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OpenAI Aftermath is a guide to mathematical research published by OpenAI and to publicly available follow-up work by others: extensions, applications, formal verifications, corrections and critiques.

How entries are written

• Each result page summarizes the published claim and cites its public source: the paper, the repository commit or the preprint version.
• Follow-up work is listed with its relation to the original result (extension, use, verification, correction or critique), its public date and its public status.
• The publication of a paper, code or a formal proof is reported as such. It is not presented as independent verification.

Browse

• All result pages
• Results by number of public follow-ups (sortable overview)
• Number theory:
  • Milne's rationality conjecture
  • Irrationality of Catalan's constant
  • The irrationality exponent of π is 2
  • Modularity of elliptic curves over imaginary quadratic fields
  • The Fontaine–Mazur conjecture at the prime 2
  • The p-adic section conjecture
  • Geometric Langlands and the generalized Ramanujan conjecture over function fields
  • The Birch and Swinnerton-Dyer formula in low Selmer corank
  • Goldfeld's conjecture for quadratic twists
  • The higher-dimensional Duke problem for torus packets
  • The quasi-Riemann hypothesis
  • Uniform exclusion of Landau–Siegel zeros
  • Prime-factor statistics of p − 1
  • Largest prime factors of consecutive integers
  • The Erdős–Hall question on totient values
  • Primitive roots for every admissible integer base
  • Bounded gaps between primes
  • Long gaps between primes
  • Positive lower density of large prime gaps
  • A quadratic bound for Jacobsthal's function
  • The inhomogeneous Duffin–Schaeffer conjecture
  • Hilbert's tenth problem over the rational numbers
  • Ostmann's inverse Goldbach conjecture
  • Erdős's conjecture on short Egyptian fractions
  • The Gaussian moat problem
  • The two-point Chowla conjecture
• Algebraic geometry:
  • The rational Hodge conjecture for CM abelian varieties
  • Griffiths' positivity conjecture
  • The Kollár–Pardon conjecture on semialgebraic universal covers
  • Anticanonical nonvanishing under smooth semipositivity
  • The abelian Zilber–Pink conjecture
  • The ordinary double point volume gap
  • The abundance conjecture for threefolds in positive characteristic
  • The hyperkähler SYZ conjecture
  • The Zariski cancellation problem over the complex numbers
  • The stable coordinate and Abhyankar–Sathaye conjectures
  • Nagata's conjecture on plane curves
  • Bloch's conjecture for surfaces with geometric genus zero
  • Zariski's multiplicity conjecture
  • The equivariant cohomological Hikita conjecture
  • Gepner and large-volume stability conditions on threefolds
  • Kuznetsov's rationality conjecture
  • The generalized Mukai conjecture
  • Fujita's freeness conjecture
  • The Lipman–Zariski conjecture
  • The Campana–Peternell conjecture
• Convex geometry:
  • The Mahler conjectures
  • The logarithmic Brunn–Minkowski conjecture
  • Petty's projection-volume conjecture
  • The optimal order of convex-body covering density
  • The generalized Lax conjecture
  • The strong isotropic constant conjecture
• Algebraic complexity:
  • Matrix multiplication with exponent at most 9/4
  • A cubic lower bound for the permanent versus the determinant
  • Black-box identity testing for noncommutative formulas
• Computational complexity:
  • The Unique Games Conjecture
  • The 2-to-1 Games Conjecture
  • The existential theory of the reals in the counting hierarchy
  • Worst-case Subset Sum below the meet-in-the-middle bound
  • Sensitivity versus block sensitivity
  • Depth-three circuit lower bounds beyond the square-root exponent
  • The Gotsman–Linial conjecture
  • Exponential semidefinite complexity of perfect matching
  • Hardness of coloring three-colorable graphs
  • The Sakoda–Sipser problem
  • Generalized star height at most three
  • Derandomization of logarithmic space
  • Moore's parity conjecture for constant-depth quantum circuits
  • The computational complexity of Weisfeiler–Leman refinement
• Group theory:
  • Classifying spaces and geometric obstructions for Artin groups
  • Thompson's group F is nonamenable
  • Infinite finitely presented simple amenable groups
  • Dixmier's unitarizability problem
  • Kaplansky's zero-divisor conjecture
  • Kaplansky's direct-finiteness conjecture
  • The Bass trace conjecture
  • The finitely presented Burnside problem
  • The Kervaire conjecture
  • Residual finiteness of hyperbolic groups
  • The Boone–Higman conjecture
  • Gersten's conjecture and virtual specialness of one-relator groups
• Algebra:
  • The Deligne–Drinfeld conjecture
  • Lech's multiplicity conjecture
  • Hochster's small Cohen–Macaulay module conjecture
  • Serre's positivity conjecture for intersection multiplicities
  • Foulkes' conjecture
  • The Benson–Etingof–Ostrik conjecture for finite tensor categories
  • The Eisenbud–Green–Harris and lex-plus-powers conjectures
  • The combinatorial invariance conjecture for Kazhdan–Lusztig polynomials
• Homotopy theory:
  • The Grothendieck homotopy hypothesis
  • Smith–Toda complexes at every height
  • The Kervaire invariant problem at the prime three
  • The Hovey–Strickland conjecture
  • The chromatic splitting conjecture
  • The Hahn–Wilson conjecture
  • Curtis's conjecture
  • Wall's D(2) problem
  • Thomason model structures on strict higher categories
  • Finite generation of the homotopy of the K(n)-local sphere
• Geometric topology:
  • The Hilbert–Smith conjecture
  • Cannon's conjecture
• Representation theory of algebras:
  • The little finitistic dimension conjecture
  • The Auslander–Reiten conjecture
  • Alperin's weight conjecture
  • Donovan's conjecture
• Mathematical logic:
  • Rigidity of the Turing degrees
  • The Barendregt–Geuvers–Klop conjecture
  • The Partition Principle and the Axiom of Choice
  • Single-fold Diophantine representations
• Algorithms:
  • A factor-two approximation for shortest common superstring
  • Exact Fourier transforms below n log n
  • Integer multiplication below n log n
  • Deterministic polynomial factorization over prime fields
  • Almost-linear-time maximum matching in general graphs
  • Approximate counting of common bases of two matroids
  • Approximate counting and entropy of perfect matchings
  • Mean-payoff games in quasipolynomial time
  • One-sample matroid prophet inequalities
  • Scheduling unit jobs with precedence constraints on three machines
  • Sampling and counting contingency tables
