OAI.StrictHotSpots.H10.subcritical: intermediate lemma from OpenAI, The hot spots conjecture for simply connected planar domains

Proved

OAI.StrictHotSpots.H10.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.H10.subcritical of lean/OAI/Analysis/HotSpots/HotSpots/Dirichlet.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StrictHotSpots.H10, with open Set MeasureTheory Filter, open scoped ENNReal InnerProductSpace Topology):

lemma subcritical (hne : Ω.Nonempty) (hb : Bornology.IsBounded Ω) :
    ∃ lam : ℝ, 0 < lam ∧ firstPositiveNeumannValue Ω < lam ∧
      ∀ v : H10 hΩ, lam * ‖value hΩ v‖^2 ≤ ‖grad hΩ v‖^2

Definitions. The statement is written with OAI.StrictHotSpots.H1, OAI.StrictHotSpots.H10, OAI.StrictHotSpots.H1Ambient, OAI.StrictHotSpots.H10.grad and OAI.StrictHotSpots.H10.value (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.Plane and OAI.StrictHotSpots.firstPositiveNeumannValue (the published bundle Def_HotSpots).

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.coordinate_null_trace, OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_identity, OAI_StrictHotSpots_Conformal_ClosedDiskChart_smooth_multiplier_kernel, OAI.StrictHotSpots.Conformal.ClosedDiskChart.subcritical and OAI.StrictHotSpots.InteriorBoundarySign.exists_boundary_pos of this split import it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho000

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 DouglasLipschitzBase
noncomputable section

section StrictGapCombinedLayer
open Set MeasureTheory Filter
open scoped ENNReal InnerProductSpace Topology
namespace StrictHotSpots
namespace H10
variable {Ω : Set Plane} (hΩ : IsOpen Ω)

lemma subcritical (hne : Ω.Nonempty) (hb : Bornology.IsBounded Ω) :
    ∃ lam : ℝ, 0 < lam ∧ firstPositiveNeumannValue Ω < lam ∧
      ∀ v : H10 hΩ, lam * ‖value hΩ v‖^2 ≤ ‖grad hΩ v‖^2 := by
  sorry
end H10
end StrictHotSpots
end StrictGapCombinedLayer

end

end DouglasLipschitzBase

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/Dirichlet.lean (declaration OAI.StrictHotSpots.H10.subcritical)

View graph