OAI.SharpNodal.Profiles.uniform_plane_length: intermediate lemma from OpenAI, Sharp nodal length on smooth surfaces

An intermediate lemma of the split of OpenAI's proof of OAI.SharpNodal.Main, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Sharp nodal length on smooth surfaces, OpenAI Math Release, September 23, 2026, result family 350, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SharpNodal.Profiles.uniform_plane_length of lean/OAI/Analysis/NodalLength/NodalLength/PlaneLength.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SharpNodal.Profiles, with open scoped ContDiff Bundle ENNReal, open Bundle Manifold MeasureTheory, open scoped ContDiff ENNReal Topology, open MeasureTheory Filter Set, open scoped Topology ENNReal, open MeasureTheory Filter Set, open scoped Topology ENNReal ContDiff, open MeasureTheory Filter Set, open scoped Topology ENNReal ContDiff, open MeasureTheory Filter Set, open scoped Topology ENNReal ContDiff, open MeasureTheory Filter Set, open scoped Topology ContDiff, open Filter Set, open scoped Topology ContDiff, open Filter Set, open scoped Topology ENNReal, open Filter Set MeasureTheory TopologicalSpace, open scoped Topology ContDiff, open Filter Set, open scoped Topology ENNReal, open Filter Set MeasureTheory TopologicalSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory TopologicalSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory Laplacian, open scoped Topology ENNReal ContDiff ComplexConjugate, open Filter Set MeasureTheory Laplacian, open scoped Topology ENNReal ContDiff ComplexConjugate, open Filter Set MeasureTheory Laplacian, open scoped Topology ENNReal NNReal, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Set MeasureTheory Filter, open scoped Topology ENNReal, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff Laplacian, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory, open scoped Topology ENNReal ContDiff CompactlySupported, open Set MeasureTheory, open scoped Topology ENNReal ContDiff CompactlySupported, open Set MeasureTheory, open scoped Topology ENNReal ContDiff CompactlySupported, open Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff Laplacian, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution ENNReal, open Filter Set MeasureTheory, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory Function, open scoped Topology, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal, open Filter Set MeasureTheory, open scoped Topology ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped BigOperators, open Filter Set MeasureTheory, open scoped BigOperators, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff BigOperators, open Filter Set MeasureTheory InnerProductSpace, open scoped Topology ENNReal ContDiff BigOperators, open Filter Set MeasureTheory InnerProductSpace, open scoped BigOperators, open MeasureTheory, open scoped BigOperators, open Set MeasureTheory, open scoped BigOperators, open scoped Classical, open scoped BigOperators Topology ENNReal, open Set MeasureTheory, open scoped BigOperators, open scoped Topology ENNReal ContDiff, open Filter Set MeasureTheory InnerProductSpace, open scoped BigOperators Classical Topology, open Filter Set MeasureTheory, open scoped BigOperators Classical Topology, open Filter Set MeasureTheory, open scoped BigOperators, open Set, open scoped BigOperators Topology, open Set MeasureTheory, open scoped BigOperators, open Set, open scoped BigOperators symmDiff, open Set, open scoped BigOperators, open Set, open scoped BigOperators symmDiff, open Set, open scoped BigOperators Classical, open Set, open scoped BigOperators, open Set, open scoped BigOperators Classical, open Set, open scoped BigOperators Classical, open Set, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set MeasureTheory, open scoped Topology ContDiff BigOperators, open Filter Set MeasureTheory, open scoped Topology ContDiff BigOperators, open Filter Set MeasureTheory, open scoped Topology ContDiff BigOperators, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff, open Filter Set MeasureTheory ComplexConjugate, open scoped Topology ContDiff, open Filter Set MeasureTheory ComplexConjugate, open scoped Topology NNReal BoundedContinuousFunction, open Filter Set Metric, open scoped Topology ContDiff, open Filter Set MeasureTheory, open scoped Topology ContDiff BigOperators, open Filter Set MeasureTheory, open scoped Topology