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

Proved

OAI.StrictHotSpots.InteriorBoundarySign.exists_boundary_pos

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.InteriorBoundarySign.exists_boundary_pos of lean/OAI/Analysis/HotSpots/HotSpots/Multiplicity.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StrictHotSpots.InteriorBoundarySign, with open Set Filter MeasureTheory, open scoped Topology ContDiff InnerProductSpace ENNReal, open StrictHotSpots.BoundarySign):

lemma exists_boundary_pos (ho : IsOpen Ω) (hbound : Bornology.IsBounded Ω)
    (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hne : ∃ x ∈ Ω, u x ≠ 0) :
    ∃ x ∈ frontier Ω, 0 < u x

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

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.noncritical, OAI.StrictHotSpots.InteriorBoundarySign.exists_boundary_neg and OAI.StrictHotSpots.closedTraceSpace_rank of this split import 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 ActualInteriorBoundarySign

noncomputable section
open Set Filter MeasureTheory
open scoped Topology ContDiff InnerProductSpace ENNReal
namespace StrictHotSpots.InteriorBoundarySign
open StrictHotSpots.BoundarySign
variable {Ω : Set Plane} {u : Plane → ℝ}

lemma exists_boundary_pos (ho : IsOpen Ω) (hbound : Bornology.IsBounded Ω)
    (hu : InInteriorNeumannEigenspace Ω u) (hc : ContinuousOn u (closure Ω)) (hne : ∃ x ∈ Ω, u x ≠ 0) :
    ∃ x ∈ frontier Ω, 0 < u x := by
  sorry

end StrictHotSpots.InteriorBoundarySign
end
end ActualInteriorBoundarySign

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/Multiplicity.lean (declaration OAI.StrictHotSpots.InteriorBoundarySign.exists_boundary_pos)

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