OAI.Percolation.critical_bounded_of_transport_finiteness: intermediate lemma from OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214)

Proved

OAI.Percolation.critical_bounded_of_transport_finiteness

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Percolation.arbitrary_cayley, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214), OpenAI Math Release, September 24, 2026, result family 214, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Percolation.critical_bounded_of_transport_finiteness of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Criticality/Operator.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Percolation, with open MeasureTheory ProbabilityTheory, open scoped ENNReal NNReal, open MeasureTheory Filter, open scoped Topology):

theorem critical_bounded_of_transport_finiteness [Infinite V]
    (G : BondGraph V E) (hG : Function.Injective G)
    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
    (htr : RootTransport root weight (vertexAut G))
    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
    OperatorBounded G (pc G)

Definitions. The statement is written with OAI.Percolation.Connected, OAI.Percolation.LocallyFinite, OAI.Percolation.QuasiTransitive and OAI.Percolation.RootTransport (definitions bundle OAIQ040C000 of this split); OAI.Percolation.hV and OAI.Percolation.vertexAut (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph, OAI.Percolation.OperatorBounded, OAI.Percolation.cluster, OAI.Percolation.law and OAI.Percolation.pc (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.arbitrary_cayley of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CayleyPercolation
import Definitions.Def_OAIQ040C002

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

open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal

namespace Percolation
open MeasureTheory Filter
open scoped Topology

universe u v
variable {V : Type u} {E : Type v}

theorem critical_bounded_of_transport_finiteness [Infinite V]
    (G : BondGraph V E) (hG : Function.Injective G)
    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G) (hh : 0 < hV G)
    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ)
    (htr : RootTransport root weight (vertexAut G))
    (hcov : ∀ x, ∃ r, ∃ g ∈ vertexAut G, g (root r) = x)
    (hw : ∀ r, 0 < weight r) (hw1 : ∑ r, weight r = 1)
    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite) :
    OperatorBounded G (pc G) := by
  sorry

end Percolation

end OAI
end

Source

OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 214; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Criticality/Operator.lean (declaration OAI.Percolation.critical_bounded_of_transport_finiteness)

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