Main Page

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
• 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 Birch and Swinnerton-Dyer formula in low Selmer corank
  • Goldfeld's conjecture for quadratic twists
  • The quasi-Riemann hypothesis
  • Prime-factor statistics of p − 1
  • Primitive roots for every admissible integer base
  • Bounded gaps between primes
  • Long gaps between primes
  • Positive lower density of large prime gaps
  • The inhomogeneous Duffin–Schaeffer conjecture
  • Hilbert's tenth problem over the rational numbers
  • Ostmann's inverse Goldbach conjecture
  • The Gaussian moat problem
• 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
• Convex geometry:
  • The Mahler conjectures
• Algebraic complexity:
  • Matrix multiplication with exponent at most 9/4
  • A cubic lower bound for the permanent versus the determinant
• Computational complexity:
  • The Unique 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
• Group theory:
  • Classifying spaces and geometric obstructions for Artin groups
  • Thompson's group F is nonamenable
  • Kaplansky's zero-divisor conjecture
  • The finitely presented Burnside problem
  • The Kervaire conjecture
• Algebra:
  • The Deligne–Drinfeld conjecture
  • Lech's multiplicity conjecture
  • Foulkes' conjecture
• 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
  • Curtis's conjecture
• Geometric topology:
  • The Hilbert–Smith conjecture
• Representation theory of algebras:
  • The little finitistic dimension conjecture
• Mathematical logic:
  • Rigidity of the Turing degrees
• 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
• Combinatorics:
  • Barnette's conjecture
  • 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
• Discrete geometry:
  • Borsuk's conjecture
  • The periodic tiling conjecture in dimension three
  • The higher-dimensional Erdős distinct-distances conjecture
  • 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
• Real analysis:
  • The Erdős similarity conjecture
• Harmonic analysis:
  • The triangular Hilbert transform at the symmetric point
  • The Falconer distance conjecture
• Functional analysis:
  • The separable quotient problem
• Partial differential equations:
  • The hot spots conjecture for simply connected planar domains
  • The critical dimension for one-phase Bernoulli minimizers
• Operator algebras:
  • Isomorphism of the free group factors
  • The Toms–Winter conjecture
  • Strong locality of unitary vertex operator algebras
• 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
• Mathematical physics:
  • Localization and delocalization in the Anderson model
  • Bose–Einstein condensation in the dilute Bose gas
  • 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
• Quantum information:
  • Classical capacity of generalized amplitude damping
  • Mutually unbiased bases in dimension six
• Information theory:
  • The Courtade–Kumar conjecture
• Probability:
  • 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
• Dynamical systems:
  • Uniform bounds for limit cycles in Hilbert's sixteenth problem
  • Permanence of weakly reversible mass-action systems

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)