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

Proved

OAI.Percolation.infinitely_many_of_pc_lt_lt_pu

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.infinitely_many_of_pc_lt_lt_pu of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Criticality/Phase.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 ProbabilityTheory, open scoped ENNReal NNReal):

theorem infinitely_many_of_pc_lt_lt_pu [Infinite V]
    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
    (hqt : QuasiTransitive G) (p : unitInterval) (hc : pc G < p) (hu : p < pu G) :
    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Infinite

Definitions. The statement is written with OAI.Percolation.Connected, OAI.Percolation.LocallyFinite and OAI.Percolation.QuasiTransitive (definitions bundle OAIQ040C000 of this split); OAI.Percolation.BondGraph, OAI.Percolation.infiniteClusters, OAI.Percolation.law, OAI.Percolation.pc and OAI.Percolation.pu (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_OAIQ040C000

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

open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal
namespace Percolation
universe u v
variable {V : Type u} {E : Type v}
attribute [local instance] Classical.propDecidable Classical.decEq

theorem infinitely_many_of_pc_lt_lt_pu [Infinite V]
    (G : BondGraph V E) (hconn : Connected G) (hloc : LocallyFinite G)
    (hqt : QuasiTransitive G) (p : unitInterval) (hc : pc G < p) (hu : p < pu G) :
    ∀ᵐ ω ∂law p, (infiniteClusters G ω).Infinite := 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/Phase.lean (declaration OAI.Percolation.infinitely_many_of_pc_lt_lt_pu)

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