OAI.PlanarFPP.exponential_coarsePair_bound: intermediate lemma from OpenAI, Strict convexity and differentiability of the planar exponential first-passage limit shape

Proved

OAI.PlanarFPP.exponential_coarsePair_bound

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.PlanarFPP.manuscriptMain, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Strict convexity and differentiability of the planar exponential first-passage limit shape, OpenAI Math Release, September 24, 2026, result family 212, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.PlanarFPP.exponential_coarsePair_bound of lean/OAI/Probability/FirstPassage/EndpointProbability.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.PlanarFPP, with open Set Finset MeasureTheory ProbabilityTheory Filter, open scoped Topology):

theorem exponential_coarsePair_bound (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b A R ε : ℝ) (hb : |b| ≤ 1) (hA : 0 < A) (hR : 0 ≤ R) (hε : 0 < ε)
    (plus : Bool) (J : ℕ) (κ : ℝ) (hκ : 0 < κ) :
    ∃ S D : ℝ, 1 ≤ S ∧ 0 < D ∧ ∀ k m : ℕ, S ≤ (2 : ℝ)^k →
      D*(2 : ℝ)^k ≤ (gridMesh m J : ℝ) → ∀ ij ∈ coarsePairIndices A J,
      P.real {ω | (fun e => τ e ω) ∈
        coarsePairEvent b (μ (direction b)) (extremeSlope μ b plus) A R ε k m J ij} < κ

Definitions. The statement is written with OAI.PlanarFPP.coarsePairIndices, OAI.PlanarFPP.direction, OAI.PlanarFPP.extremeSlope and OAI.PlanarFPP.gridMesh (definitions bundle OAISPf001 of this split); OAI.PlanarFPP.coarsePairEvent (definitions bundle OAISPf002 of this split); OAI.PlanarFPP.Edge, OAI.PlanarFPP.ExponentialEnvironment, OAI.PlanarFPP.IsTimeConstant, OAI.PlanarFPP.Plane and OAI.PlanarFPP.TimeConstantNorm (the published bundle Def_PlanarFirstPassage).

Used by. The proof of OAI.PlanarFPP.exponential_arbitraryEndpoint_tendsto_of_grids of this split imports it.

Preamble

import Mathlib
import Definitions.Def_PlanarFirstPassage
import Definitions.Def_OAISPf002

Formal statement

section

attribute [-instance] _root_.OAI.CompanionFPP.expOneProbability
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace PlanarFPP
open Set Finset MeasureTheory ProbabilityTheory Filter
open scoped Topology
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}

theorem exponential_coarsePair_bound (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b A R ε : ℝ) (hb : |b| ≤ 1) (hA : 0 < A) (hR : 0 ≤ R) (hε : 0 < ε)
    (plus : Bool) (J : ℕ) (κ : ℝ) (hκ : 0 < κ) :
    ∃ S D : ℝ, 1 ≤ S ∧ 0 < D ∧ ∀ k m : ℕ, S ≤ (2 : ℝ)^k →
      D*(2 : ℝ)^k ≤ (gridMesh m J : ℝ) → ∀ ij ∈ coarsePairIndices A J,
      P.real {ω | (fun e => τ e ω) ∈
        coarsePairEvent b (μ (direction b)) (extremeSlope μ b plus) A R ε k m J ij} < κ := by
  sorry

end PlanarFPP

end OAI
end

Source

OpenAI, Strict convexity and differentiability of the planar exponential first-passage limit shape, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 212; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/FirstPassage/EndpointProbability.lean (declaration OAI.PlanarFPP.exponential_coarsePair_bound)

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