Calabi flow breaks?
An AI preprint claims a finite-time singularity of Calabi flow on complex projective space
Covers result 352 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:04. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Calabi flow evolves a Kähler metric so as to even out its scalar curvature: it pushes the metric’s potential by the scalar curvature’s excess over its average, and the Calabi energy, the total squared deviation of scalar curvature from its average, goes down. Our computer checks, on symmetric examples, that this energy goes down. On complex projective space, the standard Fubini–Study metric has constant scalar curvature, in the paper’s normalization n(n + 1), which is 110 in complex dimension ten, and our computer finds that small symmetric perturbations of it die out, to first order.
In 2007, Chen and He recorded Chen’s conjecture that the smooth Calabi flow exists for all time from any starting metric; Chen and Cheng later listed it as a conjecture of Calabi and Chen. On Riemann surfaces it was proved. In higher dimensions, long-time existence was known only for special starting metrics, such as toric surfaces or other symmetric settings, together with criteria saying that the flow continues while the scalar curvature stays bounded. In October 2026, Li, Wang and Zheng still called the general problem open.
Result 352 is one OpenAI paper, 54 pages, dated September 24 and made public October 6, 2026. It claims a counterexample: a smooth Kähler metric on complex projective space of dimension ten, symmetric under the unitary group U(10), which fixes one point, and in the same class as Fubini–Study, whose Calabi flow stops at a finite time, with the scalar curvature at one point blowing up like one over the square root of the remaining time. Taking products extends this to every complex dimension from ten up.
What it doesn’t claim
Not claimed: anything in complex dimensions two through nine; an explicit starting metric or blowup time; anything about weak solutions of the flow; or that such singularities are typical. The blowup constant’s value is not printed; from the paper’s formula and parameters, our computer gets about 13.5.
Where it stands
- Posted as a preprint
- OpenAI, 54 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- No (an uncatalogued folder formalizes one step, assuming the profile)
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. Its computer-assisted step has been rechecked: an outside project re-derived the certificate with its own integrator, and others report re-running the shipped scripts. That checks the finite computation, not the proof. An uncatalogued Lean folder of about 35,000 lines formalizes the right-inverse theorem, assuming the profile exists; the main theorem has no Lean. An AI audit found no gap. We found no specialist comment and no reported error. OSVerify has not reviewed it.
How the proof goes
- A self-similar blowup. Symmetry reduces the flow to one fourth-order equation in one space variable, and the scalar curvature at the fixed point is 110 times one minus the solution’s value there, which our computer checks. Zooming in at that point gives a flat model, and on it the paper builds an exact shrinking self-similar solution from a profile curve. The profile is found by shooting, certified by interval arithmetic out to a fixed distance, then matched to a tail; our own rough shooting shows its shape, falling from one to about 0.18, then rising and levelling off near 0.71.
- From the model to the real flow. The paper then builds a right inverse for the linearized flat equation fixing finitely many slowly decaying or growing modes from the future, and glues the model back into the compact space with a fixed-point argument, producing an exact solution of the full flow. Uniqueness of smooth Calabi flow then makes this the flow from that starting metric, and so that flow blows up.
The numbers in the video
- Ours: the Fubini–Study metric on CP¹⁰ has scalar curvature n(n + 1) = 110 in the paper's normalization; small symmetric perturbations decay to first order.
- Ours: the scalar curvature at the fixed point is 110 (1 − H(0)), checked exactly.
- Ours: from the paper's formula a = −110 G′(0) and its printed parameters, the blowup constant is about 13.52.
- Ours: rough shooting of the profile: it falls from 1 to about 0.18, then rises and levels off near 0.714.
- Lean: an uncatalogued folder of 333 files and 35,347 lines formalizes the right-inverse step, assuming the profile (counted by us; not run by us).
Read the source
- A finite-time singularity of Calabi flow on projective space (PDF), 54 pages.
- The collection’s catalog, where this is result 352.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Calabi flow evolves a Kähler metric so as to even out its scalar curvature: it pushes the metric’s potential by the scalar curvature’s excess over its average, and the Calabi energy, the total squared deviation of scalar curvature from its average, goes down. Our computer checks, on symmetric examples, that this energy goes down. On complex projective space, the standard Fubini–Study metric has constant scalar curvature, in the paper’s normalization n(n + 1), which is 110 in complex dimension ten, and our computer finds that small symmetric perturbations of it die out, to first order. In 2007, Chen and He recorded Chen’s conjecture that the smooth Calabi flow exists for all time from any starting metric; Chen and Cheng later listed it as a conjecture of Calabi and Chen. On Riemann surfaces it was proved. In higher dimensions, long-time existence was known only for special starting metrics, such as toric surfaces or other symmetric settings, together with criteria saying that the flow continues while the scalar curvature stays bounded. In October 2026, Li, Wang and Zheng still called the general problem open.
1:17The claim
Result 352 is one OpenAI paper, 54 pages, dated September 24 and made public October 6, 2026. It claims a counterexample: a smooth Kähler metric on complex projective space of dimension ten, symmetric under the unitary group U(10), which fixes one point, and in the same class as Fubini–Study, whose Calabi flow stops at a finite time, with the scalar curvature at one point blowing up like one over the square root of the remaining time. Taking products extends this to every complex dimension from ten up. Not claimed: anything in complex dimensions two through nine; an explicit starting metric or blowup time; anything about weak solutions of the flow; or that such singularities are typical. The blowup constant’s value is not printed; from the paper’s formula and parameters, our computer gets about 13.5.
2:15How the proof goes
Symmetry reduces the flow to one fourth-order equation in one space variable, and the scalar curvature at the fixed point is 110 times one minus the solution’s value there, which our computer checks. Zooming in at that point gives a flat model, and on it the paper builds an exact shrinking self-similar solution from a profile curve. The profile is found by shooting, certified by interval arithmetic out to a fixed distance, then matched to a tail; our own rough shooting shows its shape, falling from one to about 0.18, then rising and levelling off near 0.71. The paper then builds a right inverse for the linearized flat equation fixing finitely many slowly decaying or growing modes from the future, and glues the model back into the compact space with a fixed-point argument, producing an exact solution of the full flow. Uniqueness of smooth Calabi flow then makes this the flow from that starting metric, and so that flow blows up.
3:16Where it stands
The paper is unrefereed and unchanged since release. Its computer-assisted step has been rechecked: an outside project re-derived the certificate with its own integrator, and others report re-running the shipped scripts. That checks the finite computation, not the proof. An uncatalogued Lean folder of about 35,000 lines formalizes the right-inverse theorem, assuming the profile exists; the main theorem has no Lean. An AI audit found no gap. We found no specialist comment and no reported error. If it holds, smooth Calabi flow can break down in finite time, even on complex projective space, starting in the class of its most symmetric metric.
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.