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Spectral scalar curvature, Urysohn width, and macroscopic dimension

Claims Gromov's codimension-two width bound for positive scalar curvature in every dimension, and a spectral version.

3 papers September 23, 2026 Explained by AI

Scalar curvature at least 1 is a local condition on a curved space. Gromov predicted a global effect: a map onto a space two dimensions lower where each set landing on one point is small. The earliest of three papers claims this in every dimension from four up. The other two claim it under a weaker, averaged spectral condition, with an explicit bound of 500 in dimension three. There is no Lean check.

First step
1986
Mikhail Gromov, whose macroscopic dimension program first posed the quantitative width question
Significance
Major
If correct, it settles the quantitative form of Gromov's codimension-two width conjecture and extends it to spectral curvature bounds.
Trust
Claimed, not checked
Field
Differential geometry
3 papers

The explanation

Take a round sphere of radius √2, about 1.414, and multiply it by a flat plane. You get a four-dimensional space. Its scalar curvature is 2 divided by the radius squared, which is exactly 1 everywhere. Scalar curvature measures, at each point, how the volume of small balls differs from flat space. Now project the space onto the plane. Every point of the plane comes from one sphere. Each sphere has diameter π × √2, about 4.443, measured along its surface. So the four-dimensional space is squashed onto a two-dimensional one, and every piece over a point is small.

The space itself is not small, since the plane goes on forever. Gromov's macroscopic dimension program predicts that this squashing always works. Take any complete manifold of dimension n with scalar curvature at least 1. A manifold is a space that looks like ordinary n-dimensional space near every point. It should map onto a simplicial complex, a space glued from triangles and their higher-dimensional versions, of dimension n − 2. Every fibre, the set of points sent to one place, should have diameter at most a constant C_n that depends only on n. The smallest such bound is called the Urysohn width. Earlier work reached this only in special cases, such as dimension three.

The third paper claims the full conjecture for every dimension n ≥ 4. No orientability, spin or bounded geometry is assumed. As a result, closed manifolds with scalar curvature at least σ² have filling radius at most C_n/(2σ), a case of another Gromov conjecture. It also claims that no closed aspherical manifold of dimension at least four carries positive scalar curvature. Universal covers of closed manifolds with positive scalar curvature have continuous macroscopic dimension at most n − 2.

The first two papers weaken the curvature condition. The spectral bound says −4Δ + Scal ≥ 1, an inequality averaged against every test function. It allows negative scalar curvature on parts of the space. The first paper claims the same width conclusion under this weaker bound for n ≥ 4. It uses the third paper's width theorem as its geometric input. The second paper treats dimension three. There, the target is a graph, and every fibre has diameter at most 500/√λ. For λ = 1 that is 500, and for λ = 4 it is 250. The obstacle in dimension three is that a curvature bound alone does not bound the diameter, as the sphere times a line shows. The paper notes that 500 is not optimised.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of these papers. They are long and rely on heavy geometric analysis. The spectral paper in dimensions four and up depends on the third paper. No specialist review of these papers has been published.

What it doesn't show. This does not bound the size of the manifold itself, only the size of the fibres of a map to a lower-dimensional space.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if correct: it would settle the quantitative codimension-two width conjecture and extend it to spectral bounds. There is no Lean check, the papers depend on each other, and no specialist review is published.

Words used

Manifold
A space that looks like ordinary n-dimensional space near every point.
Scalar curvature
A number at each point saying how the volume of small balls differs from flat space.
Spectral scalar curvature bound
The inequality −4Δ + Scal ≥ λ, which holds on average against every test function rather than at each point.
Simplicial complex
A space glued from points, segments, triangles and their higher-dimensional versions.
Fibre
The set of all points that a map sends to one given point.
Urysohn width
The smallest possible largest fibre diameter over maps to complexes of a given dimension.
Aspherical
A manifold whose universal cover, the fully unwrapped version, has no holes of any dimension.

The original papers (3)

All quoted lines (11)
  1. The quantitative formulation asks whether a fixed positive scalar-curvature normalization forces a dimension-dependent bound for the fibers of a map to an (n−2)(n-2)-dimensional polyhedron. This formulation goes back to; see also and.
    Spectral scalar curvature and uniform Urysohn width Back to text
  2. We prove the quantitative form of this assertion for arbitrary complete manifolds in dimensions at least four.
    Positive scalar curvature and uniform codimension-two width Back to text
  3. The next corollary gives the closed-manifold case in dimensions at least four of Gromov's filling-radius Conjecture A−A^{-}.
    Positive scalar curvature and uniform codimension-two width Back to text
  4. No closed smooth aspherical manifold of dimension n≥4n\ge4 admits a Riemannian metric of strictly positive scalar curvature.
    Positive scalar curvature and uniform codimension-two width Back to text
  5. Its universal cover, with the lifted original metric h~\widetilde h, satisfies dim⁡mc(M~,h~)≤n−2.\dim_{\mathrm{mc}}(\widetilde M,\widetilde h)\le n-2.
    Positive scalar curvature and uniform codimension-two width Back to text
  6. A spectral lower bound mixes scalar curvature with the energy of a test function and permits negative scalar curvature on parts of the manifold.
    Spectral scalar curvature and uniform Urysohn width Back to text
  7. We show that the uniform width conclusion survives this weakening. The geometric input is the stabilized width theorem of, stated precisely in Theorem …; the work here constructs its hypotheses from the spectral inequality.
    Spectral scalar curvature and uniform Urysohn width Back to text
  8. The geometric input is the stabilized width theorem of, stated precisely in Theorem …; the work here constructs its hypotheses from the spectral inequality.
    Spectral scalar curvature and uniform Urysohn width Back to text
  9. Every connected complete smooth Riemannian three-manifold without boundary satisfying −4Δ+\Scal≥λ>0-4\Delta+\Scal\ge\lambda>0 as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most 500/λ500/\sqrt\lambda in the original metric.
    Spectral scalar curvature and Urysohn width in dimension three Back to text
  10. A positive scalar-curvature lower bound need not bound the diameter of a three-manifold: the product of a round two-sphere with a line is a basic example.
    Spectral scalar curvature and Urysohn width in dimension three Back to text
  11. The constant 500500 is convenient and is not optimized.
    Spectral scalar curvature and Urysohn width in dimension three Back to text

Other sources

  1. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

Browse Differential geometry Curvature and geometric flowsOpen since the 1980sClaimed, not yet checked