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

Proved

OAI.StrictHotSpots.Conformal.ClosedDiskChart.eq_zero_of_boundary_closedGradient

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.eq_zero_of_boundary_closedGradient of lean/OAI/Analysis/HotSpots/HotSpots/Cauchy.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 Filter Metric Function, open scoped Topology ContDiff InnerProductSpace):

theorem eq_zero_of_boundary_closedGradient (hΩ : AdmissibleDomain Ω)
    {v : Plane → ℝ} (hv : InInteriorNeumannEigenspace Ω v)
    (hvc : ContinuousOn v (closure Ω)) {Y : Plane → Plane}
    (hY : ContinuousOn Y (closure Ω)) (he : EqOn (gradient v) Y Ω)
    {p : Plane} (hp : p ∈ frontier Ω) {U : Set Plane} (hU : IsOpen U) (hpU : p ∈ U)
    (hz : ∀ x ∈ U ∩ frontier Ω, v x = 0 ∧ Y x = 0) : EqOn v 0 Ω

Definitions. The statement is written with OAI.StrictHotSpots.Conformal.ClosedDiskChart (definitions bundle OAIQ3Ho000 of this split); OAI.StrictHotSpots.InInteriorNeumannEigenspace (definitions bundle OAIQ3Ho003 of this split); OAI.StrictHotSpots.AdmissibleDomain and OAI.StrictHotSpots.Plane (the published bundle Def_HotSpots).

Used by. The proof of OAI.StrictHotSpots.closedTraceSpace_local_rank 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 ClosedEigenLocalRank

noncomputable section
open Set Filter Metric Function
open scoped Topology ContDiff InnerProductSpace
namespace StrictHotSpots
namespace Conformal.ClosedDiskChart
variable {Ω : Set Plane} (c : ClosedDiskChart Ω)
include c

theorem eq_zero_of_boundary_closedGradient (hΩ : AdmissibleDomain Ω)
    {v : Plane → ℝ} (hv : InInteriorNeumannEigenspace Ω v)
    (hvc : ContinuousOn v (closure Ω)) {Y : Plane → Plane}
    (hY : ContinuousOn Y (closure Ω)) (he : EqOn (gradient v) Y Ω)
    {p : Plane} (hp : p ∈ frontier Ω) {U : Set Plane} (hU : IsOpen U) (hpU : p ∈ U)
    (hz : ∀ x ∈ U ∩ frontier Ω, v x = 0 ∧ Y x = 0) : EqOn v 0 Ω := by
  sorry
end Conformal.ClosedDiskChart

end StrictHotSpots
end
end ClosedEigenLocalRank

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/Cauchy.lean (declaration OAI.StrictHotSpots.Conformal.ClosedDiskChart.eq_zero_of_boundary_closedGradient)

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