: prompted only, no edits, no claim of authorship) to human strategy with AI-written details (Coiculescu: idea and strategy his, proofs, code and text largely by Claude; Choi: main ideas his, ChatGPT completed technical details)
Formalization of Hironaka’s resolution of singularities in characteristic zeroFormalization of Hironaka’s resolution of singularities in characteristic zero (1964)1964 (62 years)Claude Fable 5.1; GPT-6 AstraResult confirmedLean-verifiedSorry-free Lean 4 + Mathlib library (~500–600k lines) of Hironaka’s main theorems and modern refinements, for schemes and analytic spacesHumans wrote the 112-page plan, chose sources and intervened ~30 times; agents wrote the proofs and Lean code
Erdős Matching ConjectureErdős Matching Conjecture (1965); Łuczak–Mieczkowska–Šileikis tail-probability conjecture1965 (61 years)unnamed generative AIAwaiting reviewPreprint onlyTail conjecture proved for all k; Erdős Matching Conjecture proved for fixed k ≥ 3, s sufficiently large, all n ≥ k(s+1).AI-assisted: humans built the framework and the k = 2–5 and k ≥ 10^100 cases; AI produced the unified k ≥ 6 argument and the idea that closed a gap
Birch and Swinnerton-Dyer conjecture in rank ≤ 1Birch and Swinnerton-Dyer conjecture in rank ≤ 1 (full leading-term formula at every prime); Goldfeld’s conjecture (1979) on the ranks of quadratic twists1965 (61 years)OpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed: full BSD formula whenever some p-power Selmer group has corank 0 or 1; Goldfeld’s 50/50 density and mean analytic rank 1/2 for the quadratic twists of every E/ℚ; hence full BSD for a density-one set of twists of every curveAutonomous AI output released by OpenAI under its fixed procedure (README)
Lehmer’s permutation conjectureLehmer’s permutation conjecture (1965): imperfect Hamiltonian traversals of multiset permutations by adjacent swaps; an unsolved research problem in Knuth’s TAOCP1965 (61 years)Claude Opus 5.5; AristotleAwaiting reviewLean-verifiedProof of Verhoeff’s reformulation (Hamiltonicity of the non-stutter-word graph N(S), with two exceptional families), and hence of Lehmer’s conjecture.Human-led long project; the final short proof was found and developed by Claude Opus 5.5 in a multi-agent setup, then re-derived and checked by the author
Kinetic Krylov–Safonov theorem and Pucci’s conjectureA Krylov–Safonov Harnack inequality for kinetic equations in nondivergence form with measurable coefficients (missing: an Aleksandrov–Bakelman–Pucci estimate), and Pucci’s 1966 conjecture on the optimal integrability exponent in the ABP estimate1966 (60 years)GPT-6 Astra Ultra; GPT-6 Astra; GPT-6 ProAwaiting reviewPreprint onlyAleksandrov estimate sup u ≤ C‖f‖_{L^p} for Kolmogorov equations with rough coefficients, giving the kinetic Krylov–Safonov theorem; AI manuscripts also claim Pucci’s conjecture in every dimension and its parabolic version.Autonomous per the authors: each AI source is marked ‘Human contribution: None’ (the authors wrote the prompt and research instructions). The six mathematicians checked the kinetic proof, wrote a human exposition and published the chats
Lonely Runner ConjectureLonely Runner Conjecture (Wills 1967, Cusick 1973): every runner is at some time at distance ≥ 1/(n+1) from all others1967 (59 years)ChatGPT 6 Pro; GPT-6 AstraAwaiting reviewPreprint onlyML(V) > (1+c)/(2n) for an absolute constant c > 0, the first constant-factor improvement over the union bound 1/(2n)Human-led: authors developed the strategy; ChatGPT 6 Pro/Astra proposed the GCD-sum ingredient, pointed to Green–Walker, fixed gaps and turned informal proofs into rigorous ones
Stein’s Fourier restriction conjecture in ℝ³Stein’s Fourier restriction conjecture in ℝ³ (sphere, bounded data; diagonal form for positively curved surfaces); 3D Bochner–Riesz conjecture; Sogge’s local smoothing conjecture in 3D; Sobolev endpoint of Carleson’s Schrödinger convergence problem1967 (59 years)OpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed full resolution of each in three dimensions (planar endpoint for Carleson) in OpenAI manuscripts of 24 Sept 2026, released 6 Oct; not formally verified and not yet reviewedCredited to ‘OpenAI’ with no named human authors; OpenAI marks the release’s review status as unchecked
Grad’s conjectureGrad’s conjecture on the non-existence of smooth non-symmetric 3D MHD (magnetohydrostatic) equilibria with nested flux surfaces1967 (59 years)GPT-6 Astra Pro; GPT-5.6 Sol; Claude Fable 5; Claude Opus 5; GPT-6 Astra; Claude Fable 5.1Result confirmedLean-verifiedThree families of smooth, non-symmetric toroidal equilibria with nested pressure surfaces and non-constant pressure: one C_N-symmetric family proved with Nash–Moser methods and certified in Lean (Gómez-Serrano–Liehr–Taylor), and two explicit analytic families (Landreman).Landreman: solutions discovered by GPT-6 Astra Pro, equations checked by hand. Gómez-Serrano–Liehr–Taylor: human roadmap, LLMs filled in technical details and wrote the Lean proof under guidance.
Barnette’s conjectureBarnette’s conjecture (Hamiltonicity of cubic 3-connected bipartite planar graphs)1969 (57 years)OpenAI internal model (unreleased)Result confirmedLean proof, not yet reviewedEvery cubic 3-connected bipartite planar graph has a Hamiltonian cycle; indeed G − e is Hamiltonian for every edge e (claimed by OpenAI; written out and partly re-proved by Li).Autonomous AI proof (OpenAI release); Zhentao Li re-derived it by hand with combinatorial replacements, using LLMs to read the preprint and format the note
Exponent ω of matrix multiplicationExponent ω of matrix multiplication (Strassen 1969: ω < 2.81; laser-method records near 2.371)1969 (57 years)OpenAI internal model (openai/math release)Result confirmedLean-verifiedω ≤ 9/4 over ℂ (OpenAI, non-constructive, Lean-formalized); re-proved over every field and made constructive by independent authors within three weeksAutonomous AI output released by OpenAI; independent human-led re-proofs (Amanov–van den Berg–Zuiddam, Pass), both AI-assisted
Erdős problem #108Erdős problem #108: Erdős–Hajnal high-girth, high-chromatic subgraph problem1969 (57 years)ChatGPT; Codex; GPT-6 AstraResult confirmedLean-verifiedNegative answer: for every M there is a triangle-free graph with χ > M whose C4-free subgraphs are all 6-colourable (Kohlmeyer–Kruer, Lean-verified); improved to 3-colourable by Nguyen–Walczak.AI-assisted: two Purdue authors developed the construction and the Lean proof with substantial ChatGPT/Codex help. In the follow-up, GPT-6 Astra found the proof of the 3-colourable lemma and the experts streamlined it.
Matrix mortality problem in dimension 2Matrix mortality problem in dimension 2 (open since Paterson 1970)1970 (56 years)OpenAI Codex; GPT-5.6 Sol; Claude CodeAwaiting reviewLean-verifiedClaimed undecidable for finite families (5,799 matrices) of 2×2 integer matricesAI-assisted: authors chose the final proof architecture and reworked the exposition; AI used for research, proofs and Lean code
Golomb–Welch conjectureGolomb–Welch conjecture (c. 1968–1970): no perfect r-error-correcting Lee code in ℤⁿ for n ≥ 3, r ≥ 21970 (56 years)ChatGPTAwaiting reviewPreprint onlyClaimed full proof, in both the lattice and the arbitrary-centre formsAI-assisted: authors used ChatGPT to explore proof strategies and check calculations; the authors say they wrote all results and proofs
Erdős–Szemerédi sunflower conjecture, all kErdős–Szemerédi sunflower conjecture: k-sunflower-free families in 2^[n] have size at most c_k^n with c_k < 21970 (56 years)ChatGPT 6 Astra ProAwaiting reviewLean-verifiedClaimed proof for every k ≥ 3, with the bound 2^((1 − 1/(16k³))n), via a new polynomial-method bound for vector sunflowers.AI-assisted: the authors posed the generalization of Naslund–Sawin and rewrote the argument; ChatGPT 6 Astra Pro found the proof of the key Theorem 2
Open Erdős problemsOpen Erdős problems (erdosproblems.com), open OEIS conjectures, Zanello’s log-concavity conjecture, last-iterate rate of anchored GDA1970 (56 years)AlphaProof Nexus; Gemini 3.1 Pro; AlphaProofResult confirmedPeer-reviewedLean-verified resolutions of 9 of 353 attempted open Erdős (sub-)problems, 44 of 492 open OEIS conjectures, 2 of 4 open Hilbert-function problems, and an exact O(1/t) rate for anchored GDA.Autonomous on the Erdős and OEIS evaluation (humans chose and formalized problems, reviewed formalizations); AI-assisted collaboration in the deployment results
Hilbert’s tenth problem over ℚHilbert’s tenth problem over ℚ: is there an algorithm deciding whether a polynomial with integer coefficients has a rational zero?1970 (56 years)OpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed negative answer: rational solvability is undecidable (number of variables part of the input), via a Turing reduction from integer solvability to finitely many rational testsAutonomous AI output released by OpenAI under its fixed procedure (README)
Singmaster’s conjecture on multiplicities of binomial coefficients1971 (55 years)Claude Opus 5.5Awaiting reviewLean-verifiedClaimed unconditional bound N(t) = O((log t)^(1/2+o(1)))AI-generated proofs, formalization and most text; human contributor checked some parts
Chvátal’s conjectureChvátal’s conjecture on intersecting subfamilies of downsets1972 (54 years)ChatGPT; GPT-6 Astra; CodexResult confirmedLean-verifiedProof of Chvátal’s conjecture via a sharp correlation inequality, followed by an AI-found short spectral proof and a Lean formalization.AI-assisted, human-led original proof (ChatGPT tested inequalities and proved a guiding special case); the short spectral proof was found by GPT-6 Astra under human direction
Hochster’s small Cohen–Macaulay module conjectureHochster’s small Cohen–Macaulay module conjecture (1973), domain form in equal characteristic zero; Eisenbud–Schreyer–Weyman Ulrich question (2003)1973 (53 years)OpenAI internal model (unreleased); ChatGPT; ClaudeAwaiting reviewLean-verifiedA 3-dimensional complete Noetherian normal local domain over C with no nonzero finitely generated maximal Cohen–Macaulay module (OpenAI); Anghel extends the obstruction to all dimensions ≥ 3.Counterexample autonomous (OpenAI release), building on human preprints by Anghel, Lopez and Chen; Anghel’s survey re-proves and generalizes it, with proofs drafted with ChatGPT and Claude
Penrose conjecturePenrose conjecture (spacetime Penrose inequality): ADM mass bounded below by the area of the black holes, for initial data satisfying the dominant energy condition1973 (53 years)ClaudeAwaiting reviewPreprint onlySpacetime Penrose inequality in all dimensions n ≥ 3 for ‘maximally parameterized’ initial data (with a horizon nondegeneracy condition), via new generalized Schoen–Yau identities; the general case is reduced to solving a degenerate PDE systemHuman-led: the key t = 1 identity was derived by hand; Claude did and checked the algebra for general t and made ‘useful mathematical suggestions’ the authors call co-author level but ‘not decisive’
Amenability of Thompson’s group FAmenability of Thompson’s group F (Thompson 1973; Geoghegan’s conjecture)1973 (53 years)OpenAI internal model (unreleased)Result confirmedLean-verifiedClaimed by OpenAI and re-proved in streamlined form by Kim and Lodha: F is nonamenable.Autonomous AI proof (OpenAI release); human experts simplified, generalized and wrote it out without AI
Cycle double cover conjecture1973 (53 years)GPT-5.6 Sol UltraResult confirmedExpert-checkedProof that every bridgeless graph admits a cycle double cover.Autonomous proof per OpenAI; human experts wrote independent expositions
Double-cap conjecture / Witsenhausen’s problem on spherical sets without orthogonal pairs1974 (52 years)Codex; AI-assisted tools (unspecified)Awaiting reviewLean-verifiedFor n ≥ 3, sets in S^(n−1) with no orthogonal pair have measure at most that of two opposite caps of radius π/4, with equality only for double caps (claimed)Human-led paper with an assistive AI role; the separate Lean proof was produced almost entirely with AI-assisted tools
Rauch’s hot spots conjecture, simply connected planar caseRauch’s hot spots conjecture, simply connected planar case (Burdzy 2005, Conjecture 1.2(ii)), in its strict form1974 (52 years)OpenAI internal model (openai/math release); ChatGPT (follow-ups by Burdzy and Bañuelos)Awaiting reviewLean-verifiedClaimed: on every smooth bounded simply connected planar domain, every first nonconstant Neumann eigenfunction has no interior critical point, so its extrema lie only on the boundary; eigenvalue multiplicity allowedAutonomous AI output released by OpenAI under its fixed procedure (README); the follow-ups were generated with ChatGPT from prompts by Burdzy and by Bañuelos
Strong cosmic censorship near Kerr, in Christodoulou’s formulationStrong cosmic censorship (Penrose 1974) near Kerr, in Christodoulou’s formulation: generic data admit no extension with continuous metric and locally square-integrable Christoffel symbols1974 (52 years)OpenAI internal model (math release)Awaiting reviewPreprint onlyClaimed local strong cosmic censorship near every fixed subextremal rotating Kerr spacetime (C⁰ ∩ W^{1,2}_loc, plus C¹ and C² versions), without symmetryAutonomous AI output released by OpenAI; reading group of specialists organised by Sung-Jin Oh
Kung–Traub conjecture on the maximal order of multipoint iterations without memory1974 (52 years)ChatGPT 6; ChatGPT 5.6Awaiting reviewLean-verifiedLocal order ≤ 2^(n−1) for any memoryless method with n evaluations per step, for real and complex scalar evaluations; sharper bound for grouped evaluationsHuman-led: proofs by the author; ChatGPT 6 assisted the Lean formalization and ChatGPT 5.6/6 were used to discuss and check arguments
Murty–Simon conjectureMurty–Simon conjecture (Plesník 1975): diameter-2-critical graphs have at most ⌊n²/4⌋ edges, equality only for the balanced complete bipartite graph1975 (51 years)GPT-5.6 Sol; GPT-6 AstraAwaiting reviewPreprint onlyClaimed proof for every n (previously known only for n beyond a tower-type bound, Füredi 1992)AI-assisted: GPT-5.6 Sol found the n²/4 bound and the even-n equality case; the authors simplified it, proved the odd-n equality case and wrote the paper
Erdős–Lovász problemErdős–Lovász problem: the maximum size m(k) of an intersecting k-uniform family with covering number k; Frankl–Ota–Tokushige conjecture m(k) < (ck)^k1975 (51 years)unnamed generative AI toolsAwaiting reviewPreprint onlym(k) ≥ k^k·exp(−50√(k log k)) for log k ≥ 32, so m(k) = k^k·exp(−k^(1/2+o(1))) with Zakharov’s bound; the Frankl–Ota–Tokushige conjecture is false for large kAI found the construction; the authors verified the proofs, reworked the arguments and wrote the exposition
Polyakov’s conjecturePolyakov’s conjecture (1975): the 2D O(n) model with n ≥ 3 has a mass gap (exponential decay) at every positive temperature; construction of the massive continuum O(3) nonlinear sigma model1975 (51 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed: exponential decay for all n ≥ 3 and all β (Lean-formalized in finite volume); an Osterwalder–Schrader continuum O(3) theory with unique vacuum, mass gap and an isolated particle pole; exact O(4) lattice mass asymptoticsReleased as model output under OpenAI’s fixed procedure, with no human author named; first reactions by mathematical physicists quoted in the press
Erdős’s conjecture on arithmetic progressionsErdős’s conjecture on arithmetic progressions (often called the Erdős–Turán conjecture; erdosproblems.com #3, $5000): every set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions1976 (50 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed proof via quasipolynomial bounds r_k(N) ≤ C_k N exp(−c_k (log N)^ε_k) for every k ≥ 3; the qualitative conjecture is proved in Lean (with, per Bloom, a weaker quantitative bound inside the formal proof)Released as model output under OpenAI’s fixed procedure, with no human author named; no human exposition or check yet
Serre’s cyclicity problemSerre’s cyclicity problem: unconditional positive density of primes p with E(F_p) cyclic1977 (49 years)GPT-6; ChatGPTAwaiting reviewUnverified claimClaimed positive lower density c_E/(32 log 32) for every E/ℚ without full rational 2-torsion, CM and non-CM alike (known before under GRH, or for CM curves)Human-led with AI writing: Bhargava’s handwritten argument; GPT-6 added writing and details and helped with the Lean formalisation; builds on OpenAI’s AI-generated analytic estimates
Erdős–Hajnal for all pathsErdős–Hajnal conjecture for H = P_t (induced paths), open beyond P51977 (49 years)ChatGPT-6 Astra; Google DeepMind internal model; Claude Opus 5.5Awaiting reviewLean-verifiedClaimed proof that every path P_t (and the subdivided claws S_{1,2,t}) has the Erdős–Hajnal property; DeepMind’s model and a Claude-assisted manuscript claim the P_t case independently.Human-led (Nguyen): the author’s framework, with Astra proving parts of key lemmas; DeepMind’s and the Claude manuscripts are described as AI-generated
Asymptotic value of Delsarte’s linear programming bound for binary codes and spherical codesAsymptotic value of Delsarte’s linear programming bound for binary codes and spherical codes (Kalai’s question)1977 (49 years)Claude Opus 5.5; GPT-5.6 Sol; GPT-6 Astra; CodexAwaiting reviewPreprint onlyExact asymptotic rates of the binary and spherical Delsarte LPs for all distances, in closed form via elliptic integrals (matching upper and lower bounds).AI-assisted: human-led program; GPT-5.6 Sol co-formulated the upper bounds; Opus 5.5 found the key compatibility and the obstacle-problem reformulation ‘with light direction’; Codex drafted
Brennan’s conjecture on integrability of |g′|^p for conformal maps; Kraetzer’s t²/4 conjecture for the universal integral-means spectrumBrennan’s conjecture (1978) on integrability of |g′|^p for conformal maps; Kraetzer’s t²/4 conjecture for the universal integral-means spectrum1978 (48 years)OpenAI internal model (openai/math release)Result confirmedLean-verifiedOpenAI’s internal model proved Brennan’s conjecture and disproved Kraetzer’s conjecture at t = −1 (preprints dated 24 Sept, released 6 Oct). Four experts checked the work and gave a short independent proof that B_b(t) ≤ t²/4 on [−2, 2], strict except at 0 and ±2AI-generated original proof; Binder, Ivrii, Prause and Smirnov validated it and wrote a shorter, stronger proof without AI help (some used AI agents to understand the AI proof)
