Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Lean statement

Strict hot spots and absence of interior critical points

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.StrictHotSpots.main_theorem

Lean statement file: HotSpots.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 369 link it.

See the closest candidatesWho ran the proof-verification programReport a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

  • abbreviation OAI.StrictHotSpots.Plane line 11

    abbrev Plane := EuclideanSpace ℝ (Fin 2)

  • definition OAI.StrictHotSpots.SmoothBoundary line 13

    def SmoothBoundary (Ω : Set Plane) : Prop :=  ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ),    IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧    fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧    Ω ∩ U = {x | ρ x < 0} ∩ U

  • definition OAI.StrictHotSpots.AdmissibleDomain line 19

    def AdmissibleDomain (Ω : Set Plane) : Prop :=  Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧    IsSimplyConnected Ω ∧ SmoothBoundary Ω

  • definition OAI.StrictHotSpots.HasH1Gradient line 23

    def HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop :=  MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧  ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω →    ∀ e : Plane,      (∫ x in Ω, v x * (fderiv ℝ φ x) e) =        -(∫ x in Ω, (inner ℝ (g x) e) * φ x)

  • definition OAI.StrictHotSpots.rayleighValues line 30

    def rayleighValues (Ω : Set Plane) : Set ℝ :=  {r | ∃ (v : Plane → ℝ) (g : Plane → Plane),    HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧    0 < (∫ x in Ω, (v x) ^ 2) ∧    r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}

  • definition OAI.StrictHotSpots.firstPositiveNeumannValue line 36

    def firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)

  • definition OAI.StrictHotSpots.InFirstNeumannEigenspace line 38

    def InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop :=  ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧    (∫ x in Ω, u x) = 0 ∧    ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g →      (∫ x in Ω, inner ℝ (gradient u x) (g x)) =        firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)

  • definition OAI.StrictHotSpots.MainConclusion line 45

    def MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop :=  (∀ x ∈ Ω, gradient u x ≠ 0) ∧    ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Other possible paper matches, closest wording first

  • Proposition 2.5 of Strict hot spots and absence of interior critical points on smooth simply connected planar domains. Words of the Lean statement, its names split into words, that also appear in the theorem: eigenspace and real.
  • Theorem 1.1 of Strict hot spots and absence of interior critical points on smooth simply connected planar domains. Found in the paper’s title, which its main theorem states: connected, smooth, simply and domains.
  • Proposition 4.4 of Strict hot spots and absence of interior critical points on smooth simply connected planar domains. Words of the Lean statement, its names split into words, that also appear in the theorem: real.
  • Proposition 5.5 of Strict hot spots and absence of interior critical points on smooth simply connected planar domains. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: interior and points.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

  • Accepted Their term: “CERTIFIED”.

    Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.

    Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

  • Accepted Their term: “comparator-sandboxed-pass”.

    Re-run through Comparator in a sandboxed virtual machine: accepted.

    Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

  • Accepted Their term: “accepted”.

    Re-run through Comparator with Lean’s kernel: accepted (the check took 416 s; peak memory 6.9 GiB).

    Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 369’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

  • MathVet

    Finds the whole advertised result stated or implied Their term: “full”.

    The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record

    Their note: main_theorem: on every nonempty bounded open simply connected planar domain with smooth boundary, every nonzero u in the first positive Neumann eigenspace (smooth up to the boundary, mean zero, weak eigen-equation against all H¹ test pairs with eigenvalue = inf of the mean-zero Rayleigh quotient) has ∇u ≠ 0 at every interior point and inf_∂Ω u < u(x) < sup_∂Ω u inside, i.e. no interior critical point and strict hot spots; eigenvalue multiplicity is allowed because the statement is for any element of the eigenspace.

    Definitions the group asks readers to compare with the paper: firstPositiveNeumannValue Ω := sInf (rayleighValues Ω) with hand-built HasH1Gradient and SmoothBoundary (local defining functions ρ with nonzero gradient); InFirstNeumannEigenspace requires ContDiffOn ℝ ∞ u (closure Ω)

  • Carlos Toledo’s cert-machine

    Finds the whole advertised result stated or implied Their term: “MATCHES”.

    The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the stronger interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.”