ContDiff BigOperators, open Filter Set MeasureTheory, open scoped Topology ComplexConjugate BigOperators, open Filter Set Metric Complex MeromorphicOn, open scoped Topology ComplexConjugate BigOperators, open Filter Set Metric Complex MeromorphicOn, open scoped Topology ComplexConjugate BigOperators, open Filter Set Metric Complex, open scoped Topology ContDiff ENNReal, open Set MeasureTheory Metric, open scoped Topology, open Set Metric, open scoped Topology ComplexConjugate BigOperators, open Filter Set Metric Complex MeromorphicOn, open scoped Topology, open Set Metric Complex, open scoped Topology, open Set Metric, open scoped Topology ContDiff ENNReal, open Set MeasureTheory Metric, open scoped Topology, open Set Metric Complex MeasureTheory, open scoped ENNReal Topology, open Set Metric MeasureTheory TopologicalSpace Function, open scoped Topology ENNReal, open Set Metric MeasureTheory Filter, open scoped Topology ENNReal, open Set Metric MeasureTheory Filter, open scoped Topology ENNReal, open Set Metric MeasureTheory, open scoped Topology ComplexConjugate BigOperators ENNReal, open Filter Set Metric Complex MeasureTheory MeromorphicOn, open scoped Topology ContDiff Convolution ENNReal, open Filter Set MeasureTheory Metric, open scoped Topology ContDiff NNReal ENNReal, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff NNReal ENNReal, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory Metric, open scoped Topology ContDiff ENNReal, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff ENNReal, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff Convolution, open Filter Set Metric MeasureTheory, open scoped Topology ContDiff, open Filter Set Metric, open scoped Matrix, open scoped Topology ContDiff, open Filter Set Metric, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff, open Filter Set Metric, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ContDiff Bundle, open Filter Set Metric Bundle Manifold, open scoped Topology ENNReal, open Filter Set MeasureTheory TopologicalSpace, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory Metric, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory Metric, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory Metric, open scoped Topology ContDiff ENNReal, open Filter Set MeasureTheory Metric TopologicalSpace, open scoped Topology ContDiff ENNReal Bundle, open Set Filter MeasureTheory Metric Bundle Manifold, open scoped Topology ContDiff ENNReal Bundle, open Set Filter MeasureTheory Metric Bundle Manifold, open scoped Topology ContDiff ENNReal, open Set Filter Metric MeasureTheory, open scoped Topology ContDiff ENNReal Bundle, open Set Filter Metric MeasureTheory Bundle Manifold, open scoped Topology ContDiff ENNReal Bundle, open Set Filter MeasureTheory Metric Bundle Manifold, open scoped Topology ContDiff, open Set Filter Metric MeasureTheory, open scoped Topology ContDiff Bundle, open Set Filter Metric Bundle Manifold, open scoped Topology ContDiff, open Set Filter Metric MeasureTheory, open scoped Topology ContDiff, open Set Filter Metric MeasureTheory, open scoped Topology, open Set Filter Metric, open scoped Topology ContDiff ENNReal Bundle, open Set Filter MeasureTheory Metric Bundle Manifold, open scoped Topology ContDiff ENNReal NNReal Bundle Manifold, open Set Filter Metric MeasureTheory Bundle Manifold, open scoped Topology ContDiff ENNReal NNReal, open Set Filter MeasureTheory Metric, open scoped Topology ContDiff ENNReal BigOperators, open Set Filter MeasureTheory Metric, open scoped Topology ContDiff ENNReal NNReal BigOperators, open Set Filter Metric MeasureTheory, open scoped Topology ContDiff ENNReal BigOperators, open Set Filter MeasureTheory Metric):

lemma uniform_plane_length {Cp T : ℝ} (hCp : 0≤Cp) (hT : 0<T) (Cinit : ℝ) :
    ∃a₀ B : ℝ,0<a₀ ∧ a₀≤T ∧ 0≤B ∧ ∀D : ScaledWave Cp a₀,
      D.growth 0≤Cinit →
      (∀y r,0 < r → ball y r⊆ball (0:Plane) 1000 → centeredMass 0 D.U y r≠0) →
      Measure.hausdorffMeasure 1 {x | x∈ball (0:Plane) (1/4) ∧ D.U x=0}≤ENNReal.ofReal (B*D.K)

Definitions. The statement is written with OAI.SharpNodal.Profiles.ScaledWave, OAI.SharpNodal.Profiles.centeredMass, OAI.SharpNodal.Profiles.UnitWave and OAI.SharpNodal.Profiles.UnitWave.growth (definitions bundle OAIQ4Nl000 of this split); OAI.SharpNodal.Plane (the published bundle Def_NodalLength).

Used by. The proof of OAI.SharpNodal.Main of this split imports it.