De Giorgi’s conjectureDe Giorgi’s conjecture (1978): monotone bounded entire solutions of Δu = u³ − u in ℝⁿ, n ≤ 8, are one-dimensional; also stable rigidity in ℝ⁷1978 (48 years)OpenAI internal model (math release)Awaiting reviewPreprint onlyClaimed proof in the sharp dimension 8 without Savin’s limit assumption; classification of stable solutions in ℝ⁷ without energy growthAutonomous AI output released by OpenAI
Irrationality of ζ(5)Irrationality of ζ(5) = Σ 1/n⁵ (open since Apéry’s 1978 proof for ζ(3); Euler-era question)1978 (48 years)Claude Opus 5; Claude Opus 5.5; Claude Code; unnamed generative AI tool (paper)Result confirmedLean-verifiedζ(5) is irrational, with integer polynomials Q_n of degree 37n satisfying 0 < Q_n(ζ(5)) < exp(−139n²/5), and irrationality measure at most 260.Paper by a single human author who discloses AI only for editing and consistency checks (disputed by Calegari). One Lean formalization was written by Claude agents directed by Dan Romik; the other, by Moritz Firsching, does not say how it was produced.
Abundance conjectureAbundance conjecture (κ = ν; nef lc adjoints are semiample) and the nonvanishing conjecture, open in dimension ≥ 4; also Iitaka’s subadditivity conjecture C_{n,m}1980 (46 years)OpenAI internal model (math release); Liu: DeepSeek, Fable, GLM, ChatGPT, Danus, RethlasAwaiting reviewPreprint onlyOpenAI claims log abundance for rational-boundary lc pairs and canonical nonvanishing in every dimension in characteristic zero. Liu (human-led, AI-assisted) proves nonvanishing ⇒ abundance for lc pairsOpenAI: autonomous AI output released by OpenAI. Liu: human-led paper written with several AI models and agent systems
Smale’s mean value problemSmale’s mean value problem (1981): the best constant K in |p(z) − p(c)|/|z − c| ≤ K|p’(z)| at some critical point c1981 (45 years)Claude; ChatGPTAwaiting reviewLean-verifiedClaimed: the best constant over all degrees is exactly 4 (Smale’s original bound), approached but not attained as the degree growsAuthor steered AI models in dialogue; work ‘done overwhelmingly by AI models’
Smale’s mean value conjectureSmale’s mean value conjecture (1981), constant-1 form1981 (45 years)GPT-6 Astra (pre-release); Grok 4.6Result confirmedLean-verifiedDisproved: a polynomial with p(0) = 0, p’(0) = 1 and |p(c)/c| > 1 at every critical point exists (Lean). Follow-up: Smale’s constant M > 1.1018.Autonomous (benchmark run, no human steering); follow-up bound by Jatar & Ng with AI drafts rewritten by the authors
Erdős’s questionErdős’s question: is the number of C₄-free graphs on n vertices 2^{(1+o(1))ex(n,C₄)}?1982 (44 years)GPT-6 Pro; GPT-6.1; Claude Opus 5.5Awaiting reviewPreprint onlyClaimed affirmative answer (previous best upper bound 2^{1.0819…n^{3/2}}, Kleitman–Winston 1982)Human-led: main ideas by the author; GPT-6 Pro helped complete the proofs; GPT-6.1 and Opus 5.5 revised the text
Baum–Connes conjecture without coefficientsBaum–Connes conjecture without coefficients (reduced assembly map an isomorphism) and the Kadison–Kaplansky conjecture (no nontrivial projections in C*_r(G) for torsion-free G)1982 (44 years)OpenAI internal model (math release)Awaiting reviewPreprint onlyClaimed counterexamples: failure of rational injectivity for a torsion-free group, failure of surjectivity via an irrational-trace projection, and a torsion-free group with a nontrivial projection; the surjectivity failure re-proved in a stronger form by Guoliang YuAutonomous AI output released by OpenAI; follow-up construction by Guoliang Yu, who states all proofs in it are his own
Yau’s conjectureYau’s conjecture (1982): on a closed smooth Riemannian manifold, c√λ ≤ H^{n−1}(nodal set of an eigenfunction) ≤ C√λ1982 (44 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed: upper bound true on all smooth surfaces (completing the conjecture in dimension 2); upper bound false for smooth metrics in dimensions 3 and 4, and by a power of λ on S⁴×S¹Autonomous AI output released by OpenAI, with Lean formalizations
Classification of Finite Simple GroupsClassification of Finite Simple Groups (announced c. 1983, quasithin gap closed by Aschbacher–Smith 2004): formal machine-checked proof1983 (43 years)AxiomProver; Axiom agents (models not disclosed)Awaiting reviewLean-verifiedClaimed complete, sorry-free Lean 4 formalization of the first-generation CFSG proof against TauCeti’s statement, plus a repaired step in one published route to the Unbalanced Group TheoremAI agents formalized the literature from a human-written blueprint; about 34 named mathematicians and engineers; experts Lyons and Solomon consulted
Haldane conjecture / Haldane gap for the spin-one antiferromagnetic Heisenberg chainHaldane conjecture / Haldane gap for the spin-one antiferromagnetic Heisenberg chain (Haldane 1981–1983)1983 (43 years)OpenAI internal model (math release); ChatGPT Astra (Tasaki follow-up)Awaiting reviewPreprint onlyUniform positive spectral gap above a unique ground state on even periodic rings (Δ₁ ≥ log 20/784), plus a boundary-field gap for odd open chains with a nontrivial topological indexAutonomous AI output released by OpenAI; Tasaki’s follow-up is AI-assisted (he set problems and strategy, ChatGPT Astra developed proofs)
Rigorous critical storage capacity of the Hopfield associative-memory model at zero temperature1985 (41 years)Claude Opus 5.5; GPT-6.1 SolAwaiting reviewUnverified claimClaimed bracket αc ∈ [0.138198129, 0.13820106]Mainly AI-generated; human contributor checked parts
Falconer’s distance conjectureFalconer’s distance conjecture (1985): compact sets in ℝᵈ of Hausdorff dimension > d/2 have distance sets of positive Lebesgue measure1985 (41 years)OpenAI internal model (openai/math release)Awaiting reviewLean-verifiedClaimed in every dimension d ≥ 2, with a Lean proof of the exact statement (comparator accepted)Autonomous AI output released by OpenAI under its fixed procedure (README)
Large-data global classical existence for the three-dimensional relativistic Vlasov–Maxwell systemLarge-data global classical existence for the three-dimensional relativistic Vlasov–Maxwell system (open since Glassey and Strauss’s 1986 continuation criterion)1986 (40 years)OpenAI internal model (openai/math release)Awaiting reviewPeer-reviewedClaimed: every smooth compactly supported nonnegative particle density with smooth finite-energy fields satisfying the Gauss laws has a unique global classical solution, smooth on every finite time intervalAutonomous AI output released by OpenAI under its fixed procedure (README); a summary of the model’s reasoning is published
Sidorenko’s conjectureSidorenko’s conjecture (1986): every bipartite graph H satisfies t(H,G) ≥ p(G)^|E(H)| in every host graph G; also the Skokan–Thoma forcing conjecture1986 (40 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed counterexample: an explicit 35-vertex, 66-edge bipartite incidence graph H and a (non-explicit) finite host G with t(H,G) < p(G)^66; the same H is not forcingReleased as model output under OpenAI’s fixed procedure, with no human author named; no human exposition or check yet
Erdős #993: forest independence sequences are unimodalErdős Problem #993 (Alavi, Malde, Schwenk, Erdős 1987): is the independent set sequence of every tree or forest unimodal?1987 (39 years)Claude Opus 5.5; Claude Fable 5.1; GPT-6 Astra; GPT-5.6 Sol; GPT-6 Sol; GPT-6.1 SolAwaiting reviewLean-verifiedClaimed: yes for every finite forest. Two proofs: Zhang–Li (27 Sept 2026, computer-assisted, GPT-6 Astra for computations) and Li–Vallier–Zhang (6 Oct 2026, new arguments and Lean 4 proof attributed to AI agents).First proof: human strategy with GPT-6 Astra for computations. Second proof: AI-generated per the paper (all new mathematics, the Lean proof and most of the prose), in a project directed by Kevin Vallier and built on Zhang and Li’s reductions
Aggarwal–VitterAggarwal–Vitter (1988): can external-memory integer sorting beat the comparison-sorting I/O bound?1988 (38 years)ChatGPT; CodexAwaiting reviewLean-verifiedDeterministic external-memory sort of Θ(log N)-bit integers with asymptotically fewer block transfers than the comparison bound in a range of parameters, via a new network-coding-over-routing constructionAI-assisted: the authors (two of them the conjecture’s original proposers) directed the work; ChatGPT/Codex ‘materially affected’ constructions, analyses and the Lean proofs
k-server conjecture1988 (38 years)GPT-6 Astra (ChatGPT); ChatGPT 5.5 Pro; Gemini 3.1 ProAwaiting reviewPreprint onlyDeterministic k-competitiveness of the work function algorithm on every metric space.Human-designed potential and k = 3 proof; the AI generalised it to all k; humans adapted and verified the final proof
Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h)Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h) = o(π(x)) (Möbius on shifted primes; parity-problem model case)1989 (37 years)ChatGPT; OpenAI internal model (via the OpenAI Math Release preprint)Awaiting reviewPreprint onlyClaimed: Σ_{p≤x} µ(p+2) = o(π(x)), and each sign of µ(p+2) has density 1/2 among primes with p+2 squarefreeHuman-written proof; ChatGPT helped the authors understand the OpenAI preprint’s method, establish Appendix A and check the manuscript
Planar Mumford–Shah conjecturePlanar Mumford–Shah conjecture (regularity of minimizing discontinuity sets)1989 (37 years)GPT-6 Astra (ChatGPT, Work mode)Awaiting reviewPreprint onlyClaimed classification of planar Mumford–Shah generalized global minimizers, which would imply the conjecture.The human supplied his thesis, unpublished results and a research protocol; the model produced the proof text; checking by the author is ongoing
Alon–Seymour–Thomas conjectureAlon–Seymour–Thomas conjecture: K_h-minor-free graphs have balanced separators of size O(h√n)1990 (36 years)GPT-6 Astra; Claude Opus 5.5; ChatGPT 6 Pro; GPT-6.1 SolAwaiting reviewLean-verifiedClaimed proved twice: 21h√n (Lin) and O(h√n) with Lean formalization (Conroy–Le–Spalding-Jamieson)Conroy et al.: AI-originated ideas, humans verified and rewrote; Lin: human-guided AI-assisted proof checked by hand
Deligne–Drinfeld conjectureDeligne–Drinfeld conjecture: the Grothendieck–Teichmüller Lie algebra grt₁ is freely generated by one element in each odd weight 3, 5, 7, …1990 (36 years)OpenAI internal model (openai/math release)Result confirmedLean-verifiedProved in full by OpenAI’s internal model (dimension upper bound via characteristic two). An expert simplified the proof over 𝔽₂ and accepted that it settles the conjecture (Willwacher), and a second, independent proof identifies the motivic Galois group of mixed Tate motives over ℤ with GRT (Hirose)AI-generated original proof; Schneps and Enriquez reworked it, Willwacher simplified and re-proved it with AI help, Hirose gave an independent proof (discussing the mathematics with GPT-6 Astra Pro)
DelocalizationDelocalization (QUE) and the critical localization–delocalization crossover for 1-d random band matrices (Fyodorov–Mirlin 1991)1991 (35 years)AxiomProver; unnamed LLMsAwaiting reviewLean-verifiedQUE for W²/N → ∞ and a DMPK-described critical law at W² ≍ N for the Gaussian periodic block modelAuthors provided the architecture; LLMs drafted proofs, edited by authors; AxiomProver autoformalized everything
Laczkovich’s Lipschitz mapping problemLaczkovich’s Lipschitz mapping problem (1991): can every measurable set of positive measure in ℝ^d be mapped onto a cube by a Lipschitz map?1991 (35 years)ChatGPTAwaiting reviewPreprint onlyClaimed positive answer for every d (previously only d ≤ 2), plus Kolmogorov’s contraction problem for Jordan measurable setsAI-assisted: ChatGPT produced the key proposition and lemma; the author had the framework and lower bound, rewrote the proof and wrote the paper
Sogge’s local smoothing conjecture in all dimensions; consequencesSogge’s local smoothing conjecture (1991) in all dimensions; consequences: Bochner–Riesz, Stein’s restriction conjecture (interior of range), Kakeya maximal and set conjectures in ℝⁿ1991 (35 years)OpenAI internal model (openai/math release)Awaiting reviewLean-verifiedClaimed proof of local smoothing for the Euclidean wave equation in every dimension n ≥ 2 with ε loss, with Bochner–Riesz, restriction in the interior of the range, Nikodym/Kakeya maximal bounds and full Hausdorff dimension of Kakeya sets as corollariesCredited to ‘OpenAI’; released as ‘model output directly’, with no named human authors and no designated reviewers
Is ℤ a Diophantine subset of ℚ?Is ℤ a Diophantine subset of ℚ? (Mazur’s conjecture predicts no; Hilbert’s tenth problem over ℚ)1992 (34 years)GPT-6 Astra; GPT-6.1 Sol; Claude Opus 5.5; Claude Fable 5.1; Claude Fable 5.2; ChatGPT; ClaudeAwaiting reviewLean-verifiedTwo claimed proofs that ℤ is Diophantine in ℚ, with explicit quartics (Glazer: 222 variables; Burungale: degree at most four), refuting Mazur’s conjecture on the topology of rational pointsGlazer: manuscript produced with Astra, Sol and Opus, audited by Fable and by human-run AI reviews; Burungale: human proof, ChatGPT assisted Sections 4–5
Satisfiability conjecture for random k-SATSatisfiability (threshold) conjecture for random k-SAT: existence of a limiting threshold density α_k for every fixed k ≥ 31992 (34 years)OpenAI internal model (math release)Awaiting reviewPeer-reviewedExistence of α_k for every fixed k ≥ 3, proved twice: by Carenini via an O(n^{1/2+1/k}) scaling window plus Abbe–Montanari, and by OpenAI’s models via a capped last-satisfiable-index argument; OpenAI adds Θ(n) hitting-time variance and computability of α_3Carenini: human proof, no LLM-generated arguments by her statement. OpenAI: autonomous output released without named authors, crediting Carenini with priority
Odlyzko–Poonen conjectureOdlyzko–Poonen conjecture on irreducibility of random 0/1 polynomials1993 (33 years)GPT-6 AstraResult confirmedLean-verifiedUnconditional proof that random 0/1 polynomials are irreducible with probability tending to one, with the sharp asymptotic for the reducible probability.AI-generated proofs and formalisation; the author directed the work, rewrote and checked it; an expert (Breuillard) checked the proof
LeBrun–Salamon conjectureLeBrun–Salamon conjecture: every compact positive quaternion-Kähler manifold is a symmetric Wolf space1994 (32 years)ChatGPT; OpenAI internal model (unreleased)Awaiting reviewPreprint onlyClaimed proof for all real dimensions 4n with n ≥ 2 (the 4-dimensional case was already known), via transverse rational curves on the twistor space; independently claimed in OpenAI’s family 062.Human-led with AI tools for Hu–Liu–Wan (ChatGPT as an ‘auxiliary tool’, role not specified); autonomous for OpenAI’s family 062
Rapid mixing of Glauber dynamics for proper colouringsRapid mixing of Glauber dynamics for proper colourings (Jerrum 1995; conjecture q ≥ Δ+2)1995 (31 years)GPT-6 Astra ProAwaiting reviewPreprint onlyClaimed spectral gap Ω(1/n) for q ≥ (1+1/e+γ)Δ, large Δ, any graphAI-originated ideas; author simplified, verified and wrote the text
3SUM hypothesis3SUM hypothesis (no O(n^(2−ε)) algorithm) and APSP hypothesis (no O(n^(3−ε)) algorithm), plus Exact Triangle and Zero-Weight k-Clique1995 (31 years)Claude (Anthropic internal research model)Awaiting reviewLean-verifiedDeterministic O(n^1.9992) 3SUM and O(n^2.9995) integer APSP, refuting the 3SUM, APSP, Exact Triangle and Zero-Weight k-Clique hypotheses and three hinted OMv conjecturesAI-originated: the core algorithm was found autonomously by Claude (16M output tokens, no human input) during an unrelated cryptography task; Alman and Vassilevska Williams simplified, extended and wrote the paper
Kannan–Lovász–Simonovits conjectureKannan–Lovász–Simonovits conjecture (universal Cheeger/Poincaré constant for isotropic log-concave measures)1995 (31 years)GPT-6 Astra; GPT-5.6 Sol; Claude Fable 5; Claude Fable 5.1; ChatGPTAwaiting reviewPreprint onlyUniversal O(1) bound on the KLS constant ψ_n (Song–Zhang); a second, AI-generated proof via Song–Zhang’s criterion (Bizeul–Klartag–Lehec)AI-assisted (Song–Zhang: human-chosen approaches, AI co-developed); largely AI-generated (Bizeul–Klartag–Lehec: authors mainly understood and exposited ChatGPT’s proofs)
Pach–Shahrokhi–Szegedy conjecturePach–Shahrokhi–Szegedy conjecture: k-quasiplanar graphs have O_k(n) edges1996 (30 years)ChatGPT-6Awaiting reviewExpert-checkedProof produced by ChatGPT-6; simplified self-contained proof by Pach, Tardos and ZengAI produced the proof (relying on Beeker–Lazarovich); human experts rewrote and simplified it
Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal conjectured O(log k)Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal (1996) conjectured O(log k)1996 (30 years)GPT-6 Astra; Gemini 4 ArgonAwaiting reviewLean-verifiedClaimed tight bound Θ(log k log log k), so the O(log k) conjecture is falseAI-assisted: GPT Astra helped develop some proofs and the write-up, Gemini Argon assisted in parts; the authors take responsibility
Exact overlaps conjecture (dimension one)Exact overlaps conjecture for self-similar measures on ℝ (Simon 1996 for sets; Hochman 2014 programme), and the generalized entropy–Lyapunov dimension formula1996 (30 years)GPT-6 AstraAwaiting reviewLean-verifiedProof that every self-similar measure on the line has dim ν = min{1, h_μ/|χ_μ|}, so dimension drop occurs only with exact overlaps; no separation or algebraicity assumptions.AI-found, human-rewritten: GPT-6 Astra produced the first full proof in a ~2.5-hour session and the Lean formalization; Kittle found a new deduction of the main theorem and the authors rewrote the paper and checked it
Periodic tiling conjecturePeriodic tiling conjecture (explicit Euclidean formulation by Lagarias and Wang, 1996): every finite subset of ℤᵈ that tiles it by translations also tiles it periodically; likewise for bounded tiles of ℝᵈ1996 (30 years)OpenAI internal model (openai/math release)Awaiting reviewPeer-reviewedClaimed: a finite tile of ℤ³, and its unit-cube thickening in ℝ³, that tiles but has no fully periodic tiling, the lowest possible lattice dimension (Greenfeld–Tao had a counterexample only in very high dimension)Autonomous output released without named authors; method adapted from Greenfeld and Tao’s Sudoku construction
