The scalar-curvature extension problem for Ricci flow
From OpenAI Aftermath
The scalar-curvature extension problem for Ricci flow asks whether a Ricci flow on a closed manifold can be continued past any finite time at which its scalar curvature stays uniformly bounded; equivalently, whether scalar curvature must blow up at every finite-time singularity. Three manuscripts in OpenAI's public mathematics collection (github.com/openai/math), all dated 24 September 2026, claim a positive answer on closed four-manifolds and a counterexample in a sufficiently high dimension. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- Hamilton introduced Ricci flow and showed that a compact flow continues while its full curvature stays bounded; Šešum showed that a uniform bound on the Ricci tensor suffices. A scalar bound is weaker: after rescaling at a possible blowup, the scalar curvature may vanish while the full curvature does not, so the limiting geometry can be Ricci-flat without being flat.
- Bamler and Zhang developed heat-kernel, energy and compactness estimates under scalar bounds and obtained smooth extension in dimension four when the second homology vanishes over every field; Simon constructed a continuation through an orbifold Ricci flow, and Bamler's convergence theory gives smooth convergence away from a set of codimension at least four. Scalar blowup at finite-time singularities was known for closed Kähler–Ricci flows by Zhang, for flows with a global Type I bound by Enders, Müller and Topping, and at local Type I singular points by Buzano and Di Matteo.
- Stolarski constructed closed, doubly warped Ricci flows that develop conical singularities with arbitrarily fast curvature blowup and approach a Ricci-flat cone at the parabolic scale; his discussion identifies bounded scalar curvature as a possible feature of these examples. The high-dimensional manuscript uses his quantitative profile estimates as its input.
The original results
- Claim (dimension four). Theorem 1.1 of Bounded scalar curvature and smooth extension of four-dimensional Ricci flow states that a smooth Ricci flow on a closed four-manifold whose scalar curvature is uniformly bounded up to a finite time T extends smoothly on the same manifold beyond T. As a corollary, the maximum of the scalar curvature is unbounded as the flow approaches any finite maximal existence time.
- Claim (higher dimensions). A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup states that in some sufficiently high dimension there is a Ricci flow on a closed manifold whose scalar curvature stays uniformly bounded while its full curvature diverges at a finite maximal time, disproving the extension conjecture in its unrestricted form. In one fixed sufficiently high dimension the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
- Companion. Path selection and an elliptic inequality on degenerating Ricci-flat trees proves an estimate for a renormalized Einstein–Hilbert functional on finite trees of four-dimensional Ricci-flat spaces, together with a path-selection theorem for asymptotic expansions; the four-dimensional manuscript uses this estimate as an input.
- Method, as described: in dimension four, a hypothetical curvature concentration is analysed through a finite tree of Ricci-flat orbifold bubbles. A static estimate shows that the renormalized Einstein–Hilbert functional is small compared with a weighted Ricci error, this smallness is turned into vanishing Ricci energy, and small energy then prevents the outermost bubble from changing its scale by a fixed factor. In high dimension, the construction starts from Stolarski's conical flows, controls a shrinking cap inside the parabolic scale and turns a barrier for the Ricci curvature into a barrier for the scalar curvature.
- Sources (openai/math repository, commit
7ef5c0c): Bounded scalar curvature and smooth extension of four-dimensional Ricci flow; A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup; Path selection and an elliptic inequality on degenerating Ricci-flat trees; all dated 24 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Scalar curvature and Ricci-flow singularities: four-dimensional extension and higher-dimensional blowup, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the three curvatures and two kinds of continuation, the three claims, the shrinking cap in high dimension, the outermost bubble in dimension four, the static functional and its smallness, and the passage to vanishing Ricci energy, with proof outlines. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_397 (dimension four), rank_471 (high dimension) and rank_177 (Ricci-flat trees) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap in any of the three manuscripts, with minor notes that it states are not demonstrated gaps. For dimension four, the decisive estimate is imported from the companion on Ricci-flat trees, whose review only skimmed the path-selection induction, and the geometric inputs of Bamler–Zhang type were skimmed. For the high-dimensional construction, the argument depends on properties of Stolarski's 2019 flows, some taken from the proof rather than the statement of his theorem, which the reviewer could not check against the source; everything after the imported construction was checked line by line. | Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. The reviews are published under CC BY 4.0. |
| 2026-10-19 | AI Proved What?, episode 351 Is scalar enough?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claims. The episode states that no specific dimension is claimed for the counterexample, which depends on an unevaluated constant and by its reading is at least fifteen, so that dimensions five to at least fourteen stay open, and that Stolarski suggested in 2019 that these flows might have bounded scalar curvature. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, three papers dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: no, with an uncatalogued folder of about 21,000 lines formalizing some lemmas of the tree paper but not its main theorem. The page states that it found no specialist comment and no reported error. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
