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

Proved

OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_tendsto

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_tendsto 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 Topology, open DiskH10 PlaneGreen Hodge):

lemma multipliedGradient_tendsto {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u)
    {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (i : Fin 2) :
    Tendsto (fun n => c.multipliedGradient hΩ hu hμ (ambientSmooth_smooth (planeApproxBump n) hb) i)
      atTop (𝓝 (lipschitzMultipliedGradient hΩ hμ hl hD hu hb i))

Definitions. The statement is written with OAI.StrictHotSpots.H1, OAI.StrictHotSpots.H10, OAI.StrictHotSpots.H1Ambient, OAI.StrictHotSpots.Conformal.ClosedDiskChart, OAI.StrictHotSpots.H10.grad and OAI.StrictHotSpots.H10.value (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.ambientSmooth, OAI.StrictHotSpots.ambientSmooth_smooth and OAI.StrictHotSpots.planeApproxBump (definitions bundle OAIQ3Ho002 of this split); OAI.StrictHotSpots.Conformal.ClosedDiskChart.lipschitzMultipliedGradient and OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient (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 proof of OAI.StrictHotSpots.Conformal.ClosedDiskChart.lipschitz_multiplier_kernel of this split imports 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 PhysicalMultiplierLimit

noncomputable section
open Set MeasureTheory Filter
open scoped ContDiff InnerProductSpace ENNReal NNReal Topology
namespace StrictHotSpots.Conformal.ClosedDiskChart
open DiskH10 PlaneGreen Hodge
variable {Ω : Set Plane} (c : ClosedDiskChart Ω) (hΩ : AdmissibleDomain Ω)
variable (hμ : 0 < firstPositiveNeumannValue Ω)
variable {L : ℝ} (hl : firstPositiveNeumannValue Ω < L)
  (hD : ∀ j : H10 hΩ.2.1, L*‖H10.value hΩ.2.1 j‖^2 ≤ ‖H10.grad hΩ.2.1 j‖^2)

lemma multipliedGradient_tendsto {u : Plane → ℝ} (hu : InFirstNeumannEigenspace Ω u)
    {b : Plane → ℝ} {B : ℝ≥0} (hb : LipschitzWith B b) (i : Fin 2) :
    Tendsto (fun n => c.multipliedGradient hΩ hu hμ (ambientSmooth_smooth (planeApproxBump n) hb) i)
      atTop (𝓝 (lipschitzMultipliedGradient hΩ hμ hl hD hu hb i)) := by
  sorry

end StrictHotSpots.Conformal.ClosedDiskChart

end
end PhysicalMultiplierLimit

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_tendsto)

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