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

Proved

OAI.StrictHotSpots.Hodge.vectorForm_nonneg

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.Hodge.vectorForm_nonneg of lean/OAI/Analysis/HotSpots/HotSpots/VectorEnergy.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StrictHotSpots.Hodge, with open Set MeasureTheory Filter, open scoped InnerProductSpace ContDiff):

theorem vectorForm_nonneg (ho : IsOpen Ω) (hne : Ω.Nonempty) (hs : IsSimplyConnected Ω)
    (hb : Bornology.IsBounded Ω) (hμ : 0 < firstPositiveNeumannValue Ω)
    {L : ℝ} (hl : firstPositiveNeumannValue Ω < L)
    (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2)
    (X : Fin 2 → H1 Ω) (hX : TangentWeak X) :
    0 ≤ vectorForm (firstPositiveNeumannValue Ω) X X

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.Hodge.TangentWeak and OAI.StrictHotSpots.Hodge.vectorForm (definitions bundle OAIQ3Ho004 of this split); OAI.StrictHotSpots.Plane and OAI.StrictHotSpots.firstPositiveNeumannValue (the published bundle Def_HotSpots).

Used by. The proofs of OAI.StrictHotSpots.Conformal.ClosedDiskChart.lipschitzMultipliedGradient_nonneg and OAI.StrictHotSpots.Conformal.ClosedDiskChart.multipliedGradient_identity of this split import it.

Preamble

import Mathlib
import Definitions.Def_HotSpots
import Definitions.Def_OAIQ3Ho004

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 ActualVectorSupport
noncomputable section

section TangentVectorClassActualCompatibleLayer
open Set MeasureTheory Filter
open scoped InnerProductSpace ContDiff
namespace StrictHotSpots.Hodge
variable {Ω : Set Plane}

theorem vectorForm_nonneg (ho : IsOpen Ω) (hne : Ω.Nonempty) (hs : IsSimplyConnected Ω)
    (hb : Bornology.IsBounded Ω) (hμ : 0 < firstPositiveNeumannValue Ω)
    {L : ℝ} (hl : firstPositiveNeumannValue Ω < L)
    (hD : ∀ j : H10 ho, L*‖H10.value ho j‖^2 ≤ ‖H10.grad ho j‖^2)
    (X : Fin 2 → H1 Ω) (hX : TangentWeak X) :
    0 ≤ vectorForm (firstPositiveNeumannValue Ω) X X := by
  sorry

end StrictHotSpots.Hodge
end TangentVectorClassActualCompatibleLayer

open Set MeasureTheory Filter
open scoped ContDiff InnerProductSpace Topology
end
end ActualVectorSupport

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/VectorEnergy.lean (declaration OAI.StrictHotSpots.Hodge.vectorForm_nonneg)

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