Result 336, Differential geometry

Spectral scalar curvature, Urysohn width, and macroscopic dimension

Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying −4Δ+Scal≥1-4\Delta+\mathrm{Scal}\ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n−2n-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−2n-2 for every n ≥ 2.

Proof

The bigger picture

Why it matters

Curvature can constrain a space's large-scale shape even when it is not positive everywhere. These manuscripts claim that a spectral positivity condition forces such spaces to admit lower-dimensional descriptions with uniformly bounded ambiguity.

What changes?

For every complete, connected, smooth, boundaryless Riemannian n-manifold, n at least three, the manuscripts report a continuous map to a simplicial complex, built from vertices, edges and higher-dimensional simplexes, of dimension at most n minus two. Entire fibers, all points sharing an image, have diameter bounded only by n in the original metric. The assumption is that negative four times the Laplacian plus scalar curvature is at least one as a quadratic-form inequality, an integrated curvature-and-variation bound.

What does that help mathematicians do?

This extends a width conclusion from pointwise scalar-curvature positivity to spectral positivity: local curvature need not meet the lower bound everywhere. In dimension three, the reported target is a graph. With spectral lower bound lambda greater than zero, every entire fiber has diameter at most 500 divided by the square root of lambda. No orientability, spin, compactness or bounded-geometry assumption is needed. Thus the claim quantitatively limits how much geometry each graph point can hide.

Are there practical applications?

The immediate value is foundational: these claims connect curvature to the number of dimensions needed to describe geometry at large scales. The manuscripts also report that universal covers, spaces that unwrap loops, of closed positive-scalar-curvature n-manifolds have continuous macroscopic dimension at most n minus two for every n at least two. This supplies a large-scale dimensional constraint on those covering spaces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Spectral scalar curvature and uniform Urysohn width

October 5, 2026 27 pages

For every n ≥ 4, a complete connected smooth Riemannian n-manifold without boundary satisfying −4Δ+Scal≥1-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n−2n-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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026,
  author = {{OpenAI}},
  title = {{Spectral scalar curvature and uniform Urysohn width}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf}{OAI:Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026}},
  year = {2026}
}

Spectral scalar curvature and Urysohn width in dimension three

October 5, 2026 21 pages

Every connected complete smooth Riemannian three-manifold without boundary satisfying −4Δ+Scal≥λ>0-4\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/λ500/\sqrt\lambda in the original metric. No orientability, spin, compactness, or bounded-geometry assumption is required.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026,
  author = {{OpenAI}},
  title = {{Spectral scalar curvature and Urysohn width in dimension three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf}{OAI:Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026}},
  year = {2026}
}

Positive scalar curvature and uniform codimension-two width

September 23, 2026 99 pages

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−2n-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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026,
  author = {{OpenAI}},
  title = {{Positive scalar curvature and uniform codimension-two width}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf}{OAI:Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.