OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference: intermediate lemma from OpenAI, The hot spots conjecture for simply connected planar domains

Proved

OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference

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.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference of lean/OAI/Analysis/HotSpots/HotSpots/Multiplier.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma multipliedGradient_coordinate_difference {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (i : Fin 2) :
    let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1
    H1.form (firstPositiveNeumannValue Ω) (c.multipliedGradient hΩ hu hμ hb i)
      (c.multipliedGradient hΩ hu hμ hb i)-
    H1.form (firstPositiveNeumannValue Ω) (eigenGradientField hΩ hu i)
      (c.multipliedGradient hΩ hu hμ (hb.mul hb) i) =
    (1/2:ℝ)*∫ z : Boundary × Boundary,
      (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)*
      (b (c.F z.1)-b (c.F z.2))^2*
      inner ℝ (fderivWithin ℝ u (closure Ω) (c.F z.1) (Hodge.e i))
        (fderivWithin ℝ u (closure Ω) (c.F z.2) (Hodge.e i)) ∂boundaryMeasure.prod boundaryMeasure

Definitions. The statement is written with OAI.StrictHotSpots.Conformal.ClosedDiskChart, OAI.StrictHotSpots.PlaneGreen.Boundary, OAI.StrictHotSpots.Conformal.ClosedDiskChart.densityBound, OAI.StrictHotSpots.Conformal.ClosedDiskChart.densityBound_ne_top, OAI.StrictHotSpots.Conformal.ClosedDiskChart.density_bound and OAI.StrictHotSpots.Conformal.ClosedDiskChart.potential (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.Hodge.e and OAI.StrictHotSpots.Hodge.eigenGradientField (definitions bundle OAIQ3Ho002 of this split); OAI.StrictHotSpots.H1.form, OAI.StrictHotSpots.PlaneGreen.boundaryInteraction, OAI.StrictHotSpots.PlaneGreen.boundaryMeasure and OAI.StrictHotSpots.PlaneGreen.fullBoundaryR (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient and OAI.StrictHotSpots.Conformal.ClosedDiskChart.potential_finite (definitions bundle OAIQ3Ho004 of this split); OAI.StrictHotSpots.AdmissibleDomain, OAI.StrictHotSpots.InFirstNeumannEigenspace, OAI.StrictHotSpots.Plane, OAI.StrictHotSpots.SmoothBoundary and OAI.StrictHotSpots.firstPositiveNeumannValue (the published bundle Def_HotSpots).

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.lipschitz_multiplier_kernel and OAI_StrictHotSpots_Conformal_ClosedDiskChart_smooth_multiplier_kernel of this split import it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho004

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 SmoothMultiplierKernel

noncomputable section
open Set MeasureTheory Filter
open scoped ContDiff InnerProductSpace ENNReal NNReal
namespace StrictHotSpots.Conformal.ClosedDiskChart
open DiskH10 PlaneGreen Hodge
variable {Ω : Set Plane} (c : ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω)
variable {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u)
variable (hμ : 0 < firstPositiveNeumannValue Ω)

lemma multipliedGradient_coordinate_difference {b : Plane → ℝ} (hb : ContDiff ℝ ∞ b) (i : Fin 2) :
    let _ : IsFiniteMeasure c.potential := c.potential_finite hΩ.2.2.1
    H1.form (firstPositiveNeumannValue Ω) (c.multipliedGradient hΩ hu hμ hb i)
      (c.multipliedGradient hΩ hu hμ hb i)-
    H1.form (firstPositiveNeumannValue Ω) (eigenGradientField hΩ hu i)
      (c.multipliedGradient hΩ hu hμ (hb.mul hb) i) =
    (1/2:ℝ)*∫ z : Boundary × Boundary,
      (boundaryInteraction z.1 z.2+fullBoundaryR c.densityBound_ne_top (c.density_bound hμ.le) z.1 z.2)*
      (b (c.F z.1)-b (c.F z.2))^2*
      inner ℝ (fderivWithin ℝ u (closure Ω) (c.F z.1) (Hodge.e i))
        (fderivWithin ℝ u (closure Ω) (c.F z.2) (Hodge.e i)) ∂boundaryMeasure.prod boundaryMeasure := by
  sorry

end StrictHotSpots.Conformal.ClosedDiskChart
end
end SmoothMultiplierKernel

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/Multiplier.lean (declaration OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference)

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