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Gromov's integral scalar-curvature bound for simplicial volume

For a closed connected oriented nn-manifold MM let ∥M∥\|M\| be its real simplicial volume, and for a Riemannian metric gg let Scalg−=max⁡{0,−Scalg}\mathrm{Scal}_g^-=\max\{0,-\mathrm{Scal}_g\}. In 1986 Gromov (Large Riemannian manifolds, Conjecture 3.A, Remark (B), Equation (8')) proposed bounding simplicial volume by the scale-invariant integral ∫M(Scalg−)n/2dVg\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g; equivalently, Scalg≥−κ2\mathrm{Scal}_g\ge-\kappa^2 should force ∥M∥≤CnκnVolg(M)\|M\|\le C_n\kappa^n\mathrm{Vol}_g(M). Partial results covered bounded unit-ball volumes in the universal cover (Braun-Sauer), Kahler surfaces (Min-Zheng-Zhu) and qualitative vanishing for spin universal covers. Is there an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed nn-manifold and every metric?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Scalar curvature; simplicial volume and bounded cohomology
Posed by
M. Gromov, Large Riemannian manifolds, in Curvature and Topology of Riemannian Manifolds (1986), Conjecture 3.A, Remark (B), Equation (8')
Year posed
1986
Years open
40y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
38 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every n≥3n\ge3 there is an>0a_n>0 with ∫M(Scalg−)n/2dVg≥an∥M∥\int_M(\mathrm{Scal}_g^-)^{n/2}dV_g\ge a_n\|M\| for every closed connected oriented smooth nn-manifold MM and smooth metric gg; e.g. Scalg≥−n(n−1)\mathrm{Scal}_g\ge-n(n-1) gives Volg(M)≥an[n(n−1)]−n/2∥M∥\mathrm{Vol}_g(M)\ge a_n[n(n-1)]^{-n/2}\|M\|. The route is a Yamabe reduction to Scal=−1\mathrm{Scal}=-1 followed by a bounded-cohomology and graph-deformation estimate. No explicit value of ana_n is stated in the theorem, and dimension two is excluded.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. The proof uses the vanishing theorem of the companion manuscript (Positive scalar curvature forces rational inessentiality). No Lean formalization accompanies it.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source (v126-proof), read against Gromov's 1986 proposal as cited. Theorem 1.1 states the inequality for every n≥3n\ge3, every closed connected oriented smooth nn-manifold and every smooth metric, with ana_n depending only on nn. It is conditional on the companion's vanishing theorem (nonnegative scalar curvature forces ∥M∥=0\|M\|=0), itself an unrefereed claim in this release. Not refereed here. No Lean formalization.

Sources

Changelog1 change

Discussion