The scalar-curvature extension conjecture for Ricci flow in dimension four
A smooth Ricci flow on a closed manifold can stop only if the full curvature blows up. Bamler and Zhang developed heat-kernel and compactness theory for flows with bounded scalar curvature and proved smooth extension in dimension four when for all fields; Simon continued such flows through orbifold limits. In dimension four: if a smooth Ricci flow on a closed four-manifold has uniformly bounded scalar curvature on with , does it extend smoothly on the same manifold past ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric analysis; four-dimensional Ricci flow
- Posed by
- Stated as open in R. H. Bamler, Ann. of Math. 188 (2018), p. 756, with the four-dimensional theory of Bamler-Zhang (2017, 2019) and M. Simon (2020)
- 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
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: a smooth Ricci flow on a closed four-manifold with , , extends smoothly past ; Corollary: at a finite maximal time . This removes the hypothesis of Bamler-Zhang and gives smooth continuation rather than an orbifold flow. The companion proves for the renormalized Einstein-Hilbert functional on degenerating trees of Ricci-flat spaces. It is specific to dimension four; in high dimension the family's other paper shows the statement is false.
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 problem as the paper states it; it claims smooth extension on the same closed four-manifold whenever is bounded up to a finite . The long proof (blowup trees of Ricci-flat pieces, a renormalized Einstein-Hilbert functional) was not refereed and is not formalized. It depends on the companion's elliptic inequality on degenerating Ricci-flat trees as a stated input, and on Bamler-Zhang's estimates.