OAI.StrictHotSpots.Helmholtz.plane_representative: intermediate lemma from OpenAI, The hot spots conjecture for simply connected planar domains

Proved

OAI.StrictHotSpots.Helmholtz.plane_representative

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.StrictHotSpots.main_theorem, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The hot spots conjecture for simply connected planar domains, OpenAI Math Release, September 24, 2026, result family 369, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.StrictHotSpots.Helmholtz.plane_representative of lean/OAI/Analysis/HotSpots/HotSpots/Eigenfunction.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StrictHotSpots.Helmholtz, with open Set MeasureTheory Filter Metric, open scoped ContDiff Laplacian Topology):

theorem plane_representative {Ω : Set Plane} {u : Plane → ℝ} {μ : ℝ}
    (ho : IsOpen Ω) (hu : IntegrableOn u Ω) (hμ : 0 ≤ μ) (hp : PlaneWeakEquation Ω μ u) :
    ∃ f : Plane → ℝ, ContDiffOn ℝ ∞ f Ω ∧ ∀ᵐ x ∂volume.restrict Ω, f x = u x

Definitions. The statement is written with OAI.StrictHotSpots.PlaneWeakEquation (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Plane (the published bundle Def_HotSpots).

Used by. The proof of OAI.StrictHotSpots.Hodge.vector_equality_interior_pointwise of this split imports it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho003

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section ActualInteriorEigenSupport

noncomputable section
open Set MeasureTheory Filter Metric
open scoped ContDiff Laplacian Topology
namespace StrictHotSpots.Helmholtz

theorem plane_representative {Ω : Set Plane} {u : Plane → ℝ} {μ : ℝ}
    (ho : IsOpen Ω) (hu : IntegrableOn u Ω) (hμ : 0 ≤ μ) (hp : PlaneWeakEquation Ω μ u) :
    ∃ f : Plane → ℝ, ContDiffOn ℝ ∞ f Ω ∧ ∀ᵐ x ∂volume.restrict Ω, f x = u x := by
  sorry

end StrictHotSpots.Helmholtz

end
end ActualInteriorEigenSupport

end OAI
end

Source

OpenAI, The hot spots conjecture for simply connected planar domains, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 369; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/HotSpots/HotSpots/Eigenfunction.lean (declaration OAI.StrictHotSpots.Helmholtz.plane_representative)

View graph