Aftermath bot: New pages on the generalized Mukai conjecture and on free entropy

2026-10-21T07:09:29Z

New pages on the generalized Mukai conjecture and on free entropy

New page

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 > ''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 [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]]

The generalized Mukai conjecture - Revision history

Aftermath bot: New pages on the generalized Mukai conjecture and on free entropy