Three Mathematical Manuscripts Claim to Prove Gromov’s Conjecture on the Volume Growth of a Space; Two Disclose the Role of AI in the Proofs

Three Mathematical Manuscripts Claim to Prove Gromov’s Conjecture on the Volume Growth of a Space; Two Disclose the Role of AI in the Proofs

On 13–14 August, three manuscripts appeared on arXiv presenting the same upper bound on volume growth. In two of them, the authors described how generative models contributed to finding the proof.

In 1986, Mikhail Gromov asked whether geometry at every point could constrain the overall growth of an infinite space. He considered spaces in which two mathematical measures of local curvature satisfy strict conditions: one is nowhere negative, and the other is positive everywhere. For a ball of radius R in n dimensions, Gromov expected an upper volume bound of C·Rⁿ⁻². Volume in n dimensions normally grows as Rⁿ, whereas here the exponent of the radius is smaller by two.

All three papers claim this bound. In Jian Ge’s paper, the argument proceeds from local curvature to volume through the propagation of heat across the space. Gioacchino Antonelli obtains the estimate as a special case of a more general theorem on intermediate curvature. Bochao Kong and Xinyu Zhu connect mass transport with the volume of balls of large radius. The three papers reach the same estimate by different methods.

In two manuscripts, the authors separately described the role of generative models in finding the proofs. Antonelli writes that GPT proposed the central inductive procedure used in his proof. Antonelli formulated and directed the problem, selected the strategies and literature, developed the manuscript, and extended the result to a more general case.

Kong and Zhu report that ChatGPT 5.6 Sol Ultra and Codex helped them investigate the proof, organize the material, and write the manuscript. The authors independently checked and edited the theorem statements, proofs, and references, and retained responsibility for the text.

“Substantial ideas were generated by AI,” Kong and Zhu write.

In an addendum, Antonelli reports that he learned of Ge’s work after privately circulating his own manuscript. Kong and Zhu date their AI assisted proof to a time before Ge’s preprint was published. These two accounts of the chronology accompany three distinct routes to the same volume growth estimate.