Benjamini–Schramm conjectureBenjamini–Schramm conjecture: p_c < p_u on nonamenable quasi-transitive graphs1996 (30 years)OpenAI internal model; Claude Opus 5.5Awaiting reviewLean-verifiedClaimed for bond percolation (OpenAI, Lean) and for site percolation (Pei, Claude agents)OpenAI: not stated per paper; Pei: chose problem and strategy, reviewed key lemmas, AI found and wrote proofs
List total colouring conjectureList Total Colouring Conjecture (Borodin–Kostochka–Woodall; Juvan–Mohar–Škrekovski; Hilton–Johnson)1997 (29 years)GPT-6 Astra UltraAwaiting reviewPreprint onlyDisproved: an explicit 20-vertex cubic graph with total chromatic number 4 and list total chromatic number 5.AI-found counterexample: the author prompted the model to disprove the conjecture; he checked the proof and rewrote the drafts
Alon–Krivelevich conjectureAlon–Krivelevich conjecture: bipartite graphs of maximum degree Δ are O(log Δ)-choosable1998 (28 years)GPT-6 Astra; xean; Claude Opus 5.5Awaiting reviewPreprint onlyDisproved: bipartite graphs with ch(G) ≥ (1/32 − o(1))(ln Δ)²/(ln ln Δ)³AI-found: GPT-6 Astra found the counterexample and proof; authors verified it; Claude rewrote the text
Smale’s 6th problem, five bodies in the planeSmale’s sixth problem: is the number of central configurations (relative equilibria) of the n-body problem finite for every choice of positive masses? Case n = 5 in the plane1998 (28 years)unnamed generative AI toolsAwaiting reviewPeer-reviewedClaimed: finitely many planar central configurations for every five positive masses, removing the codimension-two exceptional set of Albouy–Kaloshin.AI-generated per the author’s disclosure (AI found and proved the results, wrote the programs, the Lean formalization and the draft); the human role is not described
Existence of the q-colourability threshold of G(n, d/n)Existence of the q-colourability threshold of G(n, d/n) (Achlioptas–Friedgut, 1999)1999 (27 years)Claude Opus 5.5; GPT-6 Astra; GPT-6 Luna; GPT-6.1 SolAwaiting reviewLean-verifiedClaimed: for every q ≥ 3 a constant threshold d_q exists for G(n, d/n) (d_q/2 for G(n, ⌊cn⌋)), with an O(n(log n)^(−3/2)) hitting-time windowAuthors supervised; the models did proof discovery, verification, Lean formalization and all writing; human understanding ‘some parts’
Exponential Time HypothesisExponential Time Hypothesis (Impagliazzo–Paturi 1999/2001) and SETH1999 (27 years)internal Claude research model (Anthropic)Awaiting reviewLean-verifiedClaimed deterministic 2^O(n log log n/√log n)-time (fixed k) algorithm for #k-SAT, refuting ETH and SETH, found independently of OpenAI’sAutonomous discovery (14M output tokens, ‘no human input’); Anthropic researchers worked with Claude on the presentation; Lean formalization written by AI models
Exponential Time HypothesisExponential Time Hypothesis (Impagliazzo–Paturi 1999/2001), Strong ETH, Gap-ETH, NSETH1999 (27 years)OpenAI internal model (openai/math release)Awaiting reviewLean-verifiedClaimed deterministic (n+m)^O_k(1)·2^(n/(log n)^(1/2+o(1)))-time, polynomial-space algorithm for #k-SAT, every fixed k ≥ 2, which would refute ETH and its variantsCredited to ‘OpenAI’ with no named human authors; OpenAI says it is model output. Human examination is assigned to an outside working group (Impagliazzo, Paturi, Williams)
Reed–Solomon list decoding up to capacityEfficient list decoding of Reed–Solomon codes beyond the Johnson radius up to capacity; proximity gaps / mutual correlated agreement for RS codes1999 (27 years)GPT-5.6 Sol; ChatGPT 5.6 Pro; ChatGPT 6 ProAwaiting reviewExpert-checkedDeterministic polynomial-time list decoding of RS codes over prime fields up to capacity for every constant rate and evaluation set, and near-optimal mutual correlated agreement (proximity gaps) up to capacity with fixed slack.AI-assisted: humans extracted the idea from a crowd-sourced prize submission and directed the work; GPT-5.6 Sol helped produce most of the remaining results (BCPZZ); ChatGPT 5.6 Pro interactions were ‘crucial’ (Jeronimo); authors verified everything
Barry Simon’s multidimensional L² conjectureBarry Simon’s multidimensional L² conjecture (Problem 8, ‘Schrödinger operators in the twenty-first century’)2000 (26 years)ChatGPTAwaiting reviewExpert-checkedFalse in every dimension d ≥ 2: smooth, bounded, nonnegative, arbitrarily small potentials satisfying the (even weighted) L² condition for which −Δ + V has only pure point spectrum on (0, ∞).AI-assisted: authors chose the problem and suspected falsity; ChatGPT developed the construction and the extension to the full spectrum; authors rewrote the argument
Woodin’s questionWoodin’s question: if κ is strongly compact and GCH holds below κ, does GCH hold everywhere?2000 (26 years)GPT-6 Astra (ultra reasoning); GPT-6.1 SolAwaiting reviewPreprint onlyYes, in ZFC; the argument also works for ω₁-strongly compact cardinals.AI-assisted; the author acknowledges GPT-6 Astra and GPT-6.1 Sol ‘during the investigation’ without detailing their contribution and takes responsibility
Navier–Stokes existence and smoothnessNavier–Stokes existence and smoothness (Clay Millennium Prize problem), forced-breakdown case2000 (26 years)OpenAI internal model (≈10,000 parallel agents)DisputedLean-verifiedClaimed proof, formalised in Lean, of finite-time blow-up for 3D incompressible Navier–Stokes with smooth external forcing.Largely autonomous agent swarm; builds on human techniques of Córdoba and Martínez-Zoroa
Is there a binary search tree satisfying Iacono’s unified bound?Is there a binary search tree satisfying Iacono’s unified bound? (open since 2001; Iacono’s splay-tree conjecture)2001 (25 years)Claude Opus 5.5; Claude Fable 5.1; Claude Fable 5.2; GPT-6 Astra; GPT-5.6 SolAwaiting reviewPreprint onlyA deterministic online BST with cost O(m + UB(X)); dynamic optimality now implies the unified bound for splay treesAI-assisted design, analysis, proof checking and drafting; authors take responsibility
Mézard–Parisi formula for the free energy of diluted spin-glass models, including random K-SATMézard–Parisi formula (2001) for the free energy of diluted spin-glass models, including random K-SAT2001 (25 years)ChatGPT Astra; Claude FableAwaiting reviewPreprint onlyClaimed matching lower bound for diluted p-spin models with even p and random K-sat under symmetry and integrability assumptions, completing the formula (upper bound: Franz–Leone)AI-found proof directed by the field’s main expert: Panchenko asked for a new perturbation in his earlier argument; ChatGPT Astra found the proof; Panchenko rewrote it with ChatGPT Astra and Claude Fable
Period-index conjecture2001 (25 years)ChatGPTAwaiting reviewPreprint onlyVarieties of every dimension d ≥ 3 with a Brauer class of period 2 and index 2^d, contradicting the conjecture; a corrected conjecture is proposed.Human-led: a wrong AI example inspired the construction; the human found and proved the correct one, with AI help on literature search and lattice calculations
Spectral isometry conjectureSpectral isometry conjecture (Mathieu–Schick 2002): unital surjective spectral isometries between unital C*-algebras are Jordan isomorphisms2002 (24 years)GPT-6 Astra; ChatGPT 6Awaiting reviewLean-verifiedClaimed full proof for arbitrary unital C*-algebras, plus the non-unital-map variant; Lean-checked per the authors (Palomar)AI found the proof; the authors studied it, rewrote it as a human-readable paper and checked it
Finite index over treeable (JKL 2002)Jackson–Kechris–Louveau Open Problem 6.4(B): is a countable Borel equivalence relation with a finite-index treeable subrelation treeable?2002 (24 years)ChatGPT 6 Astra UltraAwaiting reviewPreprint onlyYes, in the purely Borel setting (hence also measurably); with consequences for strongly treeable groups.Human-led: the authors designed the staged construction; GPT contributed one key coherence observation; the paper was written by the authors
Zilber–Pink conjecture for abelian varietiesZilber–Pink conjecture for abelian varieties (Zilber 2002, Pink 2005); curve case in the Siegel threefold A22002 (24 years)OpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed: finiteness of maximal atypical subvarieties for every subvariety of an abelian variety over Q̄ (hence over any characteristic-zero field by Barroero–Dill); Zilber–Pink for Hodge-generic curves in A2Autonomous AI output released by OpenAI under its fixed procedure (README)
Exact polynomial-time approximability of metric k-medianExact polynomial-time approximability of metric k-median (gap between the 1 + 2/e hardness and the ~2 upper bound)2002 (24 years)unreleased internal OpenAI modelAwaiting reviewLean-verifiedClaimed: deterministic polynomial-time (1 + 2/e + ε)-approximation for every fixed ε > 0, matching the NP-hardness bound, so the threshold is exactly 1 + 2/e; plus a randomized (2 − σ)-approximationAutonomous per OpenAI: produced by the model in OpenAI’s open-problem evaluation; humans curated and released
Unique Games ConjectureUnique Games Conjecture (Khot 2002)2002 (24 years)OpenAI internal model (unreleased)Awaiting reviewLean proof, not yet reviewedClaimed: NP-hardness of (1−ε, δ)-gap Unique Games for every fixed ε, δ; hence optimal NP-hardness for Max-Cut (Goemans–Williamson) and Vertex Cover (factor 2).Autonomous by OpenAI’s fixed procedure (per README)
Multiple-unicast network coding conjectureUndirected multiple-unicast conjecture (Li & Li 2004; Harvey, Kleinberg & Lehman 2004)2004 (22 years)GPT-6 (ChatGPT)Awaiting reviewPreprint onlyDisproved: an explicit linear network code over 𝔽₉ on a 182-vertex PG(2, 9) incidence subgraph beats fractional multicommodity flow (rate 1 vs ≤ 147/157), giving Ω((log n)^ε) coding gaps by amplification.AI found the counterexample to a strengthened version the authors formulated; humans steered the search, converted it to a causal code and checked the proof
Crouzeix’s conjecture2004 (22 years)GPT-5.6 SolResult confirmedExpert-checkedProof that the numerical range W(A) is a 2-spectral set for every matrix A.Autonomous: non-specialist prompted; experts verified
Explicit matrices with the restricted isometry property beyond the square-root bottleneckExplicit (deterministic) matrices with the restricted isometry property beyond the square-root bottleneck (Tao 2007)2007 (19 years)GPT-6 Astra; ChatGPT; Claude Opus 5.5Awaiting reviewLean-verifiedExplicit real RIP matrices with k = m^(2/3−o(1)), plus a proof that absolute-value Gram-matrix methods cannot exceed exponent 2/3AI-generated main proof; humans digested it, found the 2/3 barrier with ChatGPT and rewrote the paper
Area law for entanglement entropy of ground states of gapped local Hamiltonians in two dimensions2007 (19 years)OpenAI internal model (original proof); Axiom agent tooling; OpenAI CodexAwaiting reviewLean-verifiedLean 4 proof, with no sorry, of OpenAI’s theorem that the unique ground state of a globally gapped finite-range Hamiltonian on any finite induced square-lattice domain has entanglement entropy at most C × (number of boundary edges), with C independent of domain size and shape.AI-generated proof (OpenAI release); formalization by AI agents directed by Ruiz Chen (Axiom), who froze and audited the Lean statement layer and checked each result
Matroid secretary conjectureMatroid secretary conjecture (Babaioff–Immorlica–Kleinberg) and its strong form2007 (19 years)GPT-6 Astra; GPT-5.6 Sol; ChatGPT ProAwaiting reviewPreprint onlyO(1) (4-probability-competitive) secretary algorithm for every matroid; merged paper claims the optimal ratio e and a ratio-2 single-sample prophet inequalityAI-assisted: authors chose approaches and ran the LP computations; GPT Astra used in obtaining the main theorems; authors checked the proofs
Aaronson–Ambainis conjectureAaronson–Ambainis conjecture (2008): bounded low-degree polynomials have an influential variable2008 (18 years)non-public Anthropic model; PSI research systemAwaiting reviewLean-verifiedMax influence ≥ c·v^(5/2)/d^6 (≥ (v/d)^7) for bounded degree-d polynomials of variance v, claimed with a Lean-checked proofAI found the proof in an autonomous run; a human scoped the problem, reviewed the AI-drafted paper and advisers audited it
Gurvits’s conjecture that mixed volumes admit no FPRAS in the oracle modelGurvits’s conjecture (2009) that mixed volumes admit no FPRAS in the oracle model2009 (17 years)GPT-6 AstraAwaiting reviewPreprint onlyFPRAS for mixed volumes of n convex bodies in R^nAI-originated: main ideas from GPT-6 Astra, humans simplified and wrote the exposition
Toms–Winter conjecture (C*-algebras)Toms–Winter conjecture: for simple separable nuclear non-elementary C*-algebras, finite nuclear dimension, Z-stability and almost unperforated Cuntz semigroup (strict comparison) are equivalent2009 (17 years)ChatGPT 6 Astra (GPT-6 Astra via Codex); unnamed AI (Thiel)Awaiting reviewPreprint onlyClaimed complete: Thiel proves strict comparison implies pureness (5 Oct 2026); Evington, Hua, Schafhauser, Seth, Tikuisis and White prove pureness implies Z-stability (7 Oct 2026).Human-led with AI input: one prompt gave a 4-page proof of a lemma in a special case (W*-bundles); the authors built the general theory and the 70-page proof without further AI use. Thiel: AI collaboration at an initial stage led to the key idea of two lemmas
Courtade–Kumar conjectureCourtade–Kumar (most informative Boolean function) conjecture: dictators maximise I(f(X); Y) under binary symmetric noise2013 (13 years)Gemini (Stellar Colosseum); ChatGPT; unnamed generative AI systemsAwaiting reviewLean-verifiedProved in full (all n, all noise levels, no balance assumption) by three independent teams on 21–22 Sep 2026; the Google–CUHK proof is computer-assisted and verified end-to-end in Lean.Google–CUHK: humans supplied the reduction and the Bellman function, AI generated most new ideas and the region-by-region proof; Ky–Tran: human approach, ChatGPT developed the proofs; Mahdavifar–Beirami: AI generated the single-bit proof under human direction
Bounded gaps between primesBounded gaps between primes (toward the twin prime conjecture)2014 (12 years)GPT-6 AstraAwaiting reviewLean proof, not yet reviewedClaimed proof that infinitely many pairs of primes differ by at most 186.Largely AI-generated per OpenAI
Dimension of the SO(3) instanton homology J♯ of the dodecahedron webDimension of the SO(3) instanton homology J♯ of the dodecahedron web: 60 (number of Tait colourings) or 642015 (11 years)ChatGPT Astra 6; ChatGPT; ClaudeAwaiting reviewUnverified claimProposed proof that the dimension is 64Leading gauge theorists posed the question and supplied earlier AI-computed data; the text is primarily AI-generated and only partly checked
3-Uniform Hyperclique Hypothesis3-Uniform Hyperclique Hypothesis (Lincoln–Vassilevska Williams–Williams 2018)2018 (8 years)ChatGPT Sol 6.1Awaiting reviewPreprint onlyTruly subquartic (O(n^3.957)) algorithm for detecting or counting 4-hypercliques in 3-uniform hypergraphs; the hypothesis is falseAI-assisted: authors prompted the model with different suggestions; both proofs came from the model, the authors verified and rewrote them
Benson–Etingof–Ostrik conjecture on symmetric tensor categories of moderate growth in positive characteristicBenson–Etingof–Ostrik conjecture (2023) on symmetric tensor categories of moderate growth in positive characteristic2023 (3 years)GPT-6 AstraAwaiting reviewExpert-checkedClaimed full proof: every such category admits a symmetric tensor functor to a higher Verlinde category, possibly of infinite heightAI-generated paper; five specialists (including two of the conjecture’s authors) directed it with questions, reductions and ideas
Hyperelliptic Schottky problemHyperelliptic Schottky problem (codim-3 theta singularities); Pareschi–Popa Conjecture A; Debarre’s conjecture on base lociNot recordedunnamed AI toolsAwaiting reviewPreprint onlyCodim Sing(Θ) = 3 characterizes hyperelliptic Jacobians; IT₀ criterion for GV-subschemes; generic reducedness of codim-2 base lociHuman-led; AI suggested the final step of the main proof and constructed the sharpness example
Talagrand’s Ultimate Matching ConjectureTalagrand’s Ultimate Matching Conjecture (Research Problem 17.1.2 in ‘Upper and Lower Bounds for Stochastic Processes’)Not recordedGPT-6Awaiting reviewPreprint onlyClaimed proof for all finite critical exponent pairs α₁, α₂ > 2 with 1/α₁ + 1/α₂ = 1/2; claimed failure at the endpoints (2, ∞), (∞, 2)AI-assisted: specialists steered weeks of LLM interaction; GPT-6 found the critical and endpoint proofs; text mostly LLM-written
Rational points on X_ns+(121); the level-121 case of the Rouse–Sutherland–Zureick-Brown conjecture on ℓ-adic imagesNot recordedClaude Opus 5.5; GPT-6.1 Sol; GPT-6 Astra; Claude CodeAwaiting reviewUnverified claimClaimed: X_ns+(121)(Q) consists of 7 CM points, so the 11-adic images of non-CM elliptic curves over Q are determinedHumans chose the problem and directed; AI produced almost all the mathematics, computations and text; partial (2611.00231) or no (2611.00255) human checking
Order of the weak-coupling Hawking–Page/deconfinement transition in N=4 SYM; optimal large-c spinless modular bootstrap boundNot recordedinternal Claude research model; Claude Opus 5.5Awaiting reviewUnverified claimb = −1.48×10⁻³ λ² (first-order transition); optimal slope Λ_V = 12π²/(π²+4 arcsinh²1) ≈ 9.1266Autonomous AI result with a physicist prompting and checking
Tusnády’s problemTusnády’s problem: combinatorial discrepancy of points with respect to axis-parallel boxes in R^dNot recordedClaude; Codex; ChatGPTAwaiting reviewPreprint onlyClaimed Θ_d((log N)^(d−1/2)) in every fixed dimension d ≥ 2 (two independent proofs)AI-assisted: authors credit the tools with proof development and checking (Wengiel) and with generating ideas (Lei)
Formalizing the Stacks Project in LeanFormalizing the Stacks Project (de Jong et al.) in LeanNot recordedAxiom agentsAwaiting reviewLean-verifiedClaimed Lean 4 formalization of every formalizable Stacks tag, in human-reviewed and agent-written tiersAI agents wrote StacksFrontier; humans reviewed StacksCore
