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Spectral scalar curvature, Urysohn width, and macroscopic dimension

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying \(-4\Delta+\mathrm{Scal}\ge1\) 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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

One problem has been reported by others; the atlas has not judged it.

The paper states

Theorem 1.1

In Spectral scalar curvature and uniform Urysohn width, 5 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Five other possible matches are listed below the card, among them Corollary 1.3 (Continuous macroscopic dimension of universal covers) in Positive scalar curvature and uniform codimension-two width and Theorem 1.1 in Positive scalar curvature and uniform codimension-two width.

For every integer n\ge4 there is a finite constant C_n with the following property. Every connected complete smooth Riemannian n-manifold (M,g) without boundary satisfying (1.1) admits a continuous map f:M\to K to a simplicial complex of dimension at most n-2 such that

\diam_g f^{-1}(y)\le C_n\qquad(y\in K).

In particular, \UW_{n-2}(M,g)\le C_n.

Five more possible matches

  • Corollary 1.3 (Continuous macroscopic dimension of universal covers) in Positive scalar curvature and uniform codimension-two width.
  • Theorem 1.1 in Positive scalar curvature and uniform codimension-two width.
  • Theorem 1.1 in Spectral scalar curvature and Urysohn width in dimension three, 5 October 2026.
  • Corollary 1.4 (Aspherical obstruction) in Positive scalar curvature and uniform codimension-two width.
  • Corollary 1.2 (Closed-manifold filling-radius bound) in Positive scalar curvature and uniform codimension-two width.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Three current papers, each readable here as a page beside its PDF.

Papers

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Spectral scalar curvature and uniform Urysohn width

5 October 2026, current edition.

Read it herePDF

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying \(-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge1\) 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

5 October 2026, current edition.

Read it herePDF

Every connected complete smooth Riemannian three-manifold without boundary satisfying \(-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge\lambda\gt 0\) as a quadratic-form inequality admits a continuous map to a graph whose entire fibers have diameter at most \(500/\sqrt\lambda\) in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.

Positive scalar curvature and uniform codimension-two width

Undated, current edition.

Read it herePDF

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.

Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Problems reported by others

Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.

  • Jialong Deng, Sam Hopkins, Emil Jerabek, David White, Timothy Chow, Dimitri Vulis, TimH, MathOverflow: how can one contact the authors regarding corrections or missing references?, 8 October 2026. Source

    The author of an arXiv paper of 2 September 2026 says this family generalises two of their results without citing it, and that OpenAI Support could not pass their message to the authors; this is their account, and a later preprint makes a separate similarity claim.

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Within this family

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References outside the release

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What people have said

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 336 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 336 The release has no Lean statement for this family. Open the form