Differential geometry·OpenAI Math·Result 336
Spectral scalar curvature, Urysohn width, and macroscopic dimension
One of the 372 results in OpenAI's public mathematics catalogue, filed under differential geometry: 3 preprints, the latest dated 5 October 2026. This page quotes OpenAI's own summary of the claim, links the papers and abstracts, and will embed the animation.
Latest paper
Spectral scalar curvature and uniform Urysohn width
The animation for this result has not been released yet. Subscribe on YouTube to catch it.
What OpenAI claims
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4Δ+Scal ≥ 1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most n-2 for every n ≥ 2.
Papers (3)
Spectral scalar curvature and uniform Urysohn width
For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying -4Δ+Scal ≥ 1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only in terms of n. The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
Spectral scalar curvature and Urysohn width in dimension three
Every connected complete smooth Riemannian three-manifold without boundary satisfying -4Δ+Scal ≥ λ > 0 as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most 500/√λ in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.
Positive scalar curvature and uniform codimension-two width
We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n-manifold 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 bounded only in terms of n. We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.
Status
Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