Torsion groups of elliptic curves over number fields of fixed degree dTorsion groups of elliptic curves over number fields of fixed degree d = 6, 7, 8, 9Not recordedGPT-6.1 Sol; Claude Opus 5.5; ChatGPT-6 AstraAwaiting reviewUnverified claimComplete list of torsion groups for d = 6–9, with three groups newHuman posed questions and guided; AI produced the mathematics, computations and manuscript; human checked degree 6 only
Best proven running time of deterministic integer factoringBest proven running time of deterministic integer factoring (record N^(1/5+o(1)), Harvey 2021; Harvey–Hittmeir)Not recordedClaude Opus 5.5Awaiting reviewLean-verifiedLean-proved: deterministic factoring in time N^o(1) (C_k·2^(n/k) for every k)Authors posed the question and designed and audited the 13-line Lean statement; Claude agents produced the reduction, method, proof plan and all Lean code; the authors did not read the proof
Kervaire’s conjectureKervaire’s conjecture (1960s; Kourovka 17.94) and the Kervaire–Laudenbach conjecture (Kourovka 18.87(a))Not recordedOpenAI internal model (unreleased, unnamed)Result confirmedLean-verifiedOpenAI’s proofs accepted as solutions in the Kourovka Notebook after confirmation by N. V. BogachevProofs from OpenAI’s internal model run; human confirmation by N. V. Bogachev for the Notebook
Petty’s equilateral-set conjecturePetty’s equilateral-set conjecture (1971): every n-dimensional normed space contains n + 1 equidistant pointsNot recordedTencent Hunyuan hyproverAwaiting reviewPreprint onlyClaimed proof in dimension 5 (dimensions 3 and 4 were known)Author wrote the prompt with a methodological hint; the agent generated the proof; the author organised, verified and rewrote it
Pure semisimplicity conjecturePure semisimplicity conjecture: every right pure semisimple ring has finite representation typeNot recordedGPT-6 Astra; GPT-6.1 Sol; Claude Opus 5.5Awaiting reviewLean-verifiedClaimed counterexample via a division ring extension with right dimension sequence (1, 2, 2, 2, …), through Simson’s reductionAI-assisted: GPT-6 Astra helped choose the criterion and construct the division rings; the author verified the proofs and completed the Lean formalisation with GPT-6.1 Sol and Claude Opus 5.5
Coloring 3-colorable graphs with few colors in polynomial timeColoring 3-colorable graphs with few colors in polynomial time (best exponent of n)Not recordedGPT-6 Astra; GPT-6 ProAwaiting reviewPreprint onlyO(n^0.1537) colors, improving O(n^0.1954); also an n^(1/6+o(1)) frameworkHuman–AI collaboration: the models generated each successive algorithm; the authors studied the analyses, suggested ideas (incl. the generalization to vector k-coloring), simplified the arguments and wrote the paper
Rigorous continuum limit of three-dimensional Yang–Mills theoryRigorous continuum limit of three-dimensional Yang–Mills theory (here SU(2) on a torus, via Wilson-flowed observables)Not recordedClaude Opus 5.5; GPT-6 Astra; GPT-6.1 SolAwaiting reviewLean-verifiedClaimed convergence of flowed Wilson-loop laws of 3D SU(2) lattice Yang–Mills on a torus to a unique continuum measure on the Cao–Chatterjee state space, with short-distance and weak-coupling asymptoticsContributors posted the work; by the Hexagon label no part is understood by a human; proof discovery, verification and writing by the models
Constructing a solution of the two-dimensional KPZ equation at criticalityNot recordedChatGPT 5.5; ChatGPT 5.6; ChatGPT 6Awaiting reviewExpert-checkedConstruction of the Critical 2D KPZ Process as the unique limit of centred log partition functions of critical 2D directed polymers, a functional of the Critical 2D Stochastic Heat Flow; proved non-Gaussian.Human-led with substantial AI input: authors supplied structure, literature and judgement; ChatGPT developed key estimates and proofs, some of which failed audit
Integer programming with exponent linear in dCan integer programs in standard form with d constraints and entries ≤ N be solved in N^O(d) time instead of N^O(d²)?Not recordedGPT-6 Astra UltraAwaiting reviewLean-verifiedClaimed deterministic f(d)·N^O(d)·poly-log(c_max) + poly(input) algorithm for bounded standard-form integer programming.Human-led: the authors designed the proof route and core idea; GPT Astra Ultra wrote the complete proofs and a Lean check
FPRAS for counting perfect matchings in general graphsFPRAS for counting perfect matchings in general graphs (bipartite case: Jerrum–Sinclair–Vigoda 2004); Anari–Oveis Gharan–Vinzant entropy conjectureNot recordedOpenAI internal model (unreleased, unnamed)Result confirmedLean-verifiedClaimed FPRAS for all finite simple graphs (OpenAI), re-proved in simplified form by Liu and MengProof from OpenAI’s model run; independent human re-proof and simplification by Liu and Meng
A non-sofic hyperbolic groupIs every word-hyperbolic group sofic? (Hyperbolic groups were the leading candidates for sofic counterexamples after residual finiteness)Not recordedGPT Astra; OpenAI release model (family 252)Awaiting reviewLean-verifiedClaimed construction of a torsion-free two-dimensional word-hyperbolic group that is not sofic; independently, a claimed proof with Lean that OpenAI’s non-residually-finite hyperbolic group is not sofic.AI-assisted: AI-generated construction (Sandhu, 7 Oct) built on OpenAI’s AI result; Thom reworked and checked the proof; GPT Astra helped draft
Gauss circle problemGauss circle problem: the exponent θ in E(X) = O(X^θ)Not recordedChatGPT; CodexAwaiting reviewPreprint onlyθ ≤ 209/670 + ε ≈ 0.31194, improving Li–Yang’s 0.314483 and going below Huxley’s suggested 5/16 barrier for the Bombieri–Iwaniec method.Human-led with AI tools: authors used ChatGPT and Codex to discuss proof strategies and to draft and check parts of the arguments; final proofs written and verified by the authors
Existence of the Smith–Toda complex V(4); V(3) exists for p ≥ 7Existence of the Smith–Toda complex V(4); V(3) exists for p ≥ 7 (Toda)Not recordedOpenAI internal model (openai/math release); ChatGPTAwaiting reviewPreprint onlyV(4) exists for every prime p ≥ 11. OpenAI’s internal model had shown it only for p = 1009, plus a V(n) at some prime for each nOpenAI’s model wrote the original papers; Angeltveit checked, simplified and extended them, using ChatGPT as an interactive assistant; he checked all arguments himself
Yang–Mills existence (YM4)Construction of four-dimensional quantum Yang–Mills theory (existence part of the Yang–Mills existence and mass gap Millennium Prize Problem), via Bałaban’s renormalisation-group program (1985–1989)Not recordedClaude; Codex; GPT-5.6 SolAwaiting reviewPreprint onlyClaimed, not established: a very long Claude-generated argument for the existence of SU(2) Yang–Mills on ℝ⁴ (continuum limit with Osterwalder–Schrader properties and non-triviality), conditional on listed inputs; no mass gap, no uniqueness of the infinite-volume limit.Alpöge ‘coaxing Claude’ with ‘some small amount of direction’; Sussman proposed the program and ran the earlier Codex QED3 experiment; both say they cannot yet tell how reliable the output is
Erdős–Simonovits rational exponents conjectureErdős–Simonovits rational exponents conjecture (Erdős problem #571): every rational r in [1, 2] is the Turán exponent of some graphNot recordedGPT-6 AstraResult confirmedLean-verifiedHuman-written exposition of the AI-found proof: for every rational r in [1, 2] there is a graph H with ex(n, H) = Θ(n^r).The proof was found by GPT-6 Astra in Epoch AI’s FrontierMath Erdős runs (Sept 2026); eight human experts wrote, explained and situated it in this paper
How mathematics should respond to AI-driven ‘proof abundance’ after OpenAI’s 722-manuscript releaseNot recordedn/aResult confirmedUnverified claimPublic lecture and slides arguing that problem-solving has become a misaligned proxy and proposing ‘Math 2.0’ uses of AI (exposition problems, living result networks, formal libraries, open interpretable maths models).Human-led
Dualization lower bound under ETHIs monotone Boolean dualization (hypergraph transversal enumeration) solvable in output-polynomial time?Not recordedGPT-6 AstraAwaiting reviewPreprint onlyNo, under the Exponential Time Hypothesis: a 3SAT reduction rules out N^o(√(log N/log log N)) algorithms, leaving a gap only between √(log N) and log N in the exponent.AI found the proof of the main theorem; the authors reconstructed, revised and verified it and rewrote the exposition
Hyperkähler SYZ conjectureHyperkähler SYZ conjecture / abundance for nef line bundles on compact hyperkähler manifoldsNot recordedChatGPT Astra (proofreading only)Awaiting reviewPreprint onlyProved (preprint): every nef line bundle on a compact irreducible hyperkähler manifold is semiample; finitely many deformation classes in each dimension for b2 ≥ 5.Human-led; the proof was written by the authors, AI used only to proofread after OpenAI’s release
Artin–Davenport conjectureArtin–Davenport conjecture: integral cubic forms in at least 10 variables have nontrivial integer zerosNot recordedChatGPT 6.0 (GPT-6); ChatGPT 5.5/5.6/6.1; OpenAI CodexAwaiting reviewPreprint onlyProved in full: n ≥ 10 suffices (previous record n ≥ 14, Heath-Brown 2007); 10 is sharp.Human-led over two years; ChatGPT 6.0 supplied the improved incidence-variety dimension bound (Lemma 3.8) and the Galois-descent approach of Lemma 3.6; GPT models helped combine estimates and proofread; Codex produced a partial Lean auto-formalisation
Free group factor isomorphism problemFree group factor isomorphism problem (is L(F₂) ≅ L(F₃)?)Not recordedOpenAI internal model (unreleased)Result confirmedExpert-checkedClaimed: yes; all interpolated free group factors L(F_r), 1 < r ≤ ∞, are isomorphic, with fundamental group ℝ₊.Autonomous by OpenAI’s fixed procedure (per README)
Generalized Lax conjectureGeneralized Lax conjecture (every hyperbolicity cone is spectrahedral) and the projected Lax conjecture (every hyperbolicity cone is a spectrahedral shadow)Not recordedunnamed AI system (Kummer–Netzer); OpenAI internal modelAwaiting reviewLean-verifiedClaimed: true for strictly hyperbolic polynomials (Kummer–Netzer, AI-found); false in general, even for shadows (OpenAI family 095)Kummer–Netzer: AI produced the proof, humans posed the question, suggested approaches, checked and edited. OpenAI: autonomous model output, Lean statement
Bloch’s conjectureBloch’s conjecture (1970s): for a complex surface with p_g = 0, the Albanese map on degree-zero zero-cycles is an isomorphismNot recordedOpenAI internal model (openai/math release)Awaiting reviewExpert-checkedClaimed in full: for p_g = q = 0, CH₀(S) ≅ ℤ via the degree; hence Bloch’s conjecture for all surfaces with p_g = 0Autonomous AI output released by OpenAI under its fixed procedure (README)
Cannon’s conjectureCannon’s conjecture: does every word-hyperbolic group whose Gromov boundary is a 2-sphere act geometrically (properly, cocompactly, isometrically) on hyperbolic 3-space?Not recordedOpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed proof for all such groups (finite kernel allowed), via a fixed-scale combinatorial 2-modulus bound and the Bourdon–Kleiner criterion; Lean formalization of the conclusionReleased as model output under OpenAI’s fixed procedure, with no human author named; presented in a seminar talk by Mario Bonk; no written human check yet
Is Catalan’s constant GIs Catalan’s constant G = β(2) irrational?Not recordedOpenAI internal model (openai/math release)Awaiting reviewLean-verifiedProved irrational (OpenAI manuscript with a Lean proof of the exact statement); follow-ups give explicit irrationality measuresAutonomous AI output released by OpenAI under its fixed procedure (README)
Does every Riemannian metric on Sⁿ have infinitely many geometrically distinct closed geodesics?Does every Riemannian metric on Sⁿ (n ≥ 3) have infinitely many geometrically distinct closed geodesics?Not recordedOpenAI internal model (math release); OpenAI CodexAwaiting reviewPreprint onlyClaimed for all n ≥ 2 and every metric, plus every closed 3-manifold (OpenAI); independently re-proved and extended to all compact rank-one symmetric spaces’ underlying manifolds (Chen–Deng–Zhang)OpenAI: autonomous output. Chen–Deng–Zhang: human-verified proof developed with Codex, crediting OpenAI’s strategy
Finitely presented Burnside problemFinitely presented Burnside problem: must a finitely presented group in which every element has finite order be finite?Not recordedOpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed infinite finitely presented periodic group St₁₂(R) over an F₂-algebra, with property (T); claimed residually finite version with a finite-index 2-power-torsion subgroup; claimed finitely presented nil algebra that is not nilpotentReleased as model output under OpenAI’s fixed procedure, with no human author named; no human exposition or check yet
Gromov–Lawson / Schoen–Yau aspherical conjectureGromov–Lawson / Schoen–Yau aspherical conjecture: a closed aspherical manifold admits no metric of positive scalar curvature; Gromov’s integral scalar curvature bound for simplicial volumeNot recordedOpenAI internal model (math release)Awaiting reviewExpert-checkedClaimed in every dimension, without spin hypotheses, via rational inessentiality of positive-scalar-curvature manifolds; plus the integral inequality with a dimensional constantAutonomous AI output released by OpenAI
Long-range order of the quantum Heisenberg ferromagnet in d ≥ 3; Bose–Einstein condensation of an interacting continuum Bose gas at fixed density in the thermodynamic limitLong-range order (spontaneous magnetization) of the quantum Heisenberg ferromagnet in d ≥ 3; Bose–Einstein condensation of an interacting continuum Bose gas at fixed density in the thermodynamic limitNot recordedOpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed: magnetized KMS states at low temperature for all d ≥ 3 and all spins, Bloch’s T^{3/2} law with exact coefficient; condensation in every hard-sphere ground state and at a fixed positive temperature, with the Bogoliubov depletion lawAutonomous AI output released by OpenAI, with Lean formalizations of the two core theorems
Is BPLIs BPL = L? Can bounded-error randomized logarithmic-space computation be simulated deterministically in logarithmic space (also RL = L)?Not recordedOpenAI internal model (math release)Awaiting reviewPeer-reviewedClaimed proof of L = RL = BPL, with a Lean formalization in an explicit Turing-machine model whose statement definitions have been community-checked against the proof’sAutonomous AI output released by OpenAI; community volunteers (with Claude Code) re-checked the Lean statement
Modularity of elliptic curves over imaginary quadratic fieldsModularity of elliptic curves over imaginary quadratic fields (and, by extension, over totally real and CM fields)Not recordedOpenAI internal model (openai/math release)Awaiting reviewPreprint onlyClaimed: every elliptic curve over every imaginary quadratic field is modular; Le Hung’s note states the method also covers all totally real fields and CM fields without ζ5Autonomous AI output released by OpenAI; follow-up notes by human experts written with AI drafts
Nearby Lagrangian conjectureNearby Lagrangian conjecture (Arnold): every closed exact Lagrangian in T*Q is Hamiltonian isotopic to the zero sectionNot recordedOpenAI internal model (math release)Awaiting reviewPreprint onlyClaimed counterexample in T*(S⁹ × S^{N−1}) for some large even N: an embedded exact Lagrangian diffeomorphic to the base, not Hamiltonian isotopic to the zero sectionAutonomous AI output released by OpenAI
ε-Dvoretzky conjectureε-Dvoretzky conjecture: polynomial dependence on 1/ε in Dvoretzky’s theorem (V. Milman; Klartag–Novikov)Not recordedChatGPTAwaiting reviewPreprint onlyPolynomial bound n ≥ (C/ε)^((ℓ+1)/2)|log ε| for nearly ellipsoidal ℓ-dimensional sections of arbitrary (not necessarily symmetric) convex bodies, and a Euclidean-ball version.Human-led. The key idea (the probability distribution on the Grassmannian) came out of a discussion between ChatGPT and Shahar Moshe; ChatGPT also checked and improved proofs.
Dying percolation conjectureDying percolation conjecture θ(p_c)=0 for Bernoulli bond percolation on Z^dNot recordedClaude (Anthropic); Claude Fable 5.1; GPT-5.6 SolAwaiting reviewLean proof, not yet reviewedClaimed Lean-verified proof that θ(p_c)=0 for all d ≥ 2, via a new additive gluing inequality that settles Kozma–Nitzan Conjecture 3.Autonomous formalization: Claude wrote all the Lean code under Justin Leder’s direction. The follow-up proofs of stronger conjectures were produced by GPT-5.6 Sol + Claude Fable 5.1 with ‘minimal human intervention’ from Ahmed Bou-Rabee
Matrix Spencer conjectureNot recordedGPT-5.6 Sol ProAwaiting reviewPreprint onlyO(√n) discrepancy for n symmetric n×n matrices of norm at most 1, achievable efficiently.AI-assisted: humans chose the strategy and verified and polished; the model produced the key lemma’s proof and most calculations
Large gaps between consecutive primesLarge gaps between consecutive primes (Erdős–Rankin; Erdős problem #4)Not recordedGPT-5.6 Pro/Sol; GPT-6 AstraAwaiting reviewLean proof, not yet reviewedImproved lower bound for infinitely many large prime gaps, saving a log log log n factor over FGKMT 2018.AI-assisted: pseudonymous user DottedCalculator prompted the model; Thomas Bloom wrote the exposition
Yau–Tian–Donaldson conjecture (cscK metrics)Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metricsNot recordedDanus (Rethlas-based agent); Claude Fable 5 (Claude Code); GPT-5.6 Sol (Codex)Awaiting reviewPreprint onlyClaimed explicit K-polystable polarized smooth projective fivefold without a cscK metric.AI-generated main result with one key human redirection; the author checked and polished the text
Zeros of ζ(s) on the critical lineProportion of nontrivial zeros of ζ(s) on the critical line (toward the Riemann hypothesis)Not recordedClaude (unreleased research model) in Claude CodeResult confirmedLean-verifiedUnconditional proof that more than 67.2% of zeta zeros are simple and lie on the critical line.Near-autonomous: non-mathematician prompted; experts verified

