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

Proved

OAI.StrictHotSpots.PlaneGreen.full_embedding_coordinate_sum

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_embedding_coordinate_sum 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.PlaneGreen, with open Set MeasureTheory Filter Metric Function, open scoped ENNReal NNReal Topology InnerProductSpace, open DiskH10):

theorem full_embedding_coordinate_sum {H E ι : Type*} [NormedAddCommGroup H]
    [InnerProductSpace ℝ H] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
    [Countable ι] (e : HilbertBasis ι ℝ H)
    (p : disk) (hT : ‖weightedIntegralOperator hC hν‖ < 1) {b : Boundary → H}
    (hb : Continuous b) (hid : ∀ s t, ‖b s-b t‖^2 = fullDistanceKernel hC hν p s t)
    {g : Boundary → E} (hg : Continuous g) :
    HasSum (fun i => (1/2:ℝ)*∫ z : Boundary × Boundary,
      fullBoundaryN hC hν z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 *
        inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure)
      ((1/2:ℝ)*‖∫ s, fullBoundaryK hC hν p s • g s ∂boundaryMeasure‖^2)

Definitions. The statement is written with OAI.StrictHotSpots.DiskH10.disk and OAI.StrictHotSpots.PlaneGreen.Boundary (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.PlaneGreen.boundaryMeasure, OAI.StrictHotSpots.PlaneGreen.fullBoundaryK, OAI.StrictHotSpots.PlaneGreen.fullBoundaryN, OAI.StrictHotSpots.PlaneGreen.fullDistanceKernel and OAI.StrictHotSpots.PlaneGreen.weightedIntegralOperator (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Plane (the published bundle Def_HotSpots).

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

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho003

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 HilbertKernelCoordinates

noncomputable section
open Set MeasureTheory Filter Metric Function
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_embedding_coordinate_sum {H E ι : Type*} [NormedAddCommGroup H]
    [InnerProductSpace ℝ H] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
    [Countable ι] (e : HilbertBasis ι ℝ H)
    (p : disk) (hT : ‖weightedIntegralOperator hC hν‖ < 1) {b : Boundary → H}
    (hb : Continuous b) (hid : ∀ s t, ‖b s-b t‖^2 = fullDistanceKernel hC hν p s t)
    {g : Boundary → E} (hg : Continuous g) :
    HasSum (fun i => (1/2:ℝ)*∫ z : Boundary × Boundary,
      fullBoundaryN hC hν z.1 z.2 * (inner ℝ (e i) (b z.1-b z.2))^2 *
        inner ℝ (g z.1) (g z.2) ∂boundaryMeasure.prod boundaryMeasure)
      ((1/2:ℝ)*‖∫ s, fullBoundaryK hC hν p s • g s ∂boundaryMeasure‖^2) := by
  sorry

end StrictHotSpots.PlaneGreen
end
end HilbertKernelCoordinates

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.PlaneGreen.full_embedding_coordinate_sum)

View graph