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

Proved

OAI.PlanarFPP.exponential_arbitraryEndpoint_tendsto_of_grids

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

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

theorem exponential_arbitraryEndpoint_tendsto_of_grids {ι : Type*} {l : Filter ι}
    (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b K A R ε q₀ : ℝ) (hb : |b| ≤ 1) (hK : 0 ≤ K)
    (hA : 0 < A) (hR : 0 < R) (hε : 0 < ε) (hq : 0 < q₀) (plus : Bool)
    (k m : ι → ℕ) (hk : Tendsto k l atTop)
    (hratio : Tendsto (fun i => (2 : ℝ)^(m i)/(2 : ℝ)^(k i)) l atTop)
    (hm : ∀ᶠ i in l, m i ≤ 3*k i)
    (hgrid : ∀ J : ℕ, 1 ≤ J → ∀ᶠ i in l,
      P.real {ω | (fun e => τ e ω) ∉ gridConnectionsEvent b (μ (direction b)) K A R (k i) (m i) J} ≤
        gridFailureTail (gridIndexConstant A R) q₀ J) :
    Tendsto (fun i => P.real {ω | (fun e => τ e ω) ∈ arbitraryEndpointEvent b (μ (direction b))
      (extremeSlope μ b plus) A R ε (k i) (m i)}) l (𝓝 0)

Definitions. The statement is written with OAI.PlanarFPP.direction, OAI.PlanarFPP.extremeSlope, OAI.PlanarFPP.gridFailureTail and OAI.PlanarFPP.gridIndexConstant (definitions bundle OAISPf001 of this split); OAI.PlanarFPP.arbitraryEndpointEvent and OAI.PlanarFPP.gridConnectionsEvent (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.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 Set Filter MeasureTheory ProbabilityTheory
open scoped Topology
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}

theorem exponential_arbitraryEndpoint_tendsto_of_grids {ι : Type*} {l : Filter ι}
    (henv : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b K A R ε q₀ : ℝ) (hb : |b| ≤ 1) (hK : 0 ≤ K)
    (hA : 0 < A) (hR : 0 < R) (hε : 0 < ε) (hq : 0 < q₀) (plus : Bool)
    (k m : ι → ℕ) (hk : Tendsto k l atTop)
    (hratio : Tendsto (fun i => (2 : ℝ)^(m i)/(2 : ℝ)^(k i)) l atTop)
    (hm : ∀ᶠ i in l, m i ≤ 3*k i)
    (hgrid : ∀ J : ℕ, 1 ≤ J → ∀ᶠ i in l,
      P.real {ω | (fun e => τ e ω) ∉ gridConnectionsEvent b (μ (direction b)) K A R (k i) (m i) J} ≤
        gridFailureTail (gridIndexConstant A R) q₀ J) :
    Tendsto (fun i => P.real {ω | (fun e => τ e ω) ∈ arbitraryEndpointEvent b (μ (direction b))
      (extremeSlope μ b plus) A R ε (k i) (m i)}) l (𝓝 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/EndpointLimit.lean (declaration OAI.PlanarFPP.exponential_arbitraryEndpoint_tendsto_of_grids)

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