Hot spots on the edge?
An AI preprint claims a strict hot spots theorem for every smooth plane shape without holes
Covers result 369 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
4:02. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Heat a metal plate unevenly, insulate its edges, and wait. Where is the hottest point after a long time? In 1974 Jeffrey Rauch conjectured: on the edge. Late in the cooling, the temperature’s departure from its average follows a first non-constant Neumann eigenfunction of the plate, so the question is where that function peaks. On a rectangle and a disc, our computer finds the peaks on the edge, and on an L-shaped plate the hottest spot drifts to the outer corner of an arm, cooling at the rate our grid’s eigenvalue predicts.
In general the conjecture is false. In 1999 Burdzy and Werner found a plate with two holes where it fails, Burdzy later found one hole, and in 2024 de Dios Pont found convex counterexamples in high dimensions. It was proved for special shapes, including every triangle. In 2004 and 2005 Burdzy conjectured it for every plane shape without holes, and as late as August 2026, even the convex plane case was called a major open problem.
Result 369 is one OpenAI paper, 21 pages, dated September 24 and made public October 6, 2026. It claims that on every bounded plane shape without holes and with a smooth boundary, every eigenfunction for the first positive Neumann eigenvalue has no interior critical point at all: no peaks, no pits, no saddles. So its maximum and minimum are on the boundary, strictly, even when the eigenvalue is double. That is stronger than Burdzy’s conjecture, but only for smooth boundaries.
What it doesn’t claim
Not claimed: shapes with corners or rough edges, shapes with holes, or higher dimensions. Two days after the release, Burdzy posted a note, generated with ChatGPT, that assumes this theorem and removes the smoothness for a weaker, non-strict conclusion, when the eigenvalue is simple. It says Bañuelos independently got almost the same theorem and proof from ChatGPT, it does not assess the OpenAI proof, and it says the OpenAI model missed that theorem.
Where it stands
- Posted as a preprint
- OpenAI, 21 pages, dated September 24, 2026
- Peer reviewed
- Not yet; no specialist verdict on the proof
- Formalized in Lean
- The main theorem; the formal version assumes the eigenfunction is smooth up to the boundary, which the paper proves (scope notes)
Last checked October 19, 2026. The paper is unrefereed and unchanged since release. Its main theorem has a Lean proof, about 29,000 lines, not run by us; the formal version assumes the eigenfunction is smooth up to the boundary, which the paper proves. Burdzy’s note criticises the paper for not citing de Dios Pont, and we found no specialist verdict on the proof. OSVerify has not reviewed it.
How the proof goes
- From a variational principle to the disc. The proof starts from Rohleder’s variational principle: for vector fields tangent to the boundary, an energy defect is zero exactly for gradients of these eigenfunctions. The eigenspace has dimension at most two, a known bound the paper re-proves. A conformal map carries the shape to the disc, where the defect becomes integrals around the circle, and multiplying the boundary gradient by a scalar function gives a multiplier identity.
- A kernel with one positive direction. The engine is a theorem about kernels built from the disc’s Green and Poisson kernels, weighted by the conformal map. The key fact: matrices of their entrywise reciprocals have at most one positive eigenvalue. Our computer checks the plain disc Green kernel on random points: always one at most, while a Gaussian kernel gives up to four. That property places the boundary circle in a Hilbert space, with the kernel as squared distance, and summing gives half the squared gradient at any interior point. If the gradient vanished there, either three independent eigenfunctions would appear, which is impossible, or a combination would vanish on a boundary arc, which unique continuation forbids.
The numbers in the video
- Ours: on a 2 × 1 rectangle and on a disc, the first Neumann eigenfunctions peak on the edge.
- Ours: heat flow on an L-shaped plate (corners, outside the theorem) settles with the hot spot at an outer corner, cooling at rate 1.464 against the grid eigenvalue 1.462.
- Ours: entrywise reciprocal matrices of the plain disc Green kernel had at most one positive eigenvalue on random points; a Gaussian kernel gave up to four.
- Lean: 31 files and 29,338 lines in the import closure of the comparator's solution (counted by us; not run by us).
Read the source
- Strict hot spots and absence of interior critical points on smooth simply connected planar domains (PDF), 21 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 369.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
Heat a metal plate unevenly, insulate its edges, and wait. Where is the hottest point after a long time? In 1974 Jeffrey Rauch conjectured: on the edge. Late in the cooling, the temperature’s departure from its average follows a first non-constant Neumann eigenfunction of the plate, so the question is where that function peaks. On a rectangle and a disc, our computer finds the peaks on the edge, and on an L-shaped plate the hottest spot drifts to the outer corner of an arm, cooling at the rate our grid’s eigenvalue predicts. In general the conjecture is false. In 1999 Burdzy and Werner found a plate with two holes where it fails, Burdzy later found one hole, and in 2024 de Dios Pont found convex counterexamples in high dimensions. It was proved for special shapes, including every triangle. In 2004 and 2005 Burdzy conjectured it for every plane shape without holes, and as late as August 2026, even the convex plane case was called a major open problem.
1:06The claim
Result 369 is one OpenAI paper, 21 pages, dated September 24 and made public October 6, 2026. It claims that on every bounded plane shape without holes and with a smooth boundary, every eigenfunction for the first positive Neumann eigenvalue has no interior critical point at all: no peaks, no pits, no saddles. So its maximum and minimum are on the boundary, strictly, even when the eigenvalue is double. That is stronger than Burdzy’s conjecture, but only for smooth boundaries. Not claimed: shapes with corners or rough edges, shapes with holes, or higher dimensions. Two days after the release, Burdzy posted a note, generated with ChatGPT, that assumes this theorem and removes the smoothness for a weaker, non-strict conclusion, when the eigenvalue is simple. It says Bañuelos independently got almost the same theorem and proof from ChatGPT, it does not assess the OpenAI proof, and it says the OpenAI model missed that theorem.
2:13How the proof goes
The proof starts from Rohleder’s variational principle: for vector fields tangent to the boundary, an energy defect is zero exactly for gradients of these eigenfunctions. The eigenspace has dimension at most two, a known bound the paper re-proves. A conformal map carries the shape to the disc, where the defect becomes integrals around the circle, and multiplying the boundary gradient by a scalar function gives a multiplier identity. The engine is a theorem about kernels built from the disc’s Green and Poisson kernels, weighted by the conformal map. The key fact: matrices of their entrywise reciprocals have at most one positive eigenvalue. Our computer checks the plain disc Green kernel on random points: always one at most, while a Gaussian kernel gives up to four. That property places the boundary circle in a Hilbert space, with the kernel as squared distance, and summing gives half the squared gradient at any interior point. If the gradient vanished there, either three independent eigenfunctions would appear, which is impossible, or a combination would vanish on a boundary arc, which unique continuation forbids.
3:23Where it stands
The paper is unrefereed and unchanged since release. Its main theorem has a Lean proof, about 29,000 lines, not run by us; the formal version assumes the eigenfunction is smooth up to the boundary, which the paper proves. Burdzy’s note criticises the paper for not citing de Dios Pont, and we found no specialist verdict on the proof. If it holds, on any smooth plane shape without holes, the long-time hot and cold spots sit on the edge, and the temperature pattern that lasts has no interior peaks, pits or saddles.
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.