OpenAI · 336-02 · 2026-10-05 · Differential geometry

Spectral scalar curvature and Urysohn width in dimension three

preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026

Family 336. Spectral scalar curvature and codimension-two width.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Every complete connected smooth boundaryless $n$ -manifold, $n \geq 3$ , satisfying $- 4Δ + Scal \geq 1$ 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 $n \geq 2$ .

Every connected complete smooth Riemannian three-manifold without boundary satisfying $- 4Δ + Scal \geq λ > 0$ 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.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.