Urysohn width and macroscopic dimension under positive scalar curvature
From OpenAI Aftermath
Gromov conjectured that a lower bound on scalar curvature collapses two dimensions at the curvature scale: a complete n-manifold with scalar curvature at least one should admit a continuous map to a polyhedron of dimension n − 2 whose fibers have diameter bounded only in terms of n (a bound on the Urysohn (n − 2)-width), and universal covers of closed manifolds with positive scalar curvature should have macroscopic dimension at most n − 2. Manuscripts dated 23 September and 5 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim these statements and a spectral strengthening, without a Lean formalization. This page summarizes the published claims 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 23 September manuscript:
- Gromov introduced macroscopic dimension in his study of positive scalar curvature and asked for the quantitative codimension-two width bound. Kumar and Sen showed that the pointwise curvature hypothesis cannot be replaced by a fixed-scale macroscopic one.
- In low dimensions, Gromov and Lawson obtained width bounds for closed three-manifolds, Bolotov proved that universal covers of closed three-manifolds cannot have macroscopic dimension two, and Liokumovich–Maximo and Liokumovich–Wang constructed Morse functions with controlled level components on three-manifolds with positive scalar curvature. Chodosh and Li proved the aspherical obstruction in dimensions four and five, with an independent proof by Gromov in dimension five.
- For macroscopic dimension, Bolotov, Bolotov–Dranishnikov and Daher–Dranishnikov obtained codimension-two and dimension-drop results for closed manifolds under the Strong Novikov Conjecture or spin and fundamental-group conditions.
The original result
- Claim. Theorem 1.1 of Positive scalar curvature and uniform codimension-two width (23 September 2026) states that for every n ≥ 4 there is a constant Cn such that every connected complete smooth Riemannian n-manifold without boundary and with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most n − 2 whose entire fibers have diameter at most Cn. Corollaries give Gromov's filling-radius bound for closed manifolds in dimension at least four, macroscopic dimension at most n − 2 for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2, and the absence of positive scalar curvature metrics on closed aspherical manifolds of dimension at least four.
- Two manuscripts of 5 October 2026 replace the pointwise bound by the spectral condition −4Δ + Scal ≥ 1 in the sense of quadratic forms: Spectral scalar curvature and uniform Urysohn width for n ≥ 4, and Spectral scalar curvature and Urysohn width in dimension three, which gives maps to graphs with fibers of diameter at most 500/√λ under −4Δ + Scal ≥ λ > 0.
- Method, as described: the map is assembled from sign coordinates attached to a separated net of balls, with walls separating each inner ball from the exterior of its outer ball chosen so that no n − 1 walls of distinct labels meet. A pressure law and a local stretching estimate widen the separations within a quadratic curvature budget that does not grow with the number of labels, and the pieces produced by cuts are kept with product collars, with extra pressure near singular parts of minimizing walls.
- Sources. The manuscripts Positive scalar curvature and uniform codimension-two width, Spectral scalar curvature and uniform Urysohn width and Spectral scalar curvature and Urysohn width in dimension three at commit
1581d4c. - Formalization. The collection has no Lean scope note or Comparator statement for these results.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 21 October 2026) |
|---|---|---|---|---|
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_228 (23 September manuscript), rank_389 (spectral, n ≥ 4) and rank_374 (dimension three) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap in any of the three manuscripts. For the spectral manuscript in dimension at least four it records one minor note: a completed square in the stretching estimate gives a slightly different coefficient from the one written, so the stated bound is weaker than possible but valid. | 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) | Y. Bi, A note on the cube inequality (arXiv:2610.28390) | Extension (method) | Proves the weighted cube inequality in all dimensions, adapting ideas from this family of OpenAI's release. | arXiv preprint. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The aspherical corollary is also claimed, by a different route, on the page The Gromov–Lawson aspherical conjecture.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
