OpenAI · 336-03 · 2026-09-23 · Differential geometry

Positive scalar curvature and uniform codimension-two width

preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026

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 $n \geq 2$ .

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.

Preprint source and Lean scope

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