Is scalar enough?
Three AI preprints claim the four-dimensional scalar-curvature extension theorem and a high-dimensional counterexample
Covers result 351 in OpenAI’s catalog: all 3 papers. See its connections to other results and the work it cites.
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Chapters
The claim, in plain words
Ricci flow changes a shape’s metric according to its Ricci curvature, smoothing it rather like heat. A round sphere simply shrinks: our computer checks that its radius squared falls linearly to zero, while its scalar curvature blows up in step with its full curvature. Hamilton showed that a flow on a closed manifold keeps going as long as the full curvature stays bounded, and Šešum showed that a bound on the Ricci curvature is enough. The question is whether the scalar curvature, a single averaged number at each point, is enough on its own.
Bing Wang attributed this conjecture to Xiuxiong Chen, and Bamler called it famous: if the scalar curvature stays bounded, the flow should continue. It holds in dimensions two and three, for Kähler flows by Zhang, and for Type I flows by Enders, Müller and Topping. In dimension four, Bamler and Zhang, and Simon, showed that zooming in on any concentration gives Ricci-flat bubbles, like the Eguchi–Hanson space; our computer checks that its scalar curvature is zero while its full curvature grows like one over its size squared as it shrinks. Smooth extension in dimension four, from the scalar bound alone, was known only under a topological condition, and higher dimensions were open.
Result 351 is three OpenAI papers, all dated September 24 and made public October 6, 2026. The first, 58 pages, claims dimension four: on any closed four-manifold, a Ricci flow whose scalar curvature stays bounded up to a finite time extends smoothly past that time, on the same manifold. It relies on a 40-page companion paper, which proves an energy inequality on trees of degenerating Ricci-flat pieces. The second, 20 pages, claims a counterexample in high dimension: on a two-sphere times a high-dimensional sphere, a flow with bounded scalar curvature whose full curvature blows up at a finite time.
What it doesn’t claim
Not claimed: a specific dimension for the counterexample, which depends on an unevaluated constant; by our reading of one of its conditions it is at least fifteen. So dimensions five through at least fourteen stay open. And Stolarski suggested in 2019 that these very flows might have bounded scalar curvature; the new part is the proof.
Where it stands
- Posted as preprints
- OpenAI, 3 papers, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- No (an uncatalogued folder formalizes some companion-paper lemmas)
Last checked October 19, 2026. All three papers are unrefereed and unchanged since release. Neither main theorem has Lean; an uncatalogued folder of about 21,000 lines formalizes some lemmas of the tree paper, not its main theorem. The four-dimensional result rests on both of its papers, and the counterexample takes Stolarski’s long memoir as an input. An AI audit found no gap but skimmed the hardest parts. We found no specialist comment and no reported error. OSVerify has not reviewed it.
How the proof goes
- Dimension four: no room to concentrate. In dimension four, suppose curvature concentrates. Zooming in gives a finite tree of Ricci-flat orbifold bubbles, and the outermost bubble’s scale must shrink by a fixed factor on suitable time intervals. A renormalized Einstein–Hilbert energy, built by attaching an exterior that depends analytically on a collar around that bubble, must change by at least the Ricci energy spent there; but the tree inequality from the companion paper says this energy is tiny compared with the Ricci error. So the Ricci energy tends to zero, the bubble cannot shrink, and that contradiction rules out concentration.
- High dimension: a sharp cap on a cone. In high dimension, the paper starts from Stolarski’s doubly warped flows, which look like a Ricci-flat cone at the parabolic scale; our computer checks that this cone is exactly Ricci-flat but not flat. A tiny cap at the tip makes the full curvature blow up like one over the cap size squared. Barrier arguments show that the Ricci curvature grows much more slowly, which needs the dimension to be large, and then a scalar barrier keeps the scalar curvature bounded.
The numbers in the video
- Ours: a round 2-sphere under Ricci flow has scalar curvature 2/(1 − t/T): 2, 4, 20, 200 at t/T = 0, 0.5, 0.9, 0.99.
