OpenAI claims the aspherical positive-scalar-curvature conjecture

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Key facts

• Main claim (abstract, verbatim): ‘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.’
• Companion (verbatim): ‘We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant a_n > 0, depending only on n, such that ∫_M (Scal_g⁻)^{n/2} dV_g ≥ a_n‖M‖ for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g.’ It relies on the first paper’s vanishing theorem
• Prior state (as the paper recounts it): Schoen–Yau’s minimal-hypersurface method (non-spin, dimensions up to seven); Gromov–Lawson’s Dirac-operator method (needs spin coverings); ‘Chodosh and Li established the aspherical obstruction in dimensions four and five using generalized soap bubbles’; Gromov independently treated the closed five-dimensional case
• Reaction: Roman Sauer on Proofs and Prompts, 8 Oct: his measurable-foliation trick, from work with Sabine Braun extended by Hannah Alpert, ‘now appear[s] in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned.’
• The aspherical case includes tori (no positive scalar curvature on Tⁿ, already known) and closed hyperbolic and other negatively curved manifolds; the flatness corollary was noted by Gromov to follow from the obstruction
• Status: no Lean scope note for family 335; no revision in history.md; the integral bound is the Oct 5 manuscript (‘v126-proof.pdf’)