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

Proved

OAI.Percolation.critical_rootTriangle_subpower

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_rootTriangle_subpower of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Kernel/TraceBounds.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):

theorem critical_rootTriangle_subpower
    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
    (hqt : QuasiTransitive G