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

Proved

OAI.StrictHotSpots.PlaneGreen.full_boundary_hilbert_embedding

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.PlaneGreen.full_boundary_hilbert_embedding of lean/OAI/Analysis/HotSpots/HotSpots/KernelLimit.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem full_boundary_hilbert_embedding {d : ℝ} (hd0 : 0 ≤ d) (hd1 : d < 1)
    (hsub : ∀ u : H10 diskOpen, ‖H10.weightedValue diskOpen hC hν u‖ ^ 2 ≤
      d * ‖H10.grad diskOpen u‖ ^ 2) (p : disk) :
    ∃ (H : Type) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℝ H)
      (_ : CompleteSpace H) (_ : TopologicalSpace.SeparableSpace H) (b : Boundary → H) (L : ℝ≥0),
      Function.Injective b ∧ LipschitzWith L b ∧
        ∀ s t, ‖b s - b t‖ ^ 2 = fullDistanceKernel hC hν p s t

Definitions. The statement is written with OAI.StrictHotSpots.H1, OAI.StrictHotSpots.H10, OAI.StrictHotSpots.H1Ambient, OAI.StrictHotSpots.DiskH10.disk, OAI.StrictHotSpots.H10.grad, OAI.StrictHotSpots.PlaneGreen.Boundary and OAI.StrictHotSpots.PlaneGreen.diskOpen (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.H10.weightedValue (definitions bundle OAIQ3Ho002 of this split); OAI.StrictHotSpots.PlaneGreen.fullDistanceKernel (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Plane (the published bundle Def_HotSpots).

Used by. The proof of OAI.StrictHotSpots.Conformal.ClosedDiskChart.noncritical of this split imports it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho002
import Definitions.Def_OAIQ3Ho003

Formal statement

section

attribute [-instance] _root_.OAI.StrictHotSpots.Douglas.haar_nullSingleton _root_.OAI.StrictHotSpots.N1.completeSpace _root_.OAI.StrictHotSpots.N1.innerProductSpace _root_.OAI.StrictHotSpots.N1.normedAddCommGroup _root_.OAI.StrictHotSpots.PlaneGreen.boundaryMeasure_finite _root_.OAI.StrictHotSpots.PlaneGreen.boundaryMeasure_nullSingleton _root_.OAI.StrictHotSpots.PlaneGreen.boundaryMeasure_openPos _root_.OAI.StrictHotSpots.PlaneGreen.boundary_compactSpace
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 FullBoundaryCombinedLayer
section FullBoundaryEmbeddingLayer
noncomputable section
open MeasureTheory Set Filter
open scoped ENNReal NNReal Topology InnerProductSpace
namespace StrictHotSpots.PlaneGreen
open DiskH10
variable {ν : Measure Plane} [IsFiniteMeasure ν] {C : ℝ≥0∞}
  (hC : C ≠ ∞) (hν : ν ≤ C • volume.restrict disk)

theorem full_boundary_hilbert_embedding {d : ℝ} (hd0 : 0 ≤ d) (hd1 : d < 1)
    (hsub : ∀ u : H10 diskOpen, ‖H10.weightedValue diskOpen hC hν u‖ ^ 2 ≤
      d * ‖H10.grad diskOpen u‖ ^ 2) (p : disk) :
    ∃ (H : Type) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℝ H)
      (_ : CompleteSpace H) (_ : TopologicalSpace.SeparableSpace H) (b : Boundary → H) (L : ℝ≥0),
      Function.Injective b ∧ LipschitzWith L b ∧
        ∀ s t, ‖b s - b t‖ ^ 2 = fullDistanceKernel hC hν p s t := by
  sorry

end StrictHotSpots.PlaneGreen
end
end FullBoundaryEmbeddingLayer
end FullBoundaryCombinedLayer

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/KernelLimit.lean (declaration OAI.StrictHotSpots.PlaneGreen.full_boundary_hilbert_embedding)

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