How we count

A claim counts when an entry dated after 30 June 2026 reports an AI result on a named open problem, has importance 4 or 5, and credits an AI system. Formal proofs of theorems already proved, batches of many results and other science (forecasts, drug and protein design, competitions) are listed under ‘All AI science results’ instead. The status says where the field stands on the claim: ‘confirmed’, ‘awaiting review’ (no verdict yet from peer review or the experts in the area) or ‘disputed’. The ‘How it was checked’ column says what checking exists so far: peer review, a Lean proof, named experts, or a preprint only. A Lean proof whose formal statement nobody has reviewed is labelled ‘Lean proof, not yet reviewed’. A Lean proof can check a theorem while experts still dispute what it settles, so a row can be both ‘Lean-verified’ and ‘disputed’.

How the 194 claims were checked
How it was checkedClaims
Peer-reviewed20
Lean-verified76
Expert-checked11
Lean proof, not yet reviewed12
Preprint only67
Unverified claim8

All AI science results

Every science result we logged after 30 June 2026, 378 results in all, grouped by field and newest first, including formal proofs of known theorems and results that are not open problems. Each shows its status and how it was checked.

Mathematics

274 results

  1. 136 days after the cutoff

    AI-assisted results, 13 Nov

    Awaiting reviewLean-verified

  2. 135 days after the cutoff

    Fixed-profile soliton resolution fails (3D, 5D wave equation), with GPT-6 Astra

    Awaiting reviewPreprint only

  3. 135 days after the cutoff

    Test-free PDL lacks interpolation; counterexample by Claude Fable in Lean

    Result confirmedLean-verified

  4. 135 days after the cutoff

    AI-assisted results, 12 Nov: Hexagon

    Awaiting reviewLean-verified

  5. 134 days after the cutoff

    Band-matrix QUE and critical crossover, 1.2M lines of Lean by AxiomProver

    Awaiting reviewLean-verified

  6. 134 days after the cutoff

    Benson–Etingof–Ostrik conjecture: full proof generated by GPT-6 Astra

    Awaiting reviewExpert-checked

  7. 134 days after the cutoff

    Hyperelliptic Schottky problem solved; AI suggested the final step

    Awaiting reviewPreprint only

  8. 134 days after the cutoff

    Smale’s mean value constant = 4 (claimed, Lean-checked)

    Awaiting reviewLean-verified

  9. 134 days after the cutoff

    Talagrand’s Ultimate Matching Conjecture: GPT-6 proof (Hexagon)

    Awaiting reviewPreprint only

  10. 134 days after the cutoff

    Large triangles in planar sets of positive density: GPT-6 Astra Pro proof

    Awaiting reviewLean-verified

  11. 134 days after the cutoff

    AI-assisted results, 11 Nov: arXiv and Hexagon

    Awaiting reviewLean-verified

  12. 133 days after the cutoff

    Explicit RIP matrices reach k = m^(2/3): main theorem by GPT-6 Astra

    Awaiting reviewLean-verified

  13. 133 days after the cutoff

    AI-assisted results, 10 Nov: arXiv and Hexagon

    Awaiting reviewLean-verified

  14. 132 days after the cutoff

    X_ns+(121): two AI-generated proofs in two days

    Awaiting reviewUnverified claim

  15. 132 days after the cutoff

    Reflexive polytopes conjecture disproved, GPT-6.1 Sol

    Awaiting reviewPreprint only

  16. 132 days after the cutoff

    AI-assisted results, 9 Nov: arXiv and Hexagon

    Awaiting reviewLean-verified

  17. 131 days after the cutoff

    AI-assisted results, 8 Nov: Hexagon of 7 Nov and arXiv

    Awaiting reviewLean-verified

  18. 130 days after the cutoff

    AI-assisted results, 7 Nov: Hexagon of 6 Nov and arXiv

    Awaiting reviewLean-verified

  19. 129 days after the cutoff

    Axiom: the Weil conjectures proved in Lean

    Awaiting reviewLean-verified

  20. 129 days after the cutoff

    Navier–Stokes: unforced Leray–Hopf non-uniqueness, AI-written

    Awaiting reviewLean proof, not yet reviewed

  21. 129 days after the cutoff

    Tusnády’s problem claimed settled twice, with AI help

    Awaiting reviewPreprint only

  22. 129 days after the cutoff

    AI-assisted results, 6 Nov: arXiv of 6 Nov and Hexagon of 5 Nov

    Awaiting reviewLean-verified

  23. 128 days after the cutoff

    ℤ is Diophantine in ℚ: Glazer (Astra, Claude) and Burungale claim Mazur’s conjecture fails

    Awaiting reviewLean-verified

  24. 128 days after the cutoff

    Hopfield critical capacity bracketed, by Claude Opus 5.5

    Awaiting reviewUnverified claim

  25. 128 days after the cutoff

    Quasiplanar graph conjecture proved by ChatGPT-6 (Pach, Tardos, Zeng)

    Awaiting reviewExpert-checked

  26. 128 days after the cutoff

    Singmaster’s problem: new bound by Claude Opus 5.5

    Awaiting reviewLean-verified

  27. 128 days after the cutoff

    AI-assisted results, 5 Nov: arXiv of 5 Nov and Hexagon of 4 Nov

    Awaiting reviewLean-verified

  28. 127 days after the cutoff

    Axiom: the Stacks Project formalized in Lean

    Awaiting reviewLean-verified

  29. 127 days after the cutoff

    Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol

    Awaiting reviewUnverified claim

  30. 127 days after the cutoff

    Amplituhedron magic number conjecture: counterexample found by ChatGPT

    Awaiting reviewPreprint only

  31. 127 days after the cutoff

    Serre modularity over real quadratic fields, primarily AI-generated

    Awaiting reviewUnverified claim

  32. 127 days after the cutoff

    AI-assisted results, 4 Nov: arXiv of 4 Nov and Hexagon of 2–3 Nov

    Awaiting reviewLean-verified

  33. 126 days after the cutoff

    K_h-minor-free separators O(h√n): two AI-assisted proofs

    Awaiting reviewLean-verified

  34. 126 days after the cutoff

    Kronheimer–Mrowka: dodecahedron instanton homology is 64 (ChatGPT Astra 6)

    Awaiting reviewUnverified claim

  35. 126 days after the cutoff

    D-equivalence for generalized Kummer hyperkählers (ChatGPT-6 assisted)

    Awaiting reviewPreprint only

  36. 126 days after the cutoff

    RM(1,9) covering radius = 242, three proofs

    Awaiting reviewPreprint only

  37. 126 days after the cutoff

    AI-assisted results, 3 Nov: arXiv of 3 Nov and Hexagon of 2–3 Nov

    Awaiting reviewLean-verified

  38. 125 days after the cutoff

    AI-assisted results, 2 Nov: arXiv of 2 Nov and Hexagon of 1–2 Nov

    Awaiting reviewLean-verified

  39. 124 days after the cutoff

    AI-assisted results, 1 Nov: arXiv leftovers and result repos

    Awaiting reviewLean-verified

  40. 123 days after the cutoff

    Bhargava: elliptic curves have cyclic reduction at a positive proportion of primes (claim)

    Awaiting reviewUnverified claim

  41. 123 days after the cutoff

    AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct

    Awaiting reviewLean-verified

  42. 122 days after the cutoff

    Alon–Krivelevich conjecture disproved with GPT-6 Astra

    Awaiting reviewPreprint only

  43. 122 days after the cutoff

    Mézard–Parisi formula: Panchenko’s proof ‘found by ChatGPT Astra’

    Event confirmedAwaiting reviewPreprint only

  44. 122 days after the cutoff

    Number of C₄-free graphs is 2^{(1+o(1))ex(n,C₄)} (claim, GPT-6 Pro)

    Awaiting reviewPreprint only

  45. 122 days after the cutoff

    AFEM rate-optimal for any bulk parameter θ < 1 (ChatGPT framework)

    Awaiting reviewPreprint only

  46. 122 days after the cutoff

    Right-angled hyperbolic polytopes exist only up to dimension 8 (claim)

    Awaiting reviewLean-verified

  47. 122 days after the cutoff

    AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct

    Awaiting reviewLean-verified

  48. 121 days after the cutoff

    Sharp Berry–Esseen constant claimed: C = Esseen’s 0.4097

    Awaiting reviewExpert-checked

  49. 121 days after the cutoff

    Kervaire’s conjecture: OpenAI proof confirmed in the Kourovka Notebook

    Result confirmedLean-verified

  50. 121 days after the cutoff

    Petty’s conjecture in dimension 5: proof by Tencent’s hyprover agent

    Awaiting reviewPreprint only

  51. 121 days after the cutoff

    Pure semisimplicity conjecture: claimed counterexample, GPT-6 Astra, Lean

    Awaiting reviewLean-verified

  52. 121 days after the cutoff

    q-colourability threshold of G(n, d/n) exists, says a machine-written paper

    Awaiting reviewLean-verified

  53. 121 days after the cutoff

    Spectral isometry conjecture for C*-algebras: GPT-6 Astra proof, Lean

    Event confirmedAwaiting reviewLean-verified

  54. 121 days after the cutoff

    Gcd in O(log a / log log a) remainder calls refutes Moschovakis’s uniform-process conjecture

