---
id: "2026-10-06-openai-hot-spots-conjecture-simply-connected-lean"
url: "https://postcutoff.com/e/2026-10-06-openai-hot-spots-conjecture-simply-connected-lean/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Lean-verified
sources: 14
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-hot-spots-conjecture-simply-connected-lean/

# OpenAI claims the planar hot spots conjecture (Lean)

Full title: OpenAI's release claims the strict hot spots conjecture for smooth simply connected planar domains, with a Lean proof; Burdzy and Bañuelos extend it to rough domains with ChatGPT without checking it

"Strict hot spots and absence of interior critical points on smooth simply connected planar domains" (OpenAI, 24 Sept 2026, 21 pages, family 369) claims a result for every bounded simply connected planar domain with C^∞ boundary. Every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient inside the domain, so min over ∂Ω < u(x) < max over ∂Ω at every interior point, even when the eigenvalue is multiple. The proof builds on Rohleder's variational principle for tangent vector fields, with boundary multipliers obtained from a Green-kernel theorem. A Lean statement (HotSpots) passed the community comparator runs. Three follow-ups extend the conclusion to nonsmooth domains, assuming OpenAI's theorem: Burdzy (arXiv 2610.12622), Bañuelos (2610.32920) and Jizhou Guo (a GitHub preprint). Atar and Burdzy (2610.19908) build a starlike 3D domain where the conjecture fails.

## Key facts
- Claim (abstract, verbatim): 'We prove the strict hot spots conjecture for smooth bounded simply connected planar domains. More precisely, every nonzero eigenfunction for the first positive Neumann eigenvalue has nonvanishing gradient in the interior, so all its global maxima and minima lie on the boundary. This holds even when the eigenvalue is multiple.'
- Context (paper §1): Burdzy and Werner's 1999 counterexample with two holes and Burdzy's 2005 one-hole example show that topology matters; 'The simply connected planar conjecture is stated explicitly in [Bur05, Conjecture 1.2(ii)].' Earlier positive cases: convex domains with two axes of symmetry (Jerison–Nadirashvili), lip domains (Atar–Burdzy 2004), all triangles (Judge–Mondal), nearly circular convex domains (Miyamoto)
- Method (paper §1): the starting point is 'Rohleder’s variational principle for tangent vector fields'. Scalar boundary multipliers are constructed 'through a reciprocal Green-kernel theorem', using McCullough and Quiggin's characterization of complete Nevanlinna–Pick kernels and Schoenberg's squared-distance embedding. An interior critical point 'would thus produce too many eigenfunction gradients'
- Lean: comparator challenge HotSpots.lean (OAI.StrictHotSpots.main_theorem) covers nonempty bounded open simply connected planar sets with smooth boundary. It defines the first positive Neumann value as the infimum of mean-zero Rayleigh quotients and takes eigenfunctions in weak form; smoothness of the eigenfunction up to the boundary is a hypothesis, which MathVet lists among definitions to check. Accepted in the oai-math-revised comparator run of 25–26 Oct; MathVet verdict 'full', with a sandboxed comparator pass
- Follow-ups that use the theorem as an unchecked input: Burdzy (arXiv 2610.12622, 8 Oct; 'The paper was generated by ChatGPT with prompts from the listed author') obtains non-strict hot spots on every bounded simply connected planar domain with compact Neumann resolvent and a simple eigenvalue, and writes: 'We take this assertion as an input and do not assess or reproduce its proof.' Rodrigo Bañuelos (2610.32920, 23 Oct) proves the same for domains of finite area and adds examples with cusps and unbounded domains; his Section 1 'was generated entirely by ChatGPT (6.1 Sol) in response to a few human prompts'. Jizhou Guo (GitHub preprint, 10 Oct) deduces boundary extrema for every convex planar domain, polygons included
- Criticism (Burdzy's human commentary, 2610.12622): 'The OpenAI model that generated [3] missed the theorem presented in this note', and 'It is highly surprising that [3] did not cite the spectacular paper of de Dios Pont [2], which shows that the hot spots conjecture fails for convex domains in high dimensions'
- Specialists cite it without assessing it. Atar and Burdzy (2610.19908, 14 Oct, a starlike 3D counterexample): 'More recently, OpenAI [18] presented a proof for smooth simply connected planar domains.' Deng, Jiang and Yang (2610.18546, 13 Oct, no AI statement) prove that on every bounded convex planar domain the maximum modulus is attained only on the boundary, and write that OpenAI 'stated a theorem'. Deng, Gui, Jiang, Yang, Yao and Zou (2604.19003v4) note that OpenAI's 'setting, however, excludes polygonal domains with corners'
- Reactions: on Proofs and Prompts (8 Oct), in a reaction signed 'Mitchell Taylor' (the page lists no affiliations): 'In particular, I am happy to see that the hot spots conjecture is true for simply connected subsets of the plane'. AIM (Matthew Colbrook's tracker) marks problem 003, hot spots on smooth convex planar domains, SOLVED, after what it calls an AI statement-level comparison, not an audit of the proof

