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

Proved

OAI.PlanarFPP.exponential_source_connector_test

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_source_connector_test of lean/OAI/Probability/FirstPassage/ConnectorProbability.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, open scoped Topology):

theorem exponential_source_connector_test {Ω : Type*} [MeasurableSpace Ω]
    {P : Measure Ω} {τ : Edge → Ω → ℝ} (h : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b : ℝ) (hb : |b| ≤ 1) (plus : Bool) (M : ℕ) (hM : 0 < M)
    (R ε : ℝ) (hR : 0 ≤ R) (hε : 0 < ε) :
    ∃ D0 : ℝ, 1 ≤ D0 ∧ ∀ κ > 0, ∀ᶠ s : ℝ in atTop,
      ∀ X : ℤ, ∀ Y a : ℝ, 1 ≤ a → a ≤ 2 →
      P.real {ω | (fun e => τ e ω) ∉ allPairsRegionEvent
        (connectorVertices b X ⌈D0*s⌉ Y (trialWindow M R s) s)
        (connectorWindow b X Y (trialWindow M R s))
        (connectorWindow b (X+⌈D0*s⌉)
          (Y+contactSign plus*a*trialWindow M R s) (trialWindow M R s))
        (sourceSupportCost μ b plus ε s)} < κ

Definitions. The statement is written with OAI.PlanarFPP.contactSign and OAI.PlanarFPP.trialWindow (definitions bundle OAISPf001 of this split); OAI.PlanarFPP.allPairsRegionEvent, OAI.PlanarFPP.connectorVertices, OAI.PlanarFPP.connectorWindow and OAI.PlanarFPP.sourceSupportCost (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 proofs of OAI.PlanarFPP.exponential_characteristic_lengths_exist and OAI.PlanarFPP.exponential_coarsePair_bound of this split import 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
open scoped Topology

theorem exponential_source_connector_test {Ω : Type*} [MeasurableSpace Ω]
    {P : Measure Ω} {τ : Edge → Ω → ℝ} (h : ExponentialEnvironment P τ)
    (μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
    (b : ℝ) (hb : |b| ≤ 1) (plus : Bool) (M : ℕ) (hM : 0 < M)
    (R ε : ℝ) (hR : 0 ≤ R) (hε : 0 < ε) :
    ∃ D0 : ℝ, 1 ≤ D0 ∧ ∀ κ > 0, ∀ᶠ s : ℝ in atTop,
      ∀ X : ℤ, ∀ Y a : ℝ, 1 ≤ a → a ≤ 2 →
      P.real {ω | (fun e => τ e ω) ∉ allPairsRegionEvent
        (connectorVertices b X ⌈D0*s⌉ Y (trialWindow M R s) s)
        (connectorWindow b X Y (trialWindow M R s))
        (connectorWindow b (X+⌈D0*s⌉)
          (Y+contactSign plus*a*trialWindow M R s) (trialWindow M R s))
        (sourceSupportCost μ b plus ε s)} < 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/ConnectorProbability.lean (declaration OAI.PlanarFPP.exponential_source_connector_test)

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