    Awaiting reviewLean-verified

  55. 121 days after the cutoff

    OpenConjecture: GPT models on 200 random arXiv conjectures

    Event confirmedAwaiting reviewPreprint only

  56. 121 days after the cutoff

    AI-assisted results, 29 Oct 2026 (part 2)

    Awaiting reviewLean-verified

  57. 121 days after the cutoff

    AI-assisted results, 29 Oct (part 3): Hexagon’s evening batch

    Awaiting reviewLean-verified

  58. 121 days after the cutoff

    AI-assisted results in the 29 Oct 2026 sweep

    Awaiting reviewLean-verified

  59. 120 days after the cutoff

    Kisin proves Serre’s 1960 Artin-character conjecture with GPT Astra

    Awaiting reviewPreprint only

  60. 120 days after the cutoff

    Laczkovich’s Lipschitz mapping problem: claimed answer, ChatGPT key step

    Awaiting reviewPreprint only

  61. 120 days after the cutoff

    Murty–Simon conjecture: proof for all n, bound found by GPT-5.6 Sol

    Event confirmedAwaiting reviewPreprint only

  62. 120 days after the cutoff

    AI-assisted results in the 28 Oct 2026 sweep

    Awaiting reviewLean-verified

  63. 119 days after the cutoff

    Axiom: Classification of Finite Simple Groups formalized in Lean

    Awaiting reviewLean-verified

  64. 119 days after the cutoff

    Erdős–Lovász covering-number problem: AI-found construction disproves Frankl’s 1996 conjecture

    Awaiting reviewPreprint only

  65. 119 days after the cutoff

    Sharp exponent 2/3 for the Banach–Mazur distance to the cube

    Event confirmedAwaiting reviewPreprint only

  66. 119 days after the cutoff

    Morgan–Mullen conjecture confirmed with Claude, Lean-checked

    Awaiting reviewLean-verified

  67. 119 days after the cutoff

    R(3,3,3,3) ≤ 61 with AI, Lean and SAT

    Event confirmedAwaiting reviewLean-verified

  68. 119 days after the cutoff

    AI-assisted results in the 27 Oct 2026 sweep

    Awaiting reviewLean-verified

  69. 118 days after the cutoff

    OpenAI claims local smoothing and Kakeya in all dimensions

    Awaiting reviewLean-verified

  70. 118 days after the cutoff

    AI-assisted results in the 26 Oct 2026 sweep

    Awaiting reviewLean-verified

  71. 117 days after the cutoff

    Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT

    Awaiting reviewPreprint only

  72. 116 days after the cutoff

    AI-assisted results in the 24 Oct 2026 sweep

    Awaiting reviewPreprint only

  73. 115 days after the cutoff

    AI-assisted results in the 23 Oct 2026 arXiv listing

    Awaiting reviewPreprint only

  74. 115 days after the cutoff

    Heilman simplifies OpenAI’s propeller conjecture proof

    Result confirmedLean-verified

  75. 115 days after the cutoff

    Tokareva’s bent-function conjecture disproved by LLM agents

    Awaiting reviewPreprint only

  76. 115 days after the cutoff

    Triangle lower tail: Zhao’s conjecture settled with GPT-6 Astra and Claude

    Awaiting reviewLean-verified

  77. 114 days after the cutoff

    Eleven probabilists construct the Critical 2D KPZ process, a solution of the critical two-dimensional KPZ equation, ‘through extensive interactions’ with ChatGPT 5.5, 5.6 and 6

    Awaiting reviewExpert-checked

  78. 114 days after the cutoff

    Barry Simon’s L² conjecture (Problem 8 of his 2000 list) disproved

    Awaiting reviewExpert-checked

  79. 114 days after the cutoff

    AI-assisted results in the 22 Oct 2026 arXiv listing

    Awaiting reviewPreprint only

  80. 114 days after the cutoff

    ChatGPT 6 Astra outlines the argument that removes the Hodge-conjecture assumption from Nakayama’s local height asymptotics

    Event confirmedAwaiting reviewPreprint only

  81. 113 days after the cutoff

    Nagata’s 1959 conjecture

    Result confirmedPreprint only

  82. 113 days after the cutoff

    Barnette’s 1969 Hamiltonicity conjecture

    Result confirmedLean proof, not yet reviewed

  83. 113 days after the cutoff

    Hochster’s 1973 small Cohen–Macaulay conjecture

    Awaiting reviewLean-verified

  84. 113 days after the cutoff

    AI-assisted results in the 21 Oct 2026 arXiv listing

    Awaiting reviewLean-verified

  85. 112 days after the cutoff

    Double-cap conjecture claimed in all dimensions, with a separate AI-built Lean proof

    Awaiting reviewLean-verified

  86. 112 days after the cutoff

    MDS conjecture settled in even characteristic

    Event confirmedAwaiting reviewLean-verified

  87. 112 days after the cutoff

    AI-assisted results in the 20 Oct 2026 arXiv listing

    Awaiting reviewLean-verified

  88. 112 days after the cutoff

    Borodin–Kostochka conjecture claimed for maximum degree ≥ 4583 (down from 5.3 million), then ≥ 1000 (26 Oct), with GPT-6 Astra assisting

    Awaiting reviewPreprint only

  89. 112 days after the cutoff

    Gale’s 1953 conjecture fails in 3D

    Result confirmedUnverified claim

  90. 111 days after the cutoff

    Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3

    Awaiting reviewLean-verified

  91. 111 days after the cutoff

    Kaplansky’s quasitrace conjecture reported disproved by an unreleased OpenAI model

    Awaiting reviewPreprint only

  92. 111 days after the cutoff

    AI-assisted results in the 19 Oct 2026 arXiv listing

    Awaiting reviewLean-verified

  93. 111 days after the cutoff

    AI-assisted maths on Zenodo, 17–19 Oct 2026

    Awaiting reviewLean-verified

  94. 111 days after the cutoff

    Commutative counterexamples to Arveson’s hyperrigidity conjecture

    Awaiting reviewPreprint only

  95. 111 days after the cutoff

    Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI’s π-exponent library

    Awaiting reviewLean-verified

  96. 111 days after the cutoff

    Neumann–Zagier 1985 conjecture on tree lattices settled after an AI proof with ChatGPT 5.5 Pro and Opus 4.8

    Event confirmedAwaiting reviewPreprint only

  97. 110 days after the cutoff

    Bake AI’s Épi agent reports 120 record results with GPT-6 Astra

    Awaiting reviewPreprint only

  98. 109 days after the cutoff

    2D Euler non-uniqueness without force claimed three times, all with AI

    Awaiting reviewLean-verified

  99. 109 days after the cutoff

    Erdős–Hajnal conjecture proved for every path P_t

    Awaiting reviewLean-verified

  100. 109 days after the cutoff

    A non-sofic word-hyperbolic group

    Awaiting reviewLean-verified

  101. 109 days after the cutoff

    arXiv results with disclosed AI help, 16 Oct 2026 announcement

    Awaiting reviewLean-verified

  102. 109 days after the cutoff

    AI-assisted results, 16–17 Oct 2026 (Zenodo and arXiv leftovers)

    Awaiting reviewLean-verified

  103. 108 days after the cutoff

    Brennan’s conjecture: experts confirm and shorten OpenAI’s proof

    Result confirmedLean-verified

  104. 108 days after the cutoff

    Gauss circle problem: exponent 209/670, below 5/16

    Awaiting reviewPreprint only

  105. 108 days after the cutoff

    Lonely runner: (1+c)/2n bound with ChatGPT 6

    Event confirmedAwaiting reviewPreprint only

  106. 108 days after the cutoff

    Tournament Hamiltonian cycles: GPT-5.6 settles Alon’s question

    Event confirmedAwaiting reviewPreprint only

  107. 108 days after the cutoff

    Smith–Toda V(4) for p ≥ 11, after OpenAI’s p = 1009

    Event confirmedAwaiting reviewPreprint only

  108. 108 days after the cutoff

    arXiv results with disclosed AI help, 15 Oct 2026 listing

    Awaiting reviewPreprint only

  109. 108 days after the cutoff

    Lee–Loh–Sudakov directed-cut conjecture proved with AI, Lean-verified

    Awaiting reviewLean-verified

  110. 108 days after the cutoff

    Equiregular counterexample to the sub-Riemannian Sard conjecture

    Awaiting reviewLean-verified

  111. 107 days after the cutoff

    Exact asymptotic rate of Delsarte’s linear programming bound for binary and spherical codes found, with key steps from Claude Opus 5.5 and GPT-5.6 Sol

    Awaiting reviewPreprint only

  112. 107 days after the cutoff

    Hopf’s Euler characteristic conjecture proved under T³ symmetry

    Awaiting reviewPreprint only

  113. 107 days after the cutoff

    All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI’s Siegel-zero proof made explicit in Lean with Claude Opus 5.5

    Awaiting reviewLean-verified

  114. 107 days after the cutoff

    Ghys’s 1995 conjecture proved in the codimension-one case with Shanghai AI Lab’s Intern-S2

    Result confirmedLean-verified

  115. 106 days after the cutoff

    Lü, Wang & Wu claim µ(p+2) = o(π(x)), a model case of the parity problem open since at least 1989, building on an OpenAI preprint with ChatGPT’s help

    Awaiting reviewPreprint only

  116. 106 days after the cutoff

    Mid-October 2026: arXiv results with disclosed AI help

    Awaiting reviewPreprint only

  117. 105 days after the cutoff

    Eight extremal combinatorialists write up GPT-6 Astra’s proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)

    Result confirmedLean-verified

  118. 105 days after the cutoff

    Thompson’s group F is nonamenable

    Result confirmedLean-verified

  119. 105 days after the cutoff

    Ulam’s question answered

    Event confirmedAwaiting reviewPreprint only

  120. 105 days after the cutoff

    October 2026: the arXiv stream of conjectures settled with disclosed AI help continues

    Awaiting reviewPreprint only

  121. 104 days after the cutoff

    Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved

    Awaiting reviewLean proof, not yet reviewed

  122. 104 days after the cutoff

    Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)

    Awaiting reviewLean-verified

  123. 104 days after the cutoff

    Erdős Matching Conjecture (1965) proved for every fixed k and all large s, via a sharp tail bound whose k ≥ 6 cases generative AI completed

    Awaiting reviewPreprint only

  124. 104 days after the cutoff

    Chow polynomials of matroids are not log-concave

    Result confirmedExpert-checked

  125. 103 days after the cutoff

    William Banks writes out OpenAI’s Landau–Siegel proof with explicit constants

    Awaiting reviewLean-verified

  126. 103 days after the cutoff

    Skryabin’s 2006 conjecture proved

    Awaiting reviewPreprint only

  127. 102 days after the cutoff

    Vaught’s conjecture fails for infinitary logic

    Event confirmedAwaiting reviewLean-verified

  128. 102 days after the cutoff

    Erdős #1117 on maximum-modulus points answered negatively with GPT-6 help

    Awaiting reviewPreprint only

  129. 101 days after the cutoff

    Thurston’s geometrization conjecture formalised in Lean

    Event confirmedAwaiting reviewLean proof, not yet reviewed

  130. 101 days after the cutoff

    “We need ABP”: six PDE specialists publish GPT-6 Astra’s proofs of the kinetic Krylov–Safonov theorem and of Pucci’s 1966 conjecture, with full chats (Loher, Mooney, Mouhot, Niebel, Silvestre, Zhu)

    Awaiting reviewPreprint only

  131. 101 days after the cutoff

    Smale’s sixth problem claimed for five bodies in the plane

    Awaiting reviewPeer-reviewed

  132. 101 days after the cutoff

    Terence Tao’s Caltech lecture ‘Math 2.0’

    Result confirmedUnverified claim

  133. 101 days after the cutoff

    arXiv, 5–9 October 2026

    Awaiting reviewPreprint only

  134. 101 days after the cutoff

    Gemini Argon shrinks Goldbach, Riemann and ZF Turing machines

    Result confirmedLean-verified

  135. 101 days after the cutoff

    Erdős #32: no O(log N) complement to the primes (Radomskii)

    Awaiting reviewPreprint only

  136. 101 days after the cutoff

    Nirenberg’s 1962 conjecture on closed asymptotic curves disproved

    Event confirmedAwaiting reviewPreprint only

  137. 101 days after the cutoff

    Random permutations of quadratic length contain every pattern

    Event confirmedAwaiting reviewPreprint only

  138. 100 days after the cutoff

    Google DeepMind’s AlphaProof Nexus paper published in Science

    Result confirmedPeer-reviewed

  139. 100 days after the cutoff

    Engel & Mauri post a 7-page proof of the hyperkähler SYZ conjecture, two days after OpenAI’s catalogue claimed it; AI used only to proofread

    Event confirmedAwaiting reviewPreprint only

  140. 100 days after the cutoff

    Erdős Problem #522 (1961) resolved with a Lean-verified proof

    Result confirmedLean-verified

  141. 100 days after the cutoff

    Hilbert’s 16th problem

    Awaiting reviewLean-verified

  142. 100 days after the cutoff

    Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)

    Awaiting reviewPreprint only

  143. 100 days after the cutoff

    Set theorist Asaf Karagila refuses to review OpenAI’s claimed solution of the Partition Principle problem (does PP imply AC?): ‘It sucked’; calls the 722-paper release a ‘Denial of Service’

    Event confirmedAwaiting reviewUnverified claim

  144. 99 days after the cutoff

    Exact overlaps conjecture for self-similar measures on the line proved

    Event confirmedAwaiting reviewLean-verified

  145. 99 days after the cutoff

    Two claimed proofs of the LeBrun–Salamon conjecture

    Awaiting reviewPreprint only

  146. 99 days after the cutoff

    Toms–Winter conjecture on C*-algebras completed by two preprints in one week

    Awaiting reviewPreprint only

  147. 99 days after the cutoff

    Hovey–Strickland: author says OpenAI’s paper matches his ChatGPT-uploaded draft

    Awaiting reviewPreprint only

  148. 99 days after the cutoff

    Claimed proof of the cyclicity conjecture of Perron–Frobenius theory for positive operators (Dobrick, GPT-5.6 used to simplify arguments)

    Awaiting reviewPreprint only

  149. 98 days after the cutoff

    Artin–Davenport conjecture proved

    Awaiting reviewPreprint only

  150. 98 days after the cutoff

    OpenAI model claims the free group factor problem

    Result confirmedExpert-checked

  151. 98 days after the cutoff

    Harary–Hill and Zarankiewicz crossing-number conjectures (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  152. 98 days after the cutoff

    OpenAI release proves the rational Hodge conjecture for all CM abelian varieties, giving the Tate conjecture for abelian varieties over finite fields; Lean proof added 26 Oct, human-written proof by van Hoften and Sempliner, and Milne calls it proved

    Result confirmedLean-verified

  153. 98 days after the cutoff

    OpenAI claims Baum–Connes and Kadison–Kaplansky counterexamples

    Awaiting reviewPreprint only

  154. 98 days after the cutoff

    OpenAI claims 4D Borel counterexample and that F₂ is not good

    Awaiting reviewPreprint only

  155. 98 days after the cutoff

    Borsuk’s conjecture fails in dimension 9 (OpenAI claim, Lean-checked)

    Awaiting reviewPeer-reviewed

  156. 98 days after the cutoff

    Cannon’s conjecture claimed (OpenAI, Lean); Bonk’s seminar talk

    Awaiting reviewPeer-reviewed

  157. 98 days after the cutoff

    OpenAI: Catalan’s constant is irrational (Lean)

    Awaiting reviewLean-verified

  158. 98 days after the cutoff

    Deligne–Drinfeld conjecture proved; experts simplify and reprove OpenAI’s proof

    Result confirmedLean-verified

  159. 98 days after the cutoff

    Erdős’s arithmetic-progression conjecture claimed (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  160. 98 days after the cutoff

    Finitely presented Burnside problem answered negatively? (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  161. 98 days after the cutoff

    OpenAI claims the aspherical positive-scalar-curvature conjecture

    Awaiting reviewExpert-checked

  162. 98 days after the cutoff

    Hadwiger’s conjecture disproved? (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  163. 98 days after the cutoff

    Hilbert’s 16th problem: uniform limit-cycle bound claimed (OpenAI)

    Awaiting reviewLean-verified

  164. 98 days after the cutoff

    Hilbert–Smith conjecture claimed by OpenAI, Lean-checked by a separate AI proof

    Awaiting reviewLean-verified

  165. 98 days after the cutoff

    OpenAI claims Hilbert’s tenth problem over ℚ is undecidable

    Awaiting reviewPreprint only

  166. 98 days after the cutoff

    OpenAI claims Kadison’s similarity problem (Lean)

    Awaiting reviewPeer-reviewed

  167. 98 days after the cutoff

    Kaplansky’s zero-divisor and direct-finiteness conjectures refuted? (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  168. 98 days after the cutoff

    OpenAI proves the Mahler conjectures (Lean)