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

57 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 33 to its own declarations, 39 to Mathlib’s documentation. 9 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionopen Set MeasureTheoryopen scoped ContDiffnamespace StrictHotSpotsabbrev Plane := EuclideanSpace ℝ (Fin 2)def SmoothBoundary (Ω : Set Plane) : Prop :=  ∀ p ∈ frontier Ω, ∃ (U : Set Plane) (ρ : Plane → ℝ),    IsOpen U ∧ p ∈ U ∧ ContDiffOn ℝ ∞ ρ U ∧    fderiv ℝ ρ p ≠ 0 ∧ ρ p = 0 ∧    Ω ∩ U = {x | ρ x < 0} ∩ Udef AdmissibleDomain (Ω : Set Plane) : Prop :=  Ω.Nonempty ∧ IsOpen Ω ∧ Bornology.IsBounded Ω ∧    IsSimplyConnected Ω ∧ SmoothBoundary Ωdef HasH1Gradient (Ω : Set Plane) (v : Plane → ℝ) (g : Plane → Plane) : Prop :=  MemLp v 2 (volume.restrict Ω) ∧ MemLp g 2 (volume.restrict Ω) ∧  ∀ φ : Plane → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ Ω →    ∀ e : Plane,      (∫ x in Ω, v x * (fderiv ℝ φ x) e) =        -(∫ x in Ω, (inner ℝ (g x) e) * φ x)def rayleighValues (Ω : Set Plane) : Set ℝ :=  {r | ∃ (v : Plane → ℝ) (g : Plane → Plane),    HasH1Gradient Ω v g ∧ (∫ x in Ω, v x) = 0 ∧    0 < (∫ x in Ω, (v x) ^ 2) ∧    r = (∫ x in Ω, ‖g x‖ ^ 2) / (∫ x in Ω, (v x) ^ 2)}def firstPositiveNeumannValue (Ω : Set Plane) : ℝ := sInf (rayleighValues Ω)def InFirstNeumannEigenspace (Ω : Set Plane) (u : Plane → ℝ) : Prop :=  ContDiffOn ℝ ∞ u (closure Ω) ∧ HasH1Gradient Ω u (gradient u) ∧    (∫ x in Ω, u x) = 0 ∧    ∀ (v : Plane → ℝ) (g : Plane → Plane), HasH1Gradient Ω v g →      (∫ x in Ω, inner ℝ (gradient u x) (g x)) =        firstPositiveNeumannValue Ω * (∫ x in Ω, u x * v x)def MainConclusion (Ω : Set Plane) (u : Plane → ℝ) : Prop :=  (∀ x ∈ Ω, gradient u x ≠ 0) ∧    ∀ x ∈ Ω, sInf (u '' frontier Ω) < u x ∧ u x < sSup (u '' frontier Ω)theorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by  sorryend StrictHotSpotsendend OAI
Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Strict hot spots and absence of interior critical points on smooth simply connected planar domains

Theorem

For every 0\ne u\in\V,

\grad u(x)\ne0\qquad(x\in\Omega).

In particular,

\min_{y\in\partial\Omega}u(y)<u(x)<\max_{y\in\partial\Omega}u(y) \qquad(x\in\Omega).
Section 1, Introduction Read it in the paper TeX source, line 13
Show the TeX
For every $0\ne u\in\V$,
\[
\grad u(x)\ne0\qquad(x\in\Omega).
\]
In particular,
\begin{equation}
\min_{y\in\partial\Omega}u(y)<u(x)<\max_{y\in\partial\Omega}u(y)
\qquad(x\in\Omega).
\end{equation}
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 369 link it.
Solution module
OAI.Analysis.HotSpots.Main
Allowed axioms
propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports
Mathlib
Imports from outside Mathlib
None
Local vocabulary
8 local vocabulary declarations
Lines of Lean, without comments or blank lines
34
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas

Strict hot spots and absence of interior critical points

Family 369: The hot spots conjecture for simply connected planar domains

No paper statement selected

no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

The Lean statement says

theorem main_theorem (Ω : Set Plane) (hΩ : AdmissibleDomain Ω)    (u : Plane → ℝ) (hu : InFirstNeumannEigenspace Ω u)    (hne : ∃ x ∈ Ω, u x ≠ 0) : MainConclusion Ω u := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

No paper theorem is paired with this Lean statement.three independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

8 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/HotSpots.html

Recorded comparisons with the paper

Atlas records no contributed comparison of this statement with the paper.

Contributed Comparator run reports

Atlas records no contributed report of running Comparator for this statement file.

Compare this statement with the paperShare a Comparator run reportSuggest a change