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

Proved

OAI.PlanarFPP.ae_eventually_spatial_long_paths

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.ae_eventually_spatial_long_paths of lean/OAI/Probability/FirstPassage/WitnessBorelCantelli.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 Filter):

theorem ae_eventually_spatial_long_paths (henv : ExponentialEnvironment P τ)
    (b m₀ am ap K η r : ℝ) (hb : |b| ≤ 1) (ha : am ≤ ap) (hη : 0 < η)
    (hr : 0 < r) (hr1 : r ≤ 1) (hsmall : 196*r < 1)
    (hgap : 4*η ≤ (ap-am)/2) (k m : ℕ) (hs : 2 ≤ (2 : ℝ)^k)
    (hstep : max |m₀-((am+ap)/2)*b| |(am+ap)/2| ≤ η*(2 : ℝ)^k)
    (hscale₁ : η*(2 : ℝ)^k < 2*m₀*(2 : ℝ)^m-|((am+ap)/2)| *((2 : ℝ)^k+1))
    (hscale₂ : 3*η*(2 : ℝ)^k+2*|((am+ap)/2)| *((2 : ℝ)^k+1) < K*(2 : ℝ)^k)
    (hBK : ∀ is : List (ℤ×ℤ), P.real {ω | StrictDisjointPaths (spatialListSupport b k m is)
      (is.map (spatialWitnessFamily b m₀ am ap K η k m)) (fun e => τ e ω)} ≤ (r^16)^is.length) :
    ∀ᵐ ω ∂P, ∀ᶠ N : ℕ in atTop,
      ∀ blocks : List (List Step), ∀ rest : List Step,
      blocks.length=N → (∀ q ∈ blocks, FullSpatialChunk b k m q) →
      (∀ i ≤ rest.length, chunkInterior b k m (displacement (rest.take i))) →
      ((trace 0 (blocks.flatten++rest)).map traversalEdge).Nodup →
      N*η*(2 : ℝ)^k/2 ≤ defect (fun e => τ e ω) b m₀ ((am+ap)/2) 0 (blocks.flatten++rest)

Definitions. The statement is written with OAI.PlanarFPP.Traversal (definitions bundle OAISPf000 of this split); OAI.PlanarFPP.FullSpatialChunk, OAI.PlanarFPP.StrictDisjointPaths, OAI.PlanarFPP.StrictPathWitness, OAI.PlanarFPP.chunkInterior, OAI.PlanarFPP.defect, OAI.PlanarFPP.spatialListSupport, OAI.PlanarFPP.spatialWitnessFamily, OAI.PlanarFPP.trace and OAI.PlanarFPP.traversalEdge (definitions bundle OAISPf002 of this split); OAI.PlanarFPP.Edge, OAI.PlanarFPP.ExponentialEnvironment, OAI.PlanarFPP.Step, OAI.PlanarFPP.Vertex and OAI.PlanarFPP.displacement (the published bundle Def_PlanarFirstPassage).

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

Preamble

import Mathlib
import Lean.Elab.Tactic.Omega
import Definitions.Def_PlanarFirstPassage
import Definitions.Def_OAISPf002

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

namespace PlanarFPP
open Set Finset MeasureTheory Filter

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

theorem ae_eventually_spatial_long_paths (henv : ExponentialEnvironment P τ)
    (b m₀ am ap K η r : ℝ) (hb : |b| ≤ 1) (ha : am ≤ ap) (hη : 0 < η)
    (hr : 0 < r) (hr1 : r ≤ 1) (hsmall : 196*r < 1)
    (hgap : 4*η ≤ (ap-am)/2) (k m : ℕ) (hs : 2 ≤ (2 : ℝ)^k)
    (hstep : max |m₀-((am+ap)/2)*b| |(am+ap)/2| ≤ η*(2 : ℝ)^k)
    (hscale₁ : η*(2 : ℝ)^k < 2*m₀*(2 : ℝ)^m-|((am+ap)/2)| *((2 : ℝ)^k+1))
    (hscale₂ : 3*η*(2 : ℝ)^k+2*|((am+ap)/2)| *((2 : ℝ)^k+1) < K*(2 : ℝ)^k)
    (hBK : ∀ is : List (ℤ×ℤ), P.real {ω | StrictDisjointPaths (spatialListSupport b k m is)
      (is.map (spatialWitnessFamily b m₀ am ap K η k m)) (fun e => τ e ω)} ≤ (r^16)^is.length) :
    ∀ᵐ ω ∂P, ∀ᶠ N : ℕ in atTop,
      ∀ blocks : List (List Step), ∀ rest : List Step,
      blocks.length=N → (∀ q ∈ blocks, FullSpatialChunk b k m q) →
      (∀ i ≤ rest.length, chunkInterior b k m (displacement (rest.take i))) →
      ((trace 0 (blocks.flatten++rest)).map traversalEdge).Nodup →
      N*η*(2 : ℝ)^k/2 ≤ defect (fun e => τ e ω) b m₀ ((am+ap)/2) 0 (blocks.flatten++rest) := 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/WitnessBorelCantelli.lean (declaration OAI.PlanarFPP.ae_eventually_spatial_long_paths)

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