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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
• 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 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
• Algebraic geometry:
  • The rational Hodge conjecture for CM abelian varieties
  • Griffiths' positivity conjecture
  • The Kollár–Pardon conjecture on semialgebraic universal covers
• 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
• 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
• 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
• Combinatorics:
  • Barnette's conjecture
  • Hadwiger's conjecture
  • Monochromatic finite sums and products in the positive integers
  • The hypercube Ramsey number has linear order
  • 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
  • The Solomon–Yau least-volume conjecture
• Mathematical physics:
  • Localization and delocalization in the Anderson model
  • Bose–Einstein condensation in the dilute Bose gas
  • The spin-one Haldane gap
  • The ionization and generalized ionization conjectures
• Quantum information:
  • Classical capacity of generalized amplitude damping
• 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
• 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.

Sources

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