---
id: "2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc"
url: "https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Expert-checked
sources: 4
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/

# OpenAI claims the aspherical positive-scalar-curvature conjecture

Full title: OpenAI's release claims the Gromov–Lawson conjecture in every dimension: no closed aspherical manifold has positive scalar curvature, with Gromov's integral bound for simplicial volume

Family 335: "Positive scalar curvature forces rational inessentiality" (23 Sept 2026) claims that every closed oriented manifold with positive scalar curvature sends its rational fundamental class to zero in the classifying space, with no spin, fundamental-group or dimension restriction. Corollaries: the Gromov–Lawson aspherical conjecture (no closed aspherical manifold has positive scalar curvature) and flatness of every nonnegative-scalar-curvature metric on a closed aspherical manifold. "An integral scalar curvature bound for real simplicial volume" (5 Oct) proves Gromov's inequality ∫(Scal⁻)^{n/2} ≥ a_n‖M‖. Known before: dimensions 4 and 5 (Chodosh–Li; Gromov) and spin cases via index theory. No Lean formalization; Roman Sauer (Proofs and Prompts) called the proofs "deeply creative".

## 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')

## What happened

Scalar curvature is the weakest curvature notion, and asking which closed manifolds carry a metric of positive scalar curvature is a
central question of geometric topology. Gromov and Lawson, and Schoen and Yau, conjectured in the early 1980s that aspherical manifolds,
whose universal cover is contractible (tori, hyperbolic manifolds), never do. Two methods made partial progress. Minimal hypersurfaces
work without spin structures but originally only up to dimension seven. Dirac operators work in all dimensions but need spin assumptions and
largeness conditions. The general aspherical case was settled only in dimensions four and five, in 2020–24.

OpenAI's release claims the full statement in every dimension, through a stronger topological conclusion: positive scalar curvature
forces the fundamental class to vanish rationally in the classifying space. A second manuscript uses that vanishing to prove an
inequality Gromov proposed, bounding the simplicial volume of a manifold by an integral of its negative scalar curvature. Roman Sauer
recognised a technique from his own work with Sabine Braun in these proofs and in the release's family 207, and called them "deeply
creative".

## Why it matters

If correct, this settles one of the best-known conjectures about positive scalar curvature and removes the spin condition that has
shaped the field for forty years. It would also imply rigidity: on aspherical manifolds, nonnegative scalar curvature forces flatness.
The claims are unrefereed and unformalized, and no specialist has published a check as of 29 Oct.

## What is disputed or not yet verified

- Verification: Unrefereed manuscripts; no formalization; no published expert check

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Positive scalar curvature forces rational inessentiality (23 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI: An integral scalar curvature bound for real simplicial volume (5 Oct 2026, PDF)](https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf) (github.com, paper)
3. [Proofs and Prompts: 100+ reactions to 100+ solutions (Roman Sauer)](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/) (proofsandprompts.com, discussion)
4. [oai-math-revised: community notes on the rational-inessentiality manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 251–378; both abstracts and the first paper's introduction read, Proofs and Prompts, oai-math-revised notes)

## Related

- 2026-10-24: [oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)](https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)