Zoom in: one picture?
An AI preprint claims unique tangent flows at the first singularity of any closed embedded surface
Covers result 355 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:18. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Under mean curvature flow, every point of a surface moves in the direction that shrinks its area fastest, at a speed set by how curved the surface is there, and a closed surface always develops a singularity in finite time. A round sphere shrinks to a point: our computer checks that its squared radius drops by four per unit time, and a cylinder’s by two. Zoom in at a singular point, rescaling time by the square of the magnification, and the picture approaches a self-shrinker: a surface that only shrinks while keeping its shape, like a sphere of radius two or a cylinder of radius √2.
But different zoom sequences might give different limits: for example, a neck whose axis keeps slowly turning. Tom Ilmanen listed uniqueness of tangent flows as a problem in 2003, and Haslhofer recently called it one of the most fundamental open problems. It was known when the limit is a closed shrinker, by Schulze; a round cylinder, by Colding and Minicozzi; or cone-like far out, by Chodosh and Schulze. Bamler and Kleiner proved that every limit for a surface is a single sheet whose ends look like cones or cylinders, and the remaining case is a shrinker with a cylinder-like end that is not a cylinder. Ilmanen asked whether such a shrinker can exist, and that is still open.
Result 355 is one OpenAI paper, 100 pages, dated September 24 and made public October 6, 2026. It claims that for any smooth closed embedded surface in space, at every singular point at the first singular time, the whole family of rescalings converges to one self-shrinker, as a single sheet, fixed in the original coordinates, including its position and axes. There is no assumption that the surface moves inward everywhere, and no assumed shape for the limit.
What it doesn’t claim
Not claimed: later singularities, moving centres, or higher dimensions. It does not say which shrinkers occur, and it does not settle whether a shrinker with a cylinder-like end must be a cylinder; it works around that question. It builds on Bamler and Kleiner’s multiplicity-one theorem and their description of the ends. In 2019, Chodosh and Schulze noted that ruling such shrinkers out would give this result. We found no competing claim.
Where it stands
- Posted as a preprint
- OpenAI, 100 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- No Lean proof published
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. There is no Lean formalization and no code; the proof is 100 pages of analysis. An AI audit of part of it found no gap, noting that the result depends entirely on Bamler and Kleiner’s description of the ends, and leaving about 2,500 lines of estimates, among other parts, unchecked. We found no specialist comment and no reported error. OSVerify has not reviewed it.
How the proof goes
- Why cylinder-like ends are hard. The standard route, from Leon Simon’s work, is a Łojasiewicz inequality: if the flow’s speed is at least a power of its remaining energy, its path has finite length and cannot spiral forever. Our toy shows the danger, in a classic example: a smooth but non-analytic landscape whose downhill paths circle forever. On a cylindrical end, one deformation keeps the shrinker equation to first order but grows like z², opening the end into a cone or pinching it off far out. Its effect is smaller than any power of the opening, so the usual analytic argument breaks down in that direction.
- Two detours and a jump. The paper keeps finite Taylor jets instead and shows that no curve of critical points can open an end. It continues the problem into complex numbers along two paths around the pinch point. Their actions must agree to within an exponentially small error, but at the collapsing end the equation of the bowl soliton forces a larger, nonzero jump. Our computer checks the paper’s series for that equation, 1, 2, 11, 90, 943, against its factorial bounds. A real Nullstellensatz then gives a finite inequality, and growing graphical radii give finite length, hence one limit.
The numbers in the video
- Ours: a round sphere's squared radius drops by 4 per unit time and a cylinder's by 2; the shrinkers have radius 2 and √2.
- Ours: on the cylinder, z² − 2 (opening) and z cos φ (tilt) are exact Jacobi fields.
- Ours: the paper's bowl-equation series 1, 2, 11, 90, 943, … stays within its factorial bounds for every n up to 120.
- There is no Lean formalization and no code for this result.
Read the source
- Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow (PDF), 100 pages.
- The collection’s catalog, where this is result 355.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Under mean curvature flow, every point of a surface moves in the direction that shrinks its area fastest, at a speed set by how curved the surface is there, and a closed surface always develops a singularity in finite time. A round sphere shrinks to a point: our computer checks that its squared radius drops by four per unit time, and a cylinder’s by two. Zoom in at a singular point, rescaling time by the square of the magnification, and the picture approaches a self-shrinker: a surface that only shrinks while keeping its shape, like a sphere of radius two or a cylinder of radius √2. But different zoom sequences might give different limits: for example, a neck whose axis keeps slowly turning. Tom Ilmanen listed uniqueness of tangent flows as a problem in 2003, and Haslhofer recently called it one of the most fundamental open problems. It was known when the limit is a closed shrinker, by Schulze; a round cylinder, by Colding and Minicozzi; or cone-like far out, by Chodosh and Schulze. Bamler and Kleiner proved that every limit for a surface is a single sheet whose ends look like cones or cylinders, and the remaining case is a shrinker with a cylinder-like end that is not a cylinder. Ilmanen asked whether such a shrinker can exist, and that is still open.
1:21The claim
Result 355 is one OpenAI paper, 100 pages, dated September 24 and made public October 6, 2026. It claims that for any smooth closed embedded surface in space, at every singular point at the first singular time, the whole family of rescalings converges to one self-shrinker, as a single sheet, fixed in the original coordinates, including its position and axes. There is no assumption that the surface moves inward everywhere, and no assumed shape for the limit. Not claimed: later singularities, moving centres, or higher dimensions. It does not say which shrinkers occur, and it does not settle whether a shrinker with a cylinder-like end must be a cylinder; it works around that question. It builds on Bamler and Kleiner’s multiplicity-one theorem and their description of the ends. In 2019, Chodosh and Schulze noted that ruling such shrinkers out would give this result. We found no competing claim.
2:21How the proof goes
The standard route, from Leon Simon’s work, is a Łojasiewicz inequality: if the flow’s speed is at least a power of its remaining energy, its path has finite length and cannot spiral forever. Our toy shows the danger, in a classic example: a smooth but non-analytic landscape whose downhill paths circle forever. On a cylindrical end, one deformation keeps the shrinker equation to first order but grows like z², opening the end into a cone or pinching it off far out. Its effect is smaller than any power of the opening, so the usual analytic argument breaks down in that direction. The paper keeps finite Taylor jets instead and shows that no curve of critical points can open an end. It continues the problem into complex numbers along two paths around the pinch point. Their actions must agree to within an exponentially small error, but at the collapsing end the equation of the bowl soliton forces a larger, nonzero jump. Our computer checks the paper’s series for that equation, 1, 2, 11, 90, 943, against its factorial bounds. A real Nullstellensatz then gives a finite inequality, and growing graphical radii give finite length, hence one limit.
3:40Where it stands
The paper is unrefereed and unchanged since release. There is no Lean formalization and no code; the proof is 100 pages of analysis. An AI audit of part of it found no gap, noting that the result depends entirely on Bamler and Kleiner’s description of the ends, and leaving about 2,500 lines of estimates, among other parts, unchecked. We found no specialist comment and no reported error. If it holds, zooming in at the first singularity of any closed embedded surface always gives one picture, in one position.
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.