    Awaiting reviewLean-verified

  169. 98 days after the cutoff

    OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems

    Event confirmedAwaiting reviewLean proof, not yet reviewed

  170. 98 days after the cutoff

    Chromatic number of the plane is 6 or 7, OpenAI claims (Lean)

    Awaiting reviewPeer-reviewed

  171. 98 days after the cutoff

    OpenAI claims global smoothness for 3D relativistic Vlasov–Maxwell (Lean)

    Awaiting reviewPeer-reviewed

  172. 98 days after the cutoff

    OpenAI: Yau’s nodal bound holds on surfaces, fails in higher dimensions

    Awaiting reviewPeer-reviewed

  173. 98 days after the cutoff

    OpenAI model claims the quasi-Riemann hypothesis

    Awaiting reviewLean proof, not yet reviewed

  174. 98 days after the cutoff

    Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau ‘finite index over treeable’ problem (2002); ChatGPT 6 Astra Ultra supplied a ‘crucial observation’

    Event confirmedAwaiting reviewPreprint only

  175. 98 days after the cutoff

    Woodin’s question answered

    Awaiting reviewPreprint only

  176. 98 days after the cutoff

    Generalized Lax conjecture: OpenAI counterexample claim; AI-found generic case (Kummer–Netzer)

    Awaiting reviewLean-verified

  177. 98 days after the cutoff

    Hironaka’s resolution of singularities formalized in Lean by agents

    Result confirmedLean-verified

  178. 98 days after the cutoff

    OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges

    Awaiting reviewLean-verified

  179. 98 days after the cutoff

    OpenAI claims abelian Zilber–Pink

    Awaiting reviewPreprint only

  180. 98 days after the cutoff

    OpenAI claims Artin’s primitive root conjecture (infinitude)

    Awaiting reviewPreprint only

  181. 98 days after the cutoff

    OpenAI claims Bloch’s conjecture for surfaces

    Awaiting reviewExpert-checked

  182. 98 days after the cutoff

    OpenAI claims BSD in Selmer corank ≤ 1 and Goldfeld’s conjecture

    Awaiting reviewPreprint only

  183. 98 days after the cutoff

    Infinitely many closed geodesics on every sphere, claimed and re-proved

    Awaiting reviewPreprint only

  184. 98 days after the cutoff

    OpenAI claims De Giorgi’s conjecture in dimension eight

    Awaiting reviewPreprint only

  185. 98 days after the cutoff

    OpenAI proves the Falconer distance conjecture (Lean)

    Awaiting reviewLean-verified

  186. 98 days after the cutoff

    OpenAI claims the planar hot spots conjecture (Lean)

    Awaiting reviewLean-verified

  187. 98 days after the cutoff

    OpenAI claims the abundance conjecture; Jihao Liu raises priority

    Awaiting reviewPreprint only

  188. 98 days after the cutoff

    OpenAI: elliptic curves over imaginary quadratic fields are modular

    Awaiting reviewPreprint only

  189. 98 days after the cutoff

    OpenAI claims a counterexample to the nearby Lagrangian conjecture

    Awaiting reviewPreprint only

  190. 98 days after the cutoff

    Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)

    Awaiting reviewPeer-reviewed

  191. 98 days after the cutoff

    OpenAI claims 3D restriction and Bochner–Riesz conjectures

    Awaiting reviewPreprint only

  192. 98 days after the cutoff

    Sidorenko’s conjecture disproved? (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  193. 98 days after the cutoff

    Unit-distance bound below n^(4/3) claimed (OpenAI, Lean)

    Awaiting reviewPeer-reviewed

  194. 98 days after the cutoff

    doubleAI’s ‘Double Agent’ claims explicit regular graphs of girth (8/5)·log n, beating the 4/3 constant of Margulis and Lubotzky–Phillips–Sarnak (1988)

    Awaiting reviewPeer-reviewed

  195. 98 days after the cutoff

    Walter Trump’s 1979 packing of 11 unit squares proved optimal, with a full Lean verification (AI involvement reported, unconfirmed)

    Awaiting reviewLean-verified

  196. 98 days after the cutoff

    Optimal bound for the polynomial Littlewood–Offord problem, ‘discovered autonomously by GPT-6 Pro’ (exposition by Grebennikov)

    Awaiting reviewPreprint only

  197. 97 days after the cutoff

    Teen’s Claude-coded proof claims rhombicosidodecahedron is not Rupert

    Awaiting reviewExpert-checked

  198. 96 days after the cutoff

    Linear Hadwiger conjecture proved

    Event confirmedAwaiting reviewLean proof, not yet reviewed

  199. 94 days after the cutoff

    Kannan–Lovász–Simonovits conjecture proved in two AI-assisted preprints

    Awaiting reviewPreprint only

  200. 94 days after the cutoff

    Klartag and Moshe prove the ε-Dvoretzky conjecture

    Event confirmedAwaiting reviewPreprint only

  201. 94 days after the cutoff

    Linnik’s constant ≤ 3.99, written by Claude Opus 5.5 and GPT Astra

    Awaiting reviewLean-verified

  202. 94 days after the cutoff

    Meta publishes six maths papers made by mathematicians working with Muse Spark in ordinary meta.ai chat, saying five answer open questions

    Awaiting reviewUnverified claim

  203. 94 days after the cutoff

    ChatGPT Astra finds geometric triangle-free graphs with near-optimal chromatic number, the first improvement on Burling’s 1965 box bound

    Event confirmedAwaiting reviewPreprint only

  204. 93 days after the cutoff

    Lehmer’s 1965 permutation conjecture proved

    Event confirmedAwaiting reviewLean-verified

  205. 92 days after the cutoff

    Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help

    Awaiting reviewPreprint only

  206. 92 days after the cutoff

    mt0-svg: AI-made Tammes N=15, Robbins V>2, frog model (Lean)

    Awaiting reviewLean-verified

  207. 92 days after the cutoff

    GPT-6 Astra proves Barvinok’s log-concavity question for contingency tables on lines, giving lattice-point bounds for all totally unimodular polytopes

    Event confirmedAwaiting reviewPreprint only

  208. 91 days after the cutoff

    List Total Colouring Conjecture (late 1990s) disproved

    Event confirmedAwaiting reviewPreprint only

  209. 91 days after the cutoff

    EPFL’s LinCodeEvolve (Viazovska, Abbe) finds seven record-breaking binary linear codes with LLM-guided program search

    Result confirmedExpert-checked

  210. 91 days after the cutoff

    GPT-5.6 Sol supplies key steps in a nearly linear bound for the Lovász path conjecture

    Event confirmedAwaiting reviewPreprint only

  211. 90 days after the cutoff

    Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)

    Awaiting reviewPreprint only

  212. 90 days after the cutoff

    Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol

    Awaiting reviewPreprint only

  213. 90 days after the cutoff

    Caltech ‘Mathathon’ is reworked into ‘Old Problems, New Proofs’ after an open letter from mathematicians

    Result confirmedUnverified claim

  214. 89 days after the cutoff

    Erdős Problem #993 (1987)

    Awaiting reviewLean-verified

  215. 89 days after the cutoff

    Perelman’s proof of the Poincaré conjecture formalised in Lean, with AI-generated code

    Result confirmedUnverified claim

  216. 88 days after the cutoff

    GPT-6 Astra finds a proof improving the Kővári–Sós–Turán bound

    Event confirmedAwaiting reviewPreprint only

  217. 87 days after the cutoff

    Conjectures.io bounty platform pays out for Lean proofs of Erdős #859, #18(b), #1062(ii) and Ben Green’s problems 24, 39, 40 within ten days, mostly to Purdue’s ‘JenW1N’

    Awaiting reviewLean-verified

  218. 87 days after the cutoff

    New record bound on the irrationality measure of ζ(2), μ ≤ 5.0495243, released as a 92k-line Lean proof written by Claude; beaten a week later by a Claude-assisted paper

    Result confirmedLean-verified

  219. 86 days after the cutoff

    Benjamini–Schramm p_c < p_u: OpenAI bond proof in Lean, Claude site form

    Awaiting reviewLean-verified

  220. 84 days after the cutoff

    Mathematician posts an unchecked ChatGPT Astra proof of the planar Mumford–Shah conjecture (1989), citing OpenAI’s ‘100 open problems’ claim

    Awaiting reviewPreprint only

  221. 84 days after the cutoff

    Odlyzko–Poonen conjecture (1993) proved unconditionally

    Result confirmedLean-verified

  222. 83 days after the cutoff

    Courtade–Kumar ‘most informative Boolean function’ conjecture (2013) proved three times in two days, all with AI: Ky & Tran (ChatGPT), Google + CUHK (Gemini, Lean-verified end-to-end), Mahdavifar & Beirami

    Event confirmedAwaiting reviewLean-verified

  223. 83 days after the cutoff

    OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)

    Awaiting reviewPreprint only

  224. 82 days after the cutoff

    Kung–Traub conjecture proved, Lean check with ChatGPT 6

    Awaiting reviewLean-verified

  225. 79 days after the cutoff

    ζ(5) proved irrational

    Result confirmedLean-verified

  226. 78 days after the cutoff

    Chvátal’s 1972 conjecture proved (Chang–Liu–Liu, ChatGPT-assisted), then a GPT-6 Astra ‘proof from The Book’ and a Codex-built Lean formalization

    Result confirmedLean-verified

  227. 78 days after the cutoff

    Lean-verified ‘Liouville Goldbach’ theorem goes viral as a GPT-6 Astra ‘Goldbach breakthrough’; the AI role is unconfirmed and it is not the Goldbach conjecture

    Awaiting reviewLean-verified

  228. 77 days after the cutoff

    Erdős–Hajnal high-girth problem (Erdős #108) disproved with ChatGPT/Codex help and a Lean proof, via the Conjectures.io bounty; experts sharpen it with GPT-6 Astra

    Result confirmedLean-verified

  229. 76 days after the cutoff

    Sylvester’s 1879 conjecture on sums of two cubes is fully proved

    Awaiting reviewPreprint only

  230. 73 days after the cutoff

    Fields Medallists’ open letter ‘A Severe Misalignment of AI in Mathematics’ criticises labs’ race for famous problems

    Result confirmedUnverified claim

  231. 73 days after the cutoff

    Medvedev’s logic of finite problems (1962) shown undecidable

    Event confirmedAwaiting reviewLean-verified

  232. 72 days after the cutoff

    GPT-6 Astra’s Epoch AI run adds more Lean-checked results

    Awaiting reviewLean proof, not yet reviewed

  233. 72 days after the cutoff

    SAIR’s IGP24 challenge

    Result confirmedUnverified claim

  234. 72 days after the cutoff

    Irrationality measure of π lowered to 7.101862832357, the first improvement on Zeilberger–Zudilin (2019), with GPT-5.6 Sol and GPT-6 Astra; four weeks later OpenAI claimed the exact value 2

    Event confirmedAwaiting reviewPreprint only

  235. 71 days after the cutoff

    Pierce–Birkhoff conjecture disproved by a multi-agent GPT + Claude harness

    Event confirmedAwaiting reviewLean proof, not yet reviewed

  236. 70 days after the cutoff

    OpenAI claims a Millennium Prize problem

    DisputedLean-verified

  237. 69 days after the cutoff

    Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample

    Result confirmedLean-verified

  238. 69 days after the cutoff

    Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help

    Awaiting reviewUnverified claim

  239. 69 days after the cutoff

    Erdős’s 1946 conjecture on almost-monotone additive functions (Erdős problem #1122) proved in an anonymous Zenodo preprint whose proofs and Lean code were generated by GPT-6 Astra; Mangerel builds on it to settle Sárközy’s 2001 conjectures

    Awaiting reviewLean-verified

  240. 68 days after the cutoff

    Collatz positive-density convergence, AI-developed, Lean-checked

    Result confirmedLean-verified

  241. 68 days after the cutoff

    Preprint credits the ‘Odin’ AI research agent with a positively curved metric on S³×S³, which would disprove Hopf’s Euler-characteristic sign conjecture (unverified)

    Awaiting reviewPreprint only

  242. 66 days after the cutoff

    Claude produces the first complete machine-checked proof of Fermat’s Last Theorem in Lean, in 11 days

    Result confirmedLean-verified

  243. 65 days after the cutoff

    Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension

    Awaiting reviewLean proof, not yet reviewed

  244. 65 days after the cutoff

    GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument

    Result confirmedLean-verified

  245. 65 days after the cutoff

    Pre-release GPT-6 Astra disproves Erdős’s ‘first serious problem’ (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch’s FrontierMath Erdős runs

    Result confirmedLean-verified

  246. 65 days after the cutoff

    GPT-6 Astra autonomously disproves Smale’s mean value conjecture in Lean

    Result confirmedLean-verified

  247. 61 days after the cutoff

    GPT-6 Astra lowers the bounded prime gaps record from 246 to 186

    Awaiting reviewLean proof, not yet reviewed

  248. 59 days after the cutoff

    Matrix Spencer conjecture proved

    Awaiting reviewPreprint only

  249. 57 days after the cutoff

    GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps

    Awaiting reviewLean proof, not yet reviewed

  250. 54 days after the cutoff

    Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf’s 1947 problem (pending verification)

    Awaiting reviewLean proof, not yet reviewed

  251. 54 days after the cutoff

    Claude-assisted search breaks the elliptic curve rank record

    Result confirmedExpert-checked

  252. 50 days after the cutoff

    Smooth Carathéodory and Loewner conjectures disproved with Claude

    Awaiting reviewPreprint only

  253. 50 days after the cutoff

    Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics

    Awaiting reviewPreprint only

  254. 49 days after the cutoff

    Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims

    Result confirmedLean-verified

  255. 44 days after the cutoff

    Banach’s isometric conjecture (1932) completed in the real case with key steps from ChatGPT 5.5/5.6 Pro; complex and quaternionic cases follow five days later

    Awaiting reviewPreprint only

  256. 43 days after the cutoff

    Claude-assisted constructions complete Hadamard matrices for every order below 2000

    Result confirmedPreprint only

  257. 41 days after the cutoff

    Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%)

    Result confirmedLean-verified

  258. 36 days after the cutoff

    Sendov’s 1958 conjecture on polynomial roots proved with GPT-5.6 Pro

    Result confirmedLean-verified

  259. 36 days after the cutoff

    HRT conjecture (1996) disproved

    Event confirmedAwaiting reviewPreprint only

  260. 35 days after the cutoff

    Alexander Perry disproves the period-index conjecture

    Event confirmedAwaiting reviewPreprint only

  261. 34 days after the cutoff

    Planar Schiffer and Pompeiu conjectures disproved by two independent groups

    Event confirmedAwaiting reviewLean-verified

  262. 32 days after the cutoff

    OpenAI’s unreleased ‘Astra’ model claims ten advances in maths and theoretical CS, with Lean proofs

    Awaiting reviewLean-verified

  263. 27 days after the cutoff

    Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix’s conjecture in a 16-hour autonomous run

    Result confirmedExpert-checked

  264. 25 days after the cutoff

    Sharp thin-shell inequality

    Event confirmedAwaiting reviewPreprint only

  265. 24 days after the cutoff

    Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample

    Event confirmedAwaiting reviewPreprint only

  266. 24 days after the cutoff

    Rota’s 1970 unimodality conjecture for matroid flats disproved, with ChatGPT 5.6 Pro suggesting a key construction

    Result confirmedPreprint only

  267. 24 days after the cutoff

    Terence Tao’s ICM 2026 public lecture ‘Mathematics in the age of AI’ calls a crisis in the foundations of mathematical values

    Result confirmedUnverified claim

  268. 23 days after the cutoff

    AI systems score a perfect 42/42 at IMO 2026, officially graded

    Result confirmedExpert-checked

  269. 22 days after the cutoff

    Dmitry Rybin posts a GPT-5.6 Pro counterexample to the ~30-year-old Dinitz–Garg–Goemans (Goemans cost) conjecture on unsplittable flows

    Awaiting reviewExpert-checked

  270. 22 days after the cutoff

    Columbia PhD student Shouqiao Wang says he solved six open Erdős problems in five days with GPT-5.6 Sol in Codex, and publishes his workflow

    Awaiting reviewExpert-checked

  271. 20 days after the cutoff

    Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3

    Result confirmedUnverified claim

  272. 17 days after the cutoff

    GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture

    Result confirmedExpert-checked

  273. 14 days after the cutoff

    Berkeley’s Phillip Kerger uses GPT-5.6 Sol and a 10-page prompt to close a 30-year gap in derivative-free convex optimization

    Awaiting reviewPreprint only

  274. After June 2026

    AI-assisted counterexample answers Grothendieck’s question on finite flat group schemes, merged into Mathlib