  • Rapid mixing of the switch chain for every degree sequence
  • The randomized k-server problem on arbitrary metrics
• Combinatorics:
  • Barnette's conjecture
  • Ryser's covering conjecture
  • The Snaky achievement game
  • Hadwiger's conjecture
  • Monochromatic finite sums and products in the positive integers
  • The hypercube Ramsey number has linear order
  • The cycle–clique Ramsey conjecture
  • The Erdős–Gallai cycle-decomposition conjecture
  • Power-saving bounds for square-difference-free sets
  • Polynomial removal for ordered binary matrices
  • The sharp constant in random triangle removal
  • Seymour's second-neighborhood conjecture
  • The Ajtai–Erdős–Komlós–Szemerédi independence conjecture
  • Sidorenko's conjecture
  • The Harary–Hill and Zarankiewicz crossing-number conjectures
  • Talagrand's expectation-threshold and discrete-convexity conjectures
  • The strong thin-tree conjecture
  • Elementary positivity of chromatic quasisymmetric functions
  • Superexponential growth of van der Waerden numbers
  • Saxl's conjecture and the tensor square conjecture
  • The infinite matroid intersection and packing/covering conjectures
  • The logarithmic exponent of off-diagonal Ramsey numbers
  • The circulant Hadamard conjecture
  • Erdős's conjecture on arithmetic progressions
• Discrete geometry:
  • Borsuk's conjecture
  • The periodic tiling conjecture in dimension three
  • The higher-dimensional Erdős distinct-distances conjecture
  • The Erdős unit distance and pinned distance problems
  • The Euclidean plane is not five-colorable
  • Universal optimality of the triangular lattice
  • Classification of finite Euclidean Ramsey configurations
  • Cylinder coverings below the half-area bound
  • The Euclidean Steinitz–Bergström conjecture
  • Heilbronn's triangle problem
• Real analysis:
  • The Erdős similarity conjecture
  • Singular Bernoulli convolutions
• Complex analysis:
  • Brennan's conjecture
  • Kobayashi's canonical-ampleness conjecture
  • The Oka property of complex K3 surfaces
• Harmonic analysis:
  • Endpoint Sobolev regularity of the centered maximal function
  • The triangular Hilbert transform at the symmetric point
  • The trilinear Hilbert transform
  • The Falconer distance conjecture
  • The Kakeya conjecture in three and four dimensions
  • Ultraflat real Littlewood polynomials
  • Sogge's local smoothing conjecture in three dimensions
  • The David–Semmes problem in higher codimension
  • Fourier extension for positively curved surfaces in three dimensions
• Functional analysis:
  • The separable quotient problem
  • The hyperinvariant subspace problem
  • The Lang–Plaut problem
  • Metric Markov cotype of ℓ1
  • The complete Crouzeix conjecture
• Partial differential equations:
  • The hot spots conjecture for simply connected planar domains
  • The critical dimension for one-phase Bernoulli minimizers
  • De Giorgi's conjecture
  • Regularity of infinity-harmonic functions
  • The Mumford–Shah conjecture
  • Stable blowup for the defocusing nonlinear Schrödinger equation
  • Sharp Hölder stability of Brenier maps
  • Universal computation in forced Navier–Stokes flows
  • The Lane–Emden conjecture
• Operator algebras:
  • Isomorphism of the free group factors
  • The Toms–Winter conjecture
  • Strong locality of unitary vertex operator algebras
  • Connes' bicentralizer problem
  • Naimark's problem
  • The Phillips–Toms conjecture
  • The generator problem for von Neumann algebras
  • Kaplansky's quasitrace conjecture
  • Equality of microstates and nonmicrostates free entropy
• Differential geometry:
  • Smooth isometric immersions of surfaces into four-space
  • Spacetime Penrose inequalities
  • Bi-Lipschitz charts at regular points of RCD spaces
  • The Solomon–Yau least-volume conjecture
  • Yau's nodal conjecture for smooth metrics
  • Yau's harmonic dimension comparison problem
  • The Gromov–Lawson aspherical conjecture
  • Urysohn width and macroscopic dimension under positive scalar curvature
  • Finite-time singularities of Calabi flow
  • The LeBrun–Salamon conjecture
  • The Cartan–Hadamard isoperimetric conjecture
  • The isoperimetric conjecture for the cubic flat three-torus
  • Positively curved Einstein four-manifolds
  • Yau's uniformization conjecture
  • Strong forms of the Arnold conjecture
  • The affine Bernstein problem
  • Infinitely many closed geodesics on Riemannian spheres
  • Donaldson's tamed-to-compatible conjecture
• Mathematical physics:
  • Mass generation in two-dimensional O(n) models
  • Pure-point spectrum of the two-dimensional Anderson model
  • Absolutely continuous spectrum of the Anderson model in dimension at least three
  • Bose–Einstein condensation in the dilute Bose gas
  • Long-time derivation of the Boltzmann equation
  • The area law for gapped ground states in two dimensions
  • The spin-one Haldane gap
  • The Laughlin spectral gap
  • The ionization and generalized ionization conjectures
  • Bound states of the BFSS matrix model
  • The one-dimensional Lieb–Thirring conjecture
• Quantum information:
  • Classical capacity of generalized amplitude damping
  • Mutually unbiased bases in dimension six
  • Threshold parallel repetition for entangled games
  • Entanglement without secret key and the PPT-squared conjecture
  • The unitary synthesis problem
• Information theory:
  • The Courtade–Kumar conjecture
  • Memory–sample lower bounds for noiseless Gaussian regression
• Probability:
  • The Benjamini–Schramm percolation conjectures
  • Free energies of random perceptrons
  • Optimal mixing of the Thorp shuffle
  • The random k-SAT satisfiability threshold
  • An exact Hausdorff gauge for SLE
  • Conformal invariance of square-lattice random-cluster interfaces
  • Reconstruction thresholds for Potts broadcasting on trees
  • Scaling limits of critical Fortuin–Kasteleyn planar maps
  • The low-temperature Sherrington–Kirkpatrick fluctuation law
  • Glauber dynamics of the Sherrington–Kirkpatrick model
  • GOE bulk universality for random regular graphs
  • The second Kahn–Kalai conjecture
  • Bigeodesics in planar first-passage percolation
  • The Gaussian propeller conjecture
  • The logarithmic Sobolev conjecture for subgaussian log-concave measures
• Dynamical systems:
  • Uniform bounds for limit cycles in Hilbert's sixteenth problem
  • Permanence of weakly reversible mass-action systems
  • The near-boundary Birkhoff conjecture
  • Rokhlin's multiple-mixing problem
  • Pointwise convergence of multiple ergodic averages for mixing transformations
  • The entropy-rate dimension formula for self-similar measures

