OpenAI · 335-02 · 2026-09-23 · Differential geometry

Positive scalar curvature forces rational inessentiality

preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026

Family 335. Gromov's integral scalar-curvature inequality.

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

Proves $\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥$ for every closed connected oriented smooth $n$ -manifold, $n \geq 3$ , and every smooth metric, with $a_{n} > 0$ depending only on dimension. Here $Scal_{g}^{-} = max {0, - Scal_{g}}$ and $∥ M ∥$ is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.

We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat. We also show that every closed oriented smooth manifold of positive dimension with nonnegative scalar curvature has zero real simplicial volume, proving the qualitative vanishing consequence of Gromov's conjectural comparison.

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.