The generalized Mukai conjecture
From OpenAI Aftermath
The generalized Mukai conjecture asserts that every smooth complex Fano manifold X of dimension n, Picard number ρ and pseudoindex ι (the least anticanonical degree of a rational curve on X) satisfies ρ(ι − 1) ≤ n, with equality only for the product of ρ copies of the projective space of dimension ι − 1. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a proof in every dimension, without a Lean formalization. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The index version goes back to Mukai. Wiśniewski introduced the pseudoindex and proved that 2ι > n + 2 forces ρ = 1. Bonavero, Casagrande, Debarre and Druel formulated the pseudoindex form and proved it in dimension at most four and for smooth toric Fano manifolds in further cases; Andreatta, Chierici and Occhetta extended the low-dimensional result to dimension five.
- Casagrande settled the bound and the equality case for ℚ-factorial Gorenstein toric Fano varieties, Pasquier treated horospherical varieties, and Gagliardi, Hofscheier and Pearson proved the generalized conjecture for ℚ-factorial spherical Fano varieties.
- For large pseudoindex, Novelli and Occhetta proved both conclusions when ι ≥ (n + 3)/3, and Novelli extended this to ι > n/3 and, at the boundary n = 3ι, to manifolds with an extremal contraction of fiber type.
The original result
- Claim. Theorem 1.1 of The generalized Mukai conjecture states that every smooth connected complex projective Fano manifold of positive dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι − 1) ≤ n, with equality if and only if X is isomorphic to the product of ρ copies of the projective space of dimension ι − 1.
- Method, as described: small quantum multiplication by a basis of divisor classes gives commuting matrices whose entries are curve correspondences lowering codimension by the anticanonical degree minus one. A lower bound for one-point descendant invariants of free curve classes and a recurrence from the divisor and topological recursion equations show that one joint eigenvalue tuple has algebraically independent coordinates; an alternating trace of matrix-valued differentials converts this into a nonzero product of correspondences in independent degrees, which gives the inequality. At equality the degrees are minimal, and the ordered-chain bound of Bonavero, Casagrande, Debarre and Druel together with Occhetta's characterization of products of projective spaces gives the classification.
- Sources. The manuscript The generalized Mukai conjecture at commit
1581d4c. - Formalization. The collection has no Lean scope note or Comparator statement for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 21 October 2026) |
|---|---|---|---|---|
| 2026-10-07 (version 2; version 1: 27 April 2026) | H. Pearson, The Mukai conjecture via Cox rings for smooth canonical ambient embeddings (arXiv:2604.25023) | Parallel (Cox rings) | Proves the Mukai conjecture, with a stronger lower bound in terms of a distinguished anticanonical divisor, for locally factorial Fano varieties whose canonical toric ambient variety from the Cox ring is smooth. Version 2 states that, simultaneously and independently, an unreleased OpenAI model claimed a solution of the generalized Mukai conjecture using Gromov–Witten invariants, that the two approaches appear distinct, and that this paper also covers certain Fano varieties with mild singularities; a remark proposes connecting its conjectural refined bound with the OpenAI proof. | arXiv preprint. The paper states that its first version was AI-free and that a lemma was relaxed after a brief discussion with GPT-6 Astra. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_480 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer read all sections, re-derived the descendant recurrence, checked the steps of the descendant lower bound, the decay argument, the trace invariance (also numerically) and the equality argument; the hypotheses of the cited Bonavero–Casagrande–Debarre–Druel erratum, Occhetta's theorem and two further citations were checked only for plausibility. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-21 (listed) | K. Watanabe, Nef complexity of klt Fano varieties: the full spectrum and classification below two (arXiv:2610.27676) | Remark (note added) | Determines the set of values of the nef complexity of klt Fano varieties and classifies those with nef complexity below two. A note added states that OpenAI released this preprint, claiming a proof of the generalized Mukai conjecture for smooth complex Fano varieties, during the final checks of the manuscript. | arXiv preprint. The author states that most proofs were developed from arguments generated by OpenAI's ChatGPT in discussions initiated and guided by the author. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