- Ours: the Eguchi–Hanson space has zero scalar curvature, while its full curvature peaks at 19.6/a² for a bubble of size a.
- Ours: the cone over S² × S^q is exactly Ricci-flat but not flat; the counterexample's condition 2 < d < 5/2 needs q ≥ 12, so dimension at least 15.
- Lean: an uncatalogued folder of 58 files and 20,990 lines formalizes some lemmas of the companion paper (counted by us; not run by us).
Read the source
- A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup (PDF), 20 pages.
- Bounded scalar curvature and smooth extension of four-dimensional Ricci flow (PDF), 58 pages.
- Path selection and an elliptic inequality on degenerating Ricci-flat trees (PDF), 40 pages.
- The collection’s catalog, where this is result 351.
The 3 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Ricci flow changes a shape’s metric according to its Ricci curvature, smoothing it rather like heat. A round sphere simply shrinks: our computer checks that its radius squared falls linearly to zero, while its scalar curvature blows up in step with its full curvature. Hamilton showed that a flow on a closed manifold keeps going as long as the full curvature stays bounded, and Šešum showed that a bound on the Ricci curvature is enough. The question is whether the scalar curvature, a single averaged number at each point, is enough on its own. Bing Wang attributed this conjecture to Xiuxiong Chen, and Bamler called it famous: if the scalar curvature stays bounded, the flow should continue. It holds in dimensions two and three, for Kähler flows by Zhang, and for Type I flows by Enders, Müller and Topping. In dimension four, Bamler and Zhang, and Simon, showed that zooming in on any concentration gives Ricci-flat bubbles, like the Eguchi–Hanson space; our computer checks that its scalar curvature is zero while its full curvature grows like one over its size squared as it shrinks. Smooth extension in dimension four, from the scalar bound alone, was known only under a topological condition, and higher dimensions were open.
1:18The claim
Result 351 is three OpenAI papers, all dated September 24 and made public October 6, 2026. The first, 58 pages, claims dimension four: on any closed four-manifold, a Ricci flow whose scalar curvature stays bounded up to a finite time extends smoothly past that time, on the same manifold. It relies on a 40-page companion paper, which proves an energy inequality on trees of degenerating Ricci-flat pieces. The second, 20 pages, claims a counterexample in high dimension: on a two-sphere times a high-dimensional sphere, a flow with bounded scalar curvature whose full curvature blows up at a finite time. Not claimed: a specific dimension for the counterexample, which depends on an unevaluated constant; by our reading of one of its conditions it is at least fifteen. So dimensions five through at least fourteen stay open. And Stolarski suggested in 2019 that these very flows might have bounded scalar curvature; the new part is the proof.
2:26How the proof goes
In dimension four, suppose curvature concentrates. Zooming in gives a finite tree of Ricci-flat orbifold bubbles, and the outermost bubble’s scale must shrink by a fixed factor on suitable time intervals. A renormalized Einstein–Hilbert energy, built by attaching an exterior that depends analytically on a collar around that bubble, must change by at least the Ricci energy spent there; but the tree inequality from the companion paper says this energy is tiny compared with the Ricci error. So the Ricci energy tends to zero, the bubble cannot shrink, and that contradiction rules out concentration. In high dimension, the paper starts from Stolarski’s doubly warped flows, which look like a Ricci-flat cone at the parabolic scale; our computer checks that this cone is exactly Ricci-flat but not flat. A tiny cap at the tip makes the full curvature blow up like one over the cap size squared. Barrier arguments show that the Ricci curvature grows much more slowly, which needs the dimension to be large, and then a scalar barrier keeps the scalar curvature bounded.
3:31Where it stands
All three papers are unrefereed and unchanged since release. Neither main theorem has Lean; an uncatalogued folder of about 21,000 lines formalizes some lemmas of the tree paper, not its main theorem. The four-dimensional result rests on both of its papers, and the counterexample takes Stolarski’s long memoir as an input. An AI audit found no gap but skimmed the hardest parts. We found no specialist comment and no reported error. If they hold, bounded scalar curvature is enough to continue Ricci flow on every closed four-manifold, but not in every dimension.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.