## What happened

Rauch's hot spots problem asks whether a first nonconstant Neumann eigenfunction, the slowest-decaying mode of heat flow in an
insulated domain, takes its extreme values only on the boundary. Burdzy and Werner disproved the general version in 1999 with a
planar domain with two holes, and Burdzy later found a counterexample with one hole. The simply connected planar case, stated by
Burdzy in 2005, stayed open. Positive results covered only special classes, such as convex domains with two symmetry axes, lip
domains and triangles; general convex planar domains were among the best-known open cases.

OpenAI's release contains a 21-page manuscript dated 24 Sept that claims the strict form of the conjecture for all smooth simply
connected planar domains, for every eigenfunction even when the eigenvalue is multiple. Its Lean statement passed the community
comparator runs. Specialists reacted quickly. On 8 Oct Burdzy posted a ChatGPT-generated note deriving the non-strict conclusion
for nonsmooth simply connected domains from OpenAI's theorem, and Bañuelos posted a longer ChatGPT-assisted version on 23 Oct.
Both state that they did not examine OpenAI's proof. Burdzy also criticised the manuscript for missing this corollary and for not
citing de Dios Pont's high-dimensional convex counterexample. Other groups cite the theorem in human-written papers, and Atar and
Burdzy show the conjecture can fail in three dimensions even for starlike domains.

## Why it matters

This is a claim from OpenAI's release that the leading specialists in the field have started to build on. They treat it as an
unverified hypothesis, and they have used ChatGPT to extend it. If it holds, it settles the planar simply connected case of a
50-year-old problem, including smooth convex domains, while the follow-ups and the 3D counterexample mark where the conjecture
stops. No published expert has checked the proof, so the result still depends on a Lean statement and an unrefereed manuscript.

## What is disputed or not yet verified

- Verification: Lean formalization of the main theorem (comparator accepted); cited by specialists, but no published expert verification yet

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Strict hot spots and absence of interior critical points on smooth simply connected planar domains (24 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf) (github.com, paper)
2. [Lean scope note for family 369](https://github.com/openai/math/blob/main/lean/docs/369.md) (github.com, code)
3. [Lean comparator challenge: HotSpots.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/HotSpots.lean) (github.com, code)
4. [oai-math-revised: comparator run on all challenges (LEAN_CHECKS.md)](https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md) (github.com, code)
5. [arXiv 2610.12622: Burdzy, Removing boundary smoothness from the hot-spots conclusion (generated by ChatGPT)](https://arxiv.org/abs/2610.12622) (arxiv.org, paper)
6. [arXiv 2610.32920: Bañuelos, Hot spots: from smooth to nonsmooth simply connected planar domains](https://arxiv.org/abs/2610.32920) (arxiv.org, paper)
7. [arXiv 2610.19908: Atar and Burdzy, A starlike 3-dimensional counterexample to the hot spots conjecture](https://arxiv.org/abs/2610.19908) (arxiv.org, paper)
8. [arXiv 2610.18546: Deng, Jiang and Yang, strict maximum modulus for Neumann eigenfunctions on convex planar domains](https://arxiv.org/abs/2610.18546) (arxiv.org, paper)
9. [arXiv 2604.19003v4: Deng, Gui, Jiang, Yang, Yao and Zou, isosceles trapezoids and kites (declaration on OpenAI's paper)](https://arxiv.org/abs/2604.19003v4) (arxiv.org, paper)
10. [Jizhou Guo: First Neumann eigenfunctions of convex planar domains attain their extrema on the boundary (GitHub preprint)](https://github.com/aster2024/openai-math-followups/tree/main/hot-spots-convex-planar) (github.com, paper)
11. [oai-math-revised: community notes on the hot spots manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/AGENT_NOTES.md) (github.com, discussion)
12. [Proofs and Prompts: 100+ reactions to 100+ solutions (8 Oct)](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/) (proofsandprompts.com, discussion)
13. [AIM problem 003: Hot spots on convex planar domains (status review)](https://github.com/MColbrook/AIM/blob/main/problems/003-planar-convex-hot-spots.md) (github.com, discussion)
14. [MathVet: fidelity table for openai/math (pre-referee)](https://math.vet/openai-math/) (math.vet, discussion)

## Changes

- 2026-10-29 (filed): Created (manuscript abstract and introduction, HotSpots comparator statement and log, MathVet data, the PDFs of arXiv 2610.12622, 2610.32920, 2610.19908, 2610.18546 and 2604.19003v4, Guo's README, the AIM page, and Proofs and Prompts' reaction collection)

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