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18 results from OpenAI’s math release. 18 explained so far. OpenAI has published Lean proofs for 3: 1 of the main theorem, 2 of part of the claim.

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• 304 The Hilbert–Smith conjecture in every dimension

  Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.

  ▶Episode, 4:38

• 305 Four-dimensional disk embedding and Wall's conjecture

  The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F₂ is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.

  3 papers ▶Episode, 4:54

• 306 The purely cosmetic surgery conjecture

  Distinct Dehn surgery slopes on a nontrivial smooth knot in S³ never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.

  ▶Episode, 4:36

• 307 Failure of rational injectivity for maximal coarse assembly

  Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.

  2 papers ▶Episode, 4:47 OpenAI published a Lean proof of part of the claim.

• 308 Finite Smith–Toda complexes at every height

  For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology $BP_*/(p,v_1,\ldots,v_n)$ and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes $V(4)$ at p = 1009.

  2 papers ▶Episode, 4:26

• 309 The Kervaire invariant problem at the prime three

  Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.

  ▶Episode, 4:34

• 310 Quillen's conjecture in rational homology

  Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

  ▶Episode, 4:16

• 311 The Hovey–Strickland and Chai conjectures

  Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable $K(n)$-local spectra have exactly $n+2$ thick tensor ideals, and their Balmer spectrum is a chain of $n+1$ points.

  ▶Episode, 4:07

• 312 The Grothendieck homotopy hypothesis

  Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.

  ▶Episode, 4:31 OpenAI published a Lean proof of part of the claim.

• 313 Finite generation for the $K(n)$-local sphere

  Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the $K(n)$-local sphere is a finitely generated ℤ_p-module, for every prime, positive height and integer degree. The same conclusion holds after $K(n)$-localizing any finite p-local spectrum.

  ▶Episode, 4:20

• 314 Cyclic length and chromatic fixed-point loss

  Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

  ▶Episode, 5:06

• 315 The four-dimensional Singer conjecture

  Proves that the L²-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.

  ▶Episode, 4:48

• 316 Curtis’s conjecture

  Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.

  ▶Episode, 4:20

• 317 Thomason model structures in all strict higher dimensions

  Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.

  ▶Episode, 4:18 OpenAI published a Lean proof of the main theorem.

• 318 Chromatic splitting: filtrations and counterexamples

  Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and $p+1$ (p ≥ 7). Nevertheless, for n ≥ 1 and $p\gt n+1$, the overlap $L_{n-1}L_{K(n)}S_p^\wedge$ admits a $2^n$-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of $E(2)$-local modules over the derived completed sphere.

  5 papers ▶Episode, 4:18

• 319 Counterexamples to finite generation at chromatic height two

  Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed $\mathrm{BP}\langle2\rangle$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

  ▶Episode, 3:41

• 320 Nonhomeomorphic closed aspherical four-manifolds

  Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.

  ▶Episode, 3:43

• 321 A counterexample to Wall's finite D(2) problem

  Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.

  ▶Episode, 3:18

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.