View source for The generalized Mukai conjecture

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 ([https://github.com/openai/math 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 &gt; ''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 &gt; ''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 [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Generalized-Mukai-Conjecture-September-24-2026 The generalized Mukai conjecture] at commit <code>1581d4c</code>.
* '''Formalization.''' The collection has no Lean scope note or Comparator statement for this result.

== Public follow-up work ==
{| class="wikitable"
! 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'' ([https://arxiv.org/abs/2604.25023 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 [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_480.json rank_480] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 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'' ([https://arxiv.org/abs/2610.27676 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.

[[Category:OpenAI results]]