Proves a strict form of Burdzy’s simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. (OpenAI’s Lean note)
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The paper states
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every 0\ne u\in\V,
In particular,
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HotSpotsThe paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
The Lean statement
No paper theorem is paired with it.
HotSpots.lean, the release lists it as a supporting statement.
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 sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
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MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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)
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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. 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.
Strict hot spots and absence of interior critical points HotSpots
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: first nonconstant Neumann eigenfunctions on smooth bounded simply connected planar domains have no interior critical points, so their extrema lie on the boundary.
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HotSpotsThree reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 416 s; peak memory 6.9 GiB). | 11 October 2026 | Original report |
These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
The J₁ power series has derivative J₀ − J₁/x on (1, 2), and that derivative vanishes somewhere in (1.8411, 1.8412)
HOL Light card 369
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Jizhou Guo, Follow-ups to the OpenAI mathematics release (papers, sources and Lean ancillary files), 10 October 2026. Source
First Neumann eigenfunctions of convex planar domains attain extrema on the boundary, deduced from the release’s smooth-domain hot spots paper (10 October).
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Why this passage is shown
Theorem 1.1
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Proposition 2.5
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Theorem 1.1
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Proposition 4.4 (Multiplier identity)
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Proposition 5.5
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Proposition 3.1
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Why this passage is shown
Theorem 5.1
In Strict hot spots and absence of interior critical points on smooth simply connected planar domains, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, HotSpots
Proof-verification report
MathVet, HotSpots
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, HotSpots
Report using another proof or calculation method
The J₁ power series has derivative J₀ − J₁/x on (1, 2), and that derivative vanishes somewhere in (1.8411, 1.8412)
Proofs of selected steps in HOL Light, Ben Knill, openai-math-hol-cards
68a1228Lean statement
HotSpots