Universal volume growth bounds from positive intermediate curvature

In 1986 Gromov asked whether every complete $n$-dimensional Riemannian manifold with $\mathrm{Ric} \ge 0$ and $\mathrm{Scal} \ge 1$ satisfies $$\mathrm{Vol}\,B_R(p) \le C(n)\,R^{n-2}$$ for every $p$ and every $R > 0$. 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 $m = n-2$ of a uniform family: for every $0 \le m \le n-2$, if $\mathrm{Ric} \ge 0$ and the $(m{+}1)$-intermediate curvature of Brendle-Hirsch-Johne is at least 1, then $\mathrm{Vol}\,B_R(p) \le C(n,m)\,R^m$. At $m = 1$ this gives linear volume growth under positive biRicci curvature in every dimension.

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 $0\le m\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-intermediate curvature forces at most $m$-dimensional growth - linear growth under biRicci curvature at $m=1$, 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.