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

Proved

OAI.PlanarFPP.actual_grid_trial_probabilities

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.actual_grid_trial_probabilities of lean/OAI/Probability/FirstPassage/GridMeshes.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem actual_grid_trial_probabilities (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b q₀ : ℝ) (hb : |b| ≤ 1) (hq : 0 < q₀) (hq1 : q₀ < 1) :
    ∀ᶠ k : ℕ in atTop, ∀ j : ℕ,
      j ≤ gridLastLevel (characteristicExponent P τ μ b q₀ k) →
      ((1/16 : ℝ)^j*(characteristicLength P τ μ b q₀ k : ℝ) ≤
        (characteristicLength P τ μ b q₀ (k-j) : ℝ)) →
      q₀ < P.real {ω | (fun e => τ e ω) ∈ crossingEvent b (μ (direction b))
        (sourceCrossingK μ b) (gridWidth k j) (gridMesh (characteristicExponent P τ μ b q₀ k) j) 0}

Definitions. The statement is written with OAI.PlanarFPP.gridLastLevel (definitions bundle OAISPf000 of this split); OAI.PlanarFPP.direction, OAI.PlanarFPP.gridMesh and OAI.PlanarFPP.gridWidth (definitions bundle OAISPf001 of this split); OAI.PlanarFPP.characteristicExponent, OAI.PlanarFPP.characteristicLength, OAI.PlanarFPP.crossingEvent and OAI.PlanarFPP.sourceCrossingK (definitions bundle OAISPf002 of this split); OAI.PlanarFPP.Edge, OAI.PlanarFPP.ExponentialEnvironment, OAI.PlanarFPP.IsTimeConstant, OAI.PlanarFPP.Plane, OAI.PlanarFPP.TimeConstantNorm and OAI.PlanarFPP.Vertex (the published bundle Def_PlanarFirstPassage).

Used by. The proof of OAI.PlanarFPP.actual_spatial_rare_scales 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 Filter Set MeasureTheory
open scoped Topology

variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}

theorem actual_grid_trial_probabilities (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b q₀ : ℝ) (hb : |b| ≤ 1) (hq : 0 < q₀) (hq1 : q₀ < 1) :
    ∀ᶠ k : ℕ in atTop, ∀ j : ℕ,
      j ≤ gridLastLevel (characteristicExponent P τ μ b q₀ k) →
      ((1/16 : ℝ)^j*(characteristicLength P τ μ b q₀ k : ℝ) ≤
        (characteristicLength P τ μ b q₀ (k-j) : ℝ)) →
      q₀ < P.real {ω | (fun e => τ e ω) ∈ crossingEvent b (μ (direction b))
        (sourceCrossingK μ b) (gridWidth k j) (gridMesh (characteristicExponent P τ μ b q₀ k) j) 0} := 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/GridMeshes.lean (declaration OAI.PlanarFPP.actual_grid_trial_probabilities)

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