    Result confirmedLean-verified

Physics

25 results

  1. 130 days after the cutoff

    Anthropic’s Claude physics papers: Hawking–Page and modular bootstrap

    Awaiting reviewUnverified claim

  2. 123 days after the cutoff

    Onsager hard rods: AI-written first-order transition claim, not human-verified

    Awaiting reviewUnverified claim

  3. 122 days after the cutoff

    Hopfield model thermodynamic limit, AI-generated proofs

    Awaiting reviewPreprint only

  4. 118 days after the cutoff

    OpenAI claims 4D Yang–Mills and QCD existence

    Awaiting reviewLean-verified

  5. 118 days after the cutoff

    3D SU(2) Yang–Mills continuum limit claimed in five AI-written papers

    Awaiting reviewLean-verified

  6. 117 days after the cutoff

    AI-assisted results in the 25 Oct 2026 sweep

    Awaiting reviewPreprint only

  7. 108 days after the cutoff

    OpenAI’s 2D area law proof Lean-verified by Axiom

    Awaiting reviewLean-verified

  8. 108 days after the cutoff

    Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban’s 1980s program

    Awaiting reviewPreprint only

  9. 108 days after the cutoff

    Das Sarma paper written with Claude Opus 5.5, refereed by Codex

    Event confirmedAwaiting reviewPreprint only

  10. 107 days after the cutoff

    CERN and Mistral plan a Large Physics Model

    Event confirmedAwaiting reviewUnverified claim

  11. 106 days after the cutoff

    Bray, Dolmen, Khuri & Schmidt prove the spacetime Penrose inequality for ‘maximally parameterized’ data with Claude’s help, and say OpenAI’s Oct 6 Penrose paper appears to finish their program

    Event confirmedAwaiting reviewPreprint only

  12. 104 days after the cutoff

    “The proof presented in this paper is due to Claude Opus 5.5”

    Result confirmedPreprint only

  13. 100 days after the cutoff

    Full Lyapunov spectrum of the Sompolinsky–Crisanti–Sommers random neural network derived, a problem named open in 1988; GPT-6 Astra produced the first derivation (David G. Clark, Flatiron)

    Event confirmedAwaiting reviewPreprint only

  14. 100 days after the cutoff

    PPT-distillable entanglement can fall strictly below the regularised Rains bound

    Awaiting reviewPreprint only

  15. 98 days after the cutoff

    OpenAI claims Anderson localization in 2D and extended states in 3D

    Awaiting reviewLean proof, not yet reviewed

  16. 98 days after the cutoff

    OpenAI claims quantum ferromagnet order and interacting Bose condensation

    Awaiting reviewPeer-reviewed

  17. 98 days after the cutoff

    OpenAI claims strong cosmic censorship near Kerr

    Awaiting reviewPreprint only

  18. 98 days after the cutoff

    2D O(n) mass gap (Polyakov) and O(3) continuum limit claimed (OpenAI)

    Awaiting reviewPeer-reviewed

  19. 98 days after the cutoff

    OpenAI proves the spin-one Haldane gap (periodic chains)

    Awaiting reviewPreprint only

  20. 87 days after the cutoff

    Claude (Fable 5.1 in Claude Science) computes the nine-loop six-gluon amplitude in planar N=4 super-Yang-Mills, answering a physicist’s public challenge

    Result confirmedExpert-checked

  21. 83 days after the cutoff

    Grad’s 1967 conjecture on 3D plasma equilibria falls

    Result confirmedLean-verified

  22. 67 days after the cutoff

    The I3322 Bell inequality needs infinite dimensions

    Event confirmedAwaiting reviewPreprint only

  23. 65 days after the cutoff

    PPPL’s PACMAN framework lets multiple AI models control a tokamak in ~20 ms, preventing a tearing mode

    Result confirmedPeer-reviewed

  24. 64 days after the cutoff

    Nature: ‘Designing physics experiments with artificial intelligence’ (Krenn group) on AI-found setups that beat human designs

    Result confirmedPeer-reviewed

  25. 29 days after the cutoff

    Maxwell’s conjecture on equilibria of point charges is false

    Event confirmedAwaiting reviewPreprint only

Computer science

44 results

  1. 135 days after the cutoff

    A BST meets Iacono’s 2001 unified bound, with Claude and GPT

    Awaiting reviewPreprint only

  2. 135 days after the cutoff

    Grid-minor exponent 6 in polynomial time (Hajiaghayi with Astra)

    Awaiting reviewLean-verified

  3. 129 days after the cutoff

    PSI claims Lean-checked proof of the Aaronson–Ambainis conjecture

    Awaiting reviewLean-verified

  4. 128 days after the cutoff

    3-uniform hyperclique hypothesis refuted with ChatGPT Sol 6.1

    Event confirmedAwaiting reviewPreprint only

  5. 127 days after the cutoff

    Colourings Glauber dynamics above (1+1/e)Δ (GPT-6 Astra Pro)

    Awaiting reviewPreprint only

  6. 127 days after the cutoff

    Huffman coding with unequal letter costs is NP-complete (Codex)

    Awaiting reviewPreprint only

  7. 127 days after the cutoff

    2×2 matrix mortality claimed undecidable (AI-assisted, Lean)

    Awaiting reviewLean-verified

  8. 126 days after the cutoff

    FPRAS for mixed volumes, ideas from GPT-6 Astra

    Awaiting reviewPreprint only

  9. 121 days after the cutoff

    Deterministic factoring in N^o(1) steps, Lean-checked, ideas by Claude

    Awaiting reviewLean-verified

  10. 121 days after the cutoff

    Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra

    Event confirmedAwaiting reviewLean-verified

  11. 119 days after the cutoff

    Anthropic’s Claude also refutes ETH

    Awaiting reviewLean-verified

  12. 119 days after the cutoff

    Unbounded-error communication is not closed under AND, found with GPT-6 Astra

    Awaiting reviewPreprint only

  13. 118 days after the cutoff

    OpenAI claims ETH is false

    Awaiting reviewLean-verified

  14. 118 days after the cutoff

    3-colorable graphs: O(n^0.1537) colors with GPT-6 Astra

    Awaiting reviewPreprint only

  15. 117 days after the cutoff

    Formula minimization Σ₂ᵖ-complete (proof found by GPT-Sol-6.1)

    Awaiting reviewPreprint only

  16. 115 days after the cutoff

    Graph connectivity communication bound settled, with ChatGPT

    Awaiting reviewPreprint only

  17. 111 days after the cutoff

    Integer programming in standard form in N^O(d) time, improving Eisenbrand–Weismantel’s N^O(d²): Zhejiang team’s ideas, GPT Astra Ultra’s proofs (Chen, Mao, Zhang)

    Awaiting reviewLean-verified

  18. 111 days after the cutoff

    Counting perfect matchings in general graphs: OpenAI FPRAS, simplified proof

    Result confirmedLean-verified

  19. 106 days after the cutoff

    Li, Li & Liu: external-memory integer sorting beats the comparison bound, answering Aggarwal–Vitter (1988) by way of network coding, with ChatGPT/Codex and Lean

    Awaiting reviewLean-verified

  20. 105 days after the cutoff

    k-median breaks the factor-2 barrier

    Event confirmedAwaiting reviewLean-verified

  21. 101 days after the cutoff

    Subset Sum in O(n + N^(1+ε)) time claimed

    Awaiting reviewLean-verified

  22. 100 days after the cutoff

    GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH

    Awaiting reviewPreprint only

  23. 100 days after the cutoff

    Laderman’s 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers

    Awaiting reviewLean-verified

  24. 99 days after the cutoff

    Outside researchers using AI crowdsource a tighter exponent in OpenAI’s integer-multiplication result #109, from 2^-182 to about 2^-8.4 (conditional)

    Event confirmedAwaiting reviewLean-verified

  25. 99 days after the cutoff

    Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)

    Event confirmedAwaiting reviewPreprint only

  26. 99 days after the cutoff

    Bolzano (Charles University)

    Awaiting reviewExpert-checked

  27. 98 days after the cutoff

    OpenAI proves ω ≤ 9/4 for matrix multiplication

    Result confirmedLean-verified

  28. 98 days after the cutoff

    OpenAI model claims a proof of Khot’s Unique Games Conjecture, with Lean formalization

    Awaiting reviewLean proof, not yet reviewed

  29. 98 days after the cutoff

    OpenAI model claims tight 1 + 2/e approximation for k-median

    Awaiting reviewLean-verified

  30. 98 days after the cutoff

    OpenAI claims L = RL = BPL with a Lean proof

    Awaiting reviewPeer-reviewed

  31. 98 days after the cutoff

    Explicit tensors of border rank 3n − o(n), past the ‘linear flattening barrier’

    Event confirmedAwaiting reviewPreprint only

  32. 98 days after the cutoff

    Why SAT solvers fail on multiplication

    Awaiting reviewLean-verified

  33. 98 days after the cutoff

    Preprint claims a 4.268 upper bound for the random 3-SAT threshold, down from 4.4898, with a Lean 4 proof; ‘large language models drove the research’

    Awaiting reviewLean-verified

  34. 97 days after the cutoff

    Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses

    Event confirmedAwaiting reviewLean-verified

  35. 97 days after the cutoff

    Random k-SAT threshold exists for every k ≥ 3: a human proof and an OpenAI Lean-checked proof

    Awaiting reviewPeer-reviewed

  36. 97 days after the cutoff

    Decreasing diagrams are complete for confluence

    Awaiting reviewLean-verified

  37. 76 days after the cutoff

    The k-server conjecture, the ‘holy grail’ of online algorithms, is proved at Oxford

    Awaiting reviewPreprint only

  38. 76 days after the cutoff

    Friedgut’s 2004 influential-coalitions conjecture resolved

    Result confirmedExpert-checked

  39. 75 days after the cutoff

    Matroid secretary conjecture resolved, with GPT Astra

    Awaiting reviewPreprint only

  40. 66 days after the cutoff

    Reed–Solomon codes list-decoded up to capacity and proximity gaps settled, with GPT-5.6 Sol and ChatGPT 5.6 Pro (Brakensiek–Chen–Putterman–Zhang–Zheng; Jeronimo)

    Awaiting reviewExpert-checked

  41. After June 2026

    NVIDIA’s Nemotron-3-Ultra-CC outscores every human at IOI 2026

    Result confirmedExpert-checked

  42. 58 days after the cutoff

    Google’s Antigravity ‘Teamwork’ multi-agent framework with Gemini 3.7 Flash solves seven open CS/math problems, incl. part of Knuth’s cycles problem

    Awaiting reviewLean-verified

  43. 48 days after the cutoff

    AlphaEvolve helps lower the matrix multiplication exponent ω to below 2.371177

    Result confirmedPreprint only

  44. 28 days after the cutoff

    Claude Mythos Preview finds new cryptanalytic attacks on post-quantum HAWK and 7-round AES

    Event confirmedAwaiting reviewUnverified claim

Biology

15 results

  1. 134 days after the cutoff

    Google DeepMind launches AlphaGenome 2, predicting how DNA variants act in specific cell types

    Result confirmedUnverified claim

  2. 127 days after the cutoff

    Anthropic’s biology team, working with Claude, reports the first 3D structure of a vault particle from bacterial life

    Event confirmedAwaiting reviewPreprint only

  3. 127 days after the cutoff

    Nature: AI agents run an automated biology lab

    Result confirmedPeer-reviewed

  4. 113 days after the cutoff

    Thousands of GPT-Rosalind agents analyse millions of mouse brain cells in 32 minutes and propose a cholesterol-transport lead for Perry syndrome

    Awaiting reviewUnverified claim

  5. 112 days after the cutoff

    Claude designs binders against scorpion neurotoxins

    Event confirmedAwaiting reviewUnverified claim

  6. 106 days after the cutoff

    AI-designed proteases (Baker lab, Nature)

    Result confirmedPeer-reviewed

  7. 101 days after the cutoff

    Machine-learning-guided evolution makes a plant Rubisco 29% faster (MIT, Stanford, ANU

    Result confirmedPeer-reviewed

  8. 97 days after the cutoff

    Google DeepMind’s AlphaProtein Novo designs new-to-nature enzymes from scratch (piperidine synthesis, DEHP degradation), with Frances Arnold’s lab

    Event confirmedAwaiting reviewPreprint only

  9. 91 days after the cutoff

    Microsoft Research unveils Quine, a multimodal ‘world model of biology’ validated in pancreatic-cancer screens with the Broad Institute

    Awaiting reviewUnverified claim

  10. 86 days after the cutoff

    DeepMind, EMBL-EBI, NVIDIA and CEPI add predicted protein-complex structures for 2,800+ viruses to the AlphaFold Database

    Result confirmedUnverified claim

  11. 85 days after the cutoff

    Claude agents discover a novel CRISPR-like enzyme system

    DisputedPreprint only

  12. 81 days after the cutoff

    FutureHouse and Edison Scientific publish twelve ‘Millennium Problems for Biology’, pitched as the ‘last reasonable eval’ for AI in biology

    Event confirmedAwaiting reviewUnverified claim

  13. 79 days after the cutoff

    Science: Stanford’s ‘Virtual Biotech’ of 37,000 AI agents mines ~50,000 clinical trials and designs a lung-cancer ADC later matched by pharma

    Result confirmedPeer-reviewed

  14. 65 days after the cutoff

    Complete male fruit fly nervous-system connectome (166,000 neurons), reconstructed with Google AI, published in Cell

    Result confirmedPeer-reviewed

  15. 49 days after the cutoff

    Claude autonomously designs protein binders that work in the lab against 14 of 15 targets

    Result confirmedExpert-checked

Materials

3 results

  1. 96 days after the cutoff

    Claude agents propose room-temperature magnetic semiconductors

    Awaiting reviewUnverified claim

  2. 87 days after the cutoff

    Lila Sciences’ AI-run lab screens 2,942 catalysts and finds iridium- and ruthenium-free palladium oxides for green hydrogen

    Awaiting reviewPreprint only

  3. 77 days after the cutoff

    Periodic Labs introduces Periodic Neon, a 1T-parameter lab-trained model that beats GPT-6 Astra at X-ray diffraction analysis

    Event confirmedAwaiting reviewUnverified claim

Medicine

7 results

  1. 114 days after the cutoff

    Microsoft MAI-DxO diagnosis paper in Science

    Result confirmedPeer-reviewed

  2. 111 days after the cutoff

    FAHR-Face: Mass General Brigham foundation model estimates biological age and cancer survival from a face photo (Lancet Digital Health)

    Result confirmedPeer-reviewed

  3. 92 days after the cutoff

    Google’s ERA-built model ranks #1 of 39 in the CDC FluSight 2025–26 flu-hospitalization forecasting challenge

    Result confirmedUnverified claim

  4. 77 days after the cutoff

    OpenAI Foundation launches Public Data for Health with $125M+ in grants for open biomedical datasets

    Event confirmedAwaiting reviewUnverified claim

  5. 76 days after the cutoff

    FDA grants priority review to Takeda’s zasocitinib, a computationally designed TYK2 inhibitor, with a decision due Q1 2027

    Event confirmedAwaiting reviewUnverified claim

  6. 72 days after the cutoff

    First Phase III trial of a generative-AI-discovered drug doses first patient

    Event confirmedAwaiting reviewPeer-reviewed

  7. 70 days after the cutoff

    AlphaGenome Atlas predicts the effect of all ~9 billion possible single-letter human DNA variants

    Awaiting reviewUnverified claim

Climate and weather

6 results

  1. 109 days after the cutoff

    China’s Fengqing AI ensemble weather forecast goes operational

    Awaiting reviewUnverified claim

  2. 106 days after the cutoff

    Microsoft PBC improves sub-seasonal weather forecasts (Nature)

    Result confirmedPeer-reviewed

  3. 101 days after the cutoff

    AI weather models forecast Dubai’s unprecedented 2024 storm by ‘translocation’, not extrapolation (University of Chicago, Science Advances)

    Result confirmedPeer-reviewed

  4. 71 days after the cutoff

    Google and NASA JPL release MAPL-EMIT, an AI model that maps methane plumes worldwide

    Result confirmedPeer-reviewed

  5. 65 days after the cutoff

    Google DeepMind’s WeatherNext 3 learns from live satellite data

    Result confirmedExpert-checked

  6. 37 days after the cutoff

    DeepMind open-sources WeatherNext 2 and WeatherNext Cyclones with a Nature paper showing ~1 extra day of hurricane warning

    Result confirmedPeer-reviewed

Astronomy

1 result

  1. 100 days after the cutoff

    Claude Science builds a complete ultraviolet map of the sky

    Event confirmedAwaiting reviewUnverified claim

Other fields

3 results

  1. 133 days after the cutoff

    Viral claim that AI agents found an unmapped Maya site turns out to be a rediscovery of a site published in 2025

    DisputedUnverified claim

  2. 100 days after the cutoff

    AI archive search finds forgotten meteorite and eruptions

    Awaiting reviewUnverified claim

  3. 93 days after the cutoff

    Harvard physicist Matthew Schwartz releases BootLoops, an open-source harness that let Claude produce 36 manuscripts in 18 fields in three months

    Event confirmedAwaiting reviewPreprint only

Credits

We cross-check VibeMathed, Terence Tao’s Erdős wiki (frozen since 30 June 2026) and Epoch AI’s data on AI use in mathematics.