Collection-wide checks

• Formal-statement review. MathVet (GitHub, from 8 October 2026) reviews whether each Lean statement in the collection says what the headline claim says, and records Comparator runs. It states that it does not claim any theorem false.
• Second-kernel replay. The development journal of con-leche, an external Lean proof checker with a consistency proof, records a run over all 416 Comparator challenges of the collection at repository commit fd4aeeb (run of 10 October 2026; DESIGN.md, tasks #348 and #352–#354). Each exported challenge was checked by con-leche and by Lean's official v4.34.1 kernel, with a one-hour time limit and a 25 GB memory limit. Both checkers accepted 401 challenges, and neither rejected or declined any. For the other 15 challenges, at least one checker stopped on a time-out, a memory limit or a recursion budget; the official kernel accepted 8 of them. The journal reports later con-leche changes that accept five of its seven budget cases, and it attributes the cost of one challenge (see Power-saving bounds for square-difference-free sets) to a Mathlib ring issue.
• Audit of the manuscripts without Lean. JJardine919/openai-math-audit (Lattice24; archived at Zenodo 10.5281/zenodo.23422995) reports a referee-style review, by AI review agents with independent refutation passes for major findings, of all 547 manuscripts in the collection that have no Lean formalization, completed on 16 October 2026. It reports no gap found in 544 of them, a critical gap in the holomorphicity proof of Square-lattice FK interfaces and nested loops for 1 ≤ q < 4 (inherited by one dependent manuscript; see Conformal invariance of square-lattice random-cluster interfaces), and a verifiability gap in The maximum number of mutually unbiased bases in dimension six, whose computer proof cites code and run records that are not in the public release. Every review is published, and the audit invites corrections through repository issues.

Sources

• OpenAI mathematics collection: github.com/openai/math
• OpenAI's abridged reasoning summaries for ten results: reasoning_traces (list in the repository's README)