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

Proved
OAI.Percolation.mainConclusion_of_strict_gap

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.mainConclusion_of_strict_gap of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Graph/Paths.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):

theorem mainConclusion_of_strict_gap [Infinite V] (G : BondGraph V E)
    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (hgap : pc G < ptwo G) : MainConclusion G

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

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

section MainBridge
attribute [local instance] Classical.propDecidable

theorem mainConclusion_of_strict_gap [Infinite V] (G : BondGraph V E)
    (hconn : Connected G) (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (hgap : pc G < ptwo G) : MainConclusion G := by
  sorry

end MainBridge

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/Graph/Paths.lean (declaration OAI.Percolation.mainConclusion_of_strict_gap)

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