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

Proved

OAI.StrictHotSpots.PlaneGreen.boundarySolution_energy

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.PlaneGreen.boundarySolution_energy of lean/OAI/Analysis/HotSpots/HotSpots/GreenSolution.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StrictHotSpots.PlaneGreen, with open Set MeasureTheory Filter Metric, open scoped Topology ContDiff InnerProductSpace ENNReal NNReal, open DiskH10):

theorem boundarySolution_energy (hT : ‖weightedIntegralOperator hC hν‖ < 1)
    (f g : C(Boundary,ℝ)) {A B : ℝ≥0} (hf : LipschitzWith A f) (hg : LipschitzWith B g) :
    inner ℝ (H1.grad (boundarySolution hC hν f hf)) (H1.grad (boundarySolution hC hν g hg)) -
      inner ℝ (SubcriticalExtension.observe hC hν (boundarySolution hC hν f hf))
        (SubcriticalExtension.observe hC hν (boundarySolution hC hν g hg)) =
      (1/2:ℝ)*(∫ z : Boundary × Boundary, boundaryInteraction z.1 z.2 *
        inner ℝ (f z.1-f z.2) (g z.1-g z.2) ∂boundaryMeasure.prod boundaryMeasure) -
      (∫ z : Boundary × Boundary, fullBoundaryR hC hν z.1 z.2 * f z.1 * g z.2
        ∂boundaryMeasure.prod boundaryMeasure)

Definitions. The statement is written with OAI.StrictHotSpots.H1, OAI.StrictHotSpots.H1Ambient, OAI.StrictHotSpots.DiskH10.disk, OAI.StrictHotSpots.H1.grad and OAI.StrictHotSpots.PlaneGreen.Boundary (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.PlaneGreen.boundarySolution and OAI.StrictHotSpots.SubcriticalExtension.observe (definitions bundle OAIQ3Ho002 of this split); OAI.StrictHotSpots.PlaneGreen.boundaryInteraction, OAI.StrictHotSpots.PlaneGreen.boundaryMeasure, OAI.StrictHotSpots.PlaneGreen.fullBoundaryR and OAI.StrictHotSpots.PlaneGreen.weightedIntegralOperator (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Plane (the published bundle Def_HotSpots).

Used by. The proof of OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference of this split imports it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho002
import Definitions.Def_OAIQ3Ho003

Formal statement

section

attribute [-instance] _root_.OAI.StrictHotSpots.N1.completeSpace _root_.OAI.StrictHotSpots.N1.innerProductSpace _root_.OAI.StrictHotSpots.N1.normedAddCommGroup
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section BoundaryScalarEnergy

noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff InnerProductSpace ENNReal NNReal
namespace StrictHotSpots.PlaneGreen
open DiskH10
variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞}
  (hC : C ≠ (∞ : ℝ≥0∞)) (hν : ν ≤ C • volume.restrict disk)

theorem boundarySolution_energy (hT : ‖weightedIntegralOperator hC hν‖ < 1)
    (f g : C(Boundary,ℝ)) {A B : ℝ≥0} (hf : LipschitzWith A f) (hg : LipschitzWith B g) :
    inner ℝ (H1.grad (boundarySolution hC hν f hf)) (H1.grad (boundarySolution hC hν g hg)) -
      inner ℝ (SubcriticalExtension.observe hC hν (boundarySolution hC hν f hf))
        (SubcriticalExtension.observe hC hν (boundarySolution hC hν g hg)) =
      (1/2:ℝ)*(∫ z : Boundary × Boundary, boundaryInteraction z.1 z.2 *
        inner ℝ (f z.1-f z.2) (g z.1-g z.2) ∂boundaryMeasure.prod boundaryMeasure) -
      (∫ z : Boundary × Boundary, fullBoundaryR hC hν z.1 z.2 * f z.1 * g z.2
        ∂boundaryMeasure.prod boundaryMeasure) := by
  sorry

end StrictHotSpots.PlaneGreen
end
end BoundaryScalarEnergy

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/GreenSolution.lean (declaration OAI.StrictHotSpots.PlaneGreen.boundarySolution_energy)

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