The scalar-curvature extension conjecture for Ricci flow in all dimensions
A smooth Ricci flow on a closed manifold extends past a finite time unless the full curvature becomes unbounded (Hamilton), and a Ricci bound suffices (Sesum). The scalar-curvature extension conjecture asks whether scalar curvature alone detects singularities: is it true that every finite-time singularity of a Ricci flow on a closed manifold of dimension at least four has unbounded scalar curvature, i.e. does always allow the flow to be extended past ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometric analysis; Ricci flow singularities
- Posed by
- Stated as open in R. H. Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. of Math. 188 (2018), p. 756; known cases surveyed by C. Li and Y. Li (2026)
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for some there is a smooth Ricci flow on the closed manifold with and at the finite maximal time; a corollary gives two-sided power-law curvature blowup with arbitrarily large exponents in one fixed dimension. This refutes the unrestricted all-dimensions conjecture. It relies on Stolarski's construction and gives no explicit dimension; it does not touch dimension four, which the companion proves true.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it; it claims a Ricci flow on , , with bounded scalar curvature and at a finite maximal time. Proof not refereed; no Lean formalization. The construction imports Stolarski's (2019) doubly warped conical-singularity flows and their quantitative profile estimates as stated inputs; the new work is the scalar bound. The paper says no dimension-four conclusion is asserted.