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Gromov's codimension-two Urysohn width conjecture for positive scalar curvature, with the macroscopic dimension conjecture for universal covers

For a metric space XX the Urysohn dd-width UWd(X)\mathrm{UW}_d(X) is the infimum, over continuous maps f:X→Kf:X\to K to simplicial complexes of dimension at most dd, of sup⁡ydiam⁡f−1(y)\sup_y\operatorname{diam}f^{-1}(y). Gromov proposed that positive scalar curvature collapses two dimensions at the curvature scale: a complete Riemannian nn-manifold with Scal≥1\mathrm{Scal}\ge1 should map to an (n−2)(n-2)-dimensional polyhedron with all point inverses of diameter bounded by a constant depending only on nn, and the universal cover of a closed manifold with positive scalar curvature should have macroscopic dimension at most n−2n-2. Known: closed and complete three-manifolds (Gromov-Lawson, Liokumovich-Maximo, Liokumovich-Wang), spin cases under the Strong Novikov Conjecture (Bolotov, Bolotov-Dranishnikov), and the aspherical consequence in dimensions four and five (Chodosh-Li, Gromov). Is UWn−2(M)≤Cn\mathrm{UW}_{n-2}(M)\le C_n for every complete nn-manifold with Scal≥1\mathrm{Scal}\ge1?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Scalar curvature; Urysohn width and macroscopic dimension
Posed by
Mikhael Gromov (Large Riemannian manifolds, 1986, Section 2.A(c); Positive curvature, macroscopic dimension, spectral gaps and higher signatures, 1996, Section 2 1/2; A dozen problems, 2017)
Year posed
1986
Years open
40y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal manuscript, Theorem 1.1: for every n≥4n\ge4 there is CnC_n such that every connected complete boundaryless Riemannian nn-manifold with Scal≥1\mathrm{Scal}\ge1 admits a continuous map to a simplicial complex of dimension at most n−2n-2 whose entire fibers have diameter at most CnC_n; no orientability, spin, compactness or bounded-geometry assumption. Corollaries: Gromov's filling-radius bound FillRad≤Cn/(2σ)\mathrm{FillRad}\le C_n/(2\sigma) for closed manifolds with Scal≥σ2\mathrm{Scal}\ge\sigma^2, dim⁡mc≤n−2\dim_{mc}\le n-2 for universal covers of closed positive-scalar-curvature manifolds in every n≥2n\ge2, and an independent proof that closed aspherical manifolds of dimension ≥4\ge4 carry no positive scalar curvature. The October 5 companions extend the width bound to the spectral condition −4Δ+Scal≥1-4\Delta+\mathrm{Scal}\ge1 (n≥4n\ge4, and n=3n=3 with a graph target and bound 500/λ500/\sqrt\lambda). Not shown: the Lipschitz form dim⁡MC\dim_{MC}, or any explicit constant.

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: the introductions and main theorems of all three manuscripts, read against Gromov's conjecture as they quote it; the proofs were not refereed. No Lean formalization exists for this family (lean/docs/336.md is absent at the pinned commit). Scope limits the papers state: the uniform width theorem is for n≥4n\ge4 (dimension three rests on Liokumovich-Wang and the spectral three-manifold companion); the macroscopic-dimension corollary is the continuous invariant dim⁡mc\dim_{mc}, which the paper distinguishes from Dranishnikov's Lipschitz invariant dim⁡MC\dim_{MC}; the constant Cn=4D+2C_n=4D+2 is not made explicit. The proof is long (a countable-label minimal-hypersurface cutting scheme with singular-set shielding in appendices of over 200 kB of TeX), so independent checking will take time.

Sources

Changelog1 change

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