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

Proved

OAI.PlanarFPP.actual_differentiable_horizontal

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

The statement, as published (namespace OAI.PlanarFPP, with open MeasureTheory):

lemma actual_differentiable_horizontal (henv : ExponentialEnvironment P τ)
    (v : Plane) (hv : v ≠ 0) (hdom : |v 1| ≤ |v 0|) :
    DifferentiableAt ℝ actualSourceTimeConstant v

Definitions. The statement is written with OAI.PlanarFPP.actualSourceTimeConstant (definitions bundle OAISPf002 of this split); OAI.PlanarFPP.Edge, OAI.PlanarFPP.ExponentialEnvironment and OAI.PlanarFPP.Plane (the published bundle Def_PlanarFirstPassage).

Used by. The proof of OAI.PlanarFPP.manuscriptMain 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 MeasureTheory

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

lemma actual_differentiable_horizontal (henv : ExponentialEnvironment P τ)
    (v : Plane) (hv : v ≠ 0) (hdom : |v 1| ≤ |v 0|) :
    DifferentiableAt ℝ actualSourceTimeConstant v := 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/GlobalDifferentiability.lean (declaration OAI.PlanarFPP.actual_differentiable_horizontal)

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