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

Proved

OAI.StrictHotSpots.H10.minimum

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.minimum 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 Topology InnerProductSpace):

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

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 (the published bundle Def_HotSpots).

Used by. The proof of OAI.StrictHotSpots.H10.subcritical of this split imports 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 DirichletMinimumCombinedLayer
open Set MeasureTheory Filter
open scoped ENNReal Topology InnerProductSpace
namespace StrictHotSpots.H10
variable {Ω : Set Plane} (hΩ : IsOpen Ω)

lemma minimum (hne : Ω.Nonempty) (hb : Bornology.IsBounded Ω) :
    ∃ (lam : ℝ) (φ : H10 hΩ), 0 < lam ∧ value hΩ φ ≠ 0 ∧
      (∀ v, lam * ‖value hΩ v‖^2 ≤ ‖grad hΩ v‖^2) ∧
      (∀ v, inner ℝ (grad hΩ φ) (grad hΩ v) =
        lam * inner ℝ (value hΩ φ) (value hΩ v)) := by
  sorry

end StrictHotSpots.H10
end DirichletMinimumCombinedLayer

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.minimum)

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