OAI.SourcePatternTail.uniform: intermediate lemma from OpenAI, The optimal order of convex-body covering density

An intermediate lemma of the split of OpenAI's proof of OAI.CoveringOrder.optimal_order, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The optimal order of convex-body covering density, OpenAI Math Release, September 23, 2026, result family 92, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SourcePatternTail.uniform of lean/OAI/Geometry/TranslativeCovering/SourcePatternTail.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SourcePatternTail, with open Set Filter MeasureTheory, open scoped ENNReal, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal Topology, open Set Filter MeasureTheory, open scoped ENNReal Topology, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal Topology, open Set Filter MeasureTheory, open scoped ENNReal Topology, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped BigOperators Classical, open MeasureTheory ProbabilityTheory Set, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open Set MeasureTheory, open scoped ENNReal Classical, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal Topology, open Set Filter MeasureTheory, open scoped ENNReal Topology, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Filter MeasureTheory, open scoped ENNReal Topology, open Set Filter MeasureTheory, open scoped ENNReal Topology, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open scoped BigOperators Classical, open MeasureTheory ProbabilityTheory Set, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open scoped Classical BigOperators, open scoped Classical BigOperators, open scoped BigOperators Classical, open Set MeasureTheory, open scoped ENNReal Classical, open Set Filter MeasureTheory, open scoped ENNReal, open Set MeasureTheory ProbabilityTheory, open scoped Classical BigOperators ENNReal, open Set Metric Filter MeasureTheory SphericalLaw SourceParameters, open scoped Topology ENNReal Classical):

theorem uniform : ∃ c : ℝ,0 < c ∧ ∃ n₀ : ℕ,∀ n : ℕ,n₀ ≤ n →
    ∀ [NeZero n] [Fact (2 ≤ n)] (J : Type u_1) [Fintype J] [Nonempty J]
      (e : Sphere n) (p : J → Space n) (R : ℝ),
    0 ≤ R → R ≤ c*(n:ℝ)*Real.log n → R ≤ (n:ℝ)^2 →
    Fintype.card J ≤ N n R →
    volume (closedBall (0 : Space n) D) ≤ 4*volume
      (PatternGeometry.roundedGood p a D (RadialShell.low a n) (RadialShell.high a n) (2*Cloc*R)) →
    (BodyVolume.law e a).real (PatternTail.coverEvent p a D) ≤
      Real.exp (-Real.exp ((1/2:ℝ)*(n:ℝ)))

Definitions. The statement is written with OAI.BodyVolume.law, OAI.PatternGeometry.Space, OAI.PatternGeometry.roundedGood, OAI.PatternTail.coverEvent, OAI.PoissonConfig.Config, OAI.RadialShell.high, OAI.RadialShell.low, OAI.SourceParameters.Cloc, OAI.SourceParameters.D, OAI.SourceParameters.N, OAI.SourceParameters.a, OAI.SphericalLaw.Space and OAI.SphericalLaw.Sphere (definitions bundle OAIQ16Cd000 of this split).

Used by. The proof of OAI.TranslativeCovering.main of this split imports it.