Universal volume growth bounds from positive intermediate curvature
In 1986 Gromov asked whether every complete -dimensional Riemannian manifold with and satisfiesfor every and every . The three-dimensional case had been settled, and higher dimensions were known only under extra hypotheses such as nonnegative sectional curvature, noncollapsing or an injectivity-radius bound.
This paper answers the question affirmatively, as the case of a uniform family: for every , if and the -intermediate curvature of Brendle-Hirsch-Johne is at least 1, then . At this gives linear volume growth under positive biRicci curvature in every dimension.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Differential geometry
- Posed by
- Mikhail Gromov
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-08-13
- Model
- GPT-5.6 Sol
- Vendor
- OpenAI
- Collaborators
- Gioacchino Antonelli
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 38 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every , nonnegative Ricci plus positive -intermediate curvature forces at most -dimensional growth - linear growth under biRicci curvature at , plus a noncollapsed Urysohn-width bound; the author states this extension is his own, not the model's. Kong and Zhu's proof (also 14 August; GPT-5.6 Sol Ultra and Codex, "essential ideas were generated by AI") proves the same case plus a related codimension-one conjecture via a heat-transport deficit, close to Ge's method by its own account, produced before Ge's went public and not derived from it. Headline axes stay Antonelli's, for the broader scope.
What the AI did
From the paper's own disclosure of AI tools. The work made substantial use of OpenAI's GPT-5.6 Sol at Ultra reasoning effort. GPT proposed the central inductive procedure, based on the Hodge obstruction and rank improvement, behind the proof of Theorem 1.1 - the scalar-curvature case, which is Gromov's question. Antonelli formulated and guided the problem, suggested strategies and literature, and developed the note from that strategy.
He is explicit about how far the published proof moved from the model's: it is effective, where the suggested strategy proceeded by contradiction, and it is much less reliant on Kapovitch-Wilking's Theorem 5.1 - both changes emerging from discussions with Daniele Semola, Elia Bruè and Kai Xu. He also states that the extension to the full Brendle-Hirsch-Johne intermediate-curvature family, which is the paper's general theorem, is his own contribution and not the model's.
Kong and Zhu's independent same-day proof (arXiv:2608.14438) discloses its own AI use more bluntly: "Generative AI tools, more explicitly, ChatGPT 5.6 Sol Ultra and Codex, assisted with proof exploration, organization, and drafting. Essential ideas were generated by AI." That is the stronger of the two claims by wording, though for a narrower result than Antonelli's general family; the headline axes stay Antonelli's for that reason.
Verification
A five-day-old arXiv preprint, unrefereed and not formally endorsed, so this stays Unreviewed. It is not unexamined, though: the author thanks Elia Bruè, Otis Chodosh, Alessandro Cucinotta, Chao Li, Aaron Naber, Daniele Semola and Kai Xu for comments on preliminary versions, and the acknowledgments record two specific ways their comments changed the argument. That is a stronger signal than most preprints of this age carry, but comments are not endorsement, and this site ran no independent check of its own.
The strongest external evidence is indirect and worth stating: three independent routes converge on the same 1986 conjecture within about 24 hours - Jian Ge's heat-kernel Fisher-metric argument (13 August, no AI), this paper's Hodge-obstruction argument (14 August, GPT-5.6 Sol), and Kong-Zhu's heat-transport-deficit argument (also 14 August, GPT-5.6 Sol Ultra and Codex) - the last stating its own method is close to Ge's but developed independently and before Ge's preprint was public. None of the three has been independently checked, but three unrelated-to-mostly-unrelated routes agreeing is meaningful corroboration of the statement regardless.
Sources
Submitted by VibeGene on