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

Proved

OAI.StrictHotSpots.Conformal.ClosedDiskChart.operator_norm_lt_one

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.operator_norm_lt_one 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 operator_norm_lt_one (hb : Bornology.IsBounded Ω) (hne : Ω.Nonempty)
    (hμ : 0 < firstPositiveNeumannValue Ω) :
    ‖PlaneGreen.weightedIntegralOperator c.densityBound_ne_top (c.density_bound hμ.le)‖ < 1

Definitions. The statement is written with OAI.StrictHotSpots.Conformal.ClosedDiskChart, 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.PlaneGreen.weightedIntegralOperator (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.Plane and OAI.StrictHotSpots.firstPositiveNeumannValue (the published bundle Def_HotSpots).

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.boundary_gradient_integral_zero, OAI.StrictHotSpots.Conformal.ClosedDiskChart.coordinate_null_trace, OAI.StrictHotSpots.Conformal.ClosedDiskChart.lipschitzMultipliedGradient_closed, OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_coordinate_difference and OAI.StrictHotSpots.Conformal.ClosedDiskChart.scalarSolution_weak of this split import it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho003

Formal statement

section

attribute [-instance] _root_.OAI.StrictHotSpots.N1.completeSpace _root_.OAI.StrictHotSpots.N1.innerProductSpace _root_.OAI.StrictHotSpots.N1.normedAddCommGroup
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 operator_norm_lt_one (hb : Bornology.IsBounded Ω) (hne : Ω.Nonempty)
    (hμ : 0 < firstPositiveNeumannValue Ω) :
    ‖PlaneGreen.weightedIntegralOperator c.densityBound_ne_top (c.density_bound hμ.le)‖ < 1 := 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.operator_norm_lt_one)

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