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

Proved

OAI.StrictHotSpots.Conformal.ClosedDiskChart.subcritical

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.subcritical of lean/OAI/Analysis/HotSpots/HotSpots/Pullback.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 ENNReal InnerProductSpace, open DiskH10):

theorem subcritical (hb : Bornology.IsBounded Ω) (hne : Ω.Nonempty)
    (hμ : 0 < firstPositiveNeumannValue Ω) :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ v : H10 PlaneGreen.diskOpen,
      ‖H10.weightedValue PlaneGreen.diskOpen c.densityBound_ne_top (c.density_bound hμ.le) v‖^2 ≤
        δ*‖H10.grad PlaneGreen.diskOpen v‖^2

Definitions. The statement is written with OAI.StrictHotSpots.H1, OAI.StrictHotSpots.H10, OAI.StrictHotSpots.H1Ambient, OAI.StrictHotSpots.Conformal.ClosedDiskChart, OAI.StrictHotSpots.DiskH10.disk, OAI.StrictHotSpots.H10.grad, OAI.StrictHotSpots.PlaneGreen.diskOpen, 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.H10.weightedValue (definitions bundle OAIQ3Ho002 of this split); OAI.StrictHotSpots.Plane and OAI.StrictHotSpots.firstPositiveNeumannValue (the published bundle Def_HotSpots).

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.gradient_K_representation, OAI.StrictHotSpots.Conformal.ClosedDiskChart.noncritical, OAI.StrictHotSpots.Conformal.ClosedDiskChart.operator_norm_lt_one, OAI.StrictHotSpots.Conformal.ClosedDiskChart.scalarSolution_eq_of_helmholtz and OAI_StrictHotSpots_Conformal_ClosedDiskChart_scalarSolution_form of this split import it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho002

Formal statement

section

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

namespace OAI

section ActualPotential

noncomputable section
open Set MeasureTheory Filter
open scoped ENNReal InnerProductSpace
namespace StrictHotSpots.Conformal
open DiskH10

namespace ClosedDiskChart
variable {Ω : Set Plane} (c : ClosedDiskChart Ω)

theorem subcritical (hb : Bornology.IsBounded Ω) (hne : Ω.Nonempty)
    (hμ : 0 < firstPositiveNeumannValue Ω) :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ v : H10 PlaneGreen.diskOpen,
      ‖H10.weightedValue PlaneGreen.diskOpen c.densityBound_ne_top (c.density_bound hμ.le) v‖^2 ≤
        δ*‖H10.grad PlaneGreen.diskOpen v‖^2 := by
  sorry

end ClosedDiskChart
end StrictHotSpots.Conformal
end
end ActualPotential

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/Pullback.lean (declaration OAI.StrictHotSpots.Conformal.ClosedDiskChart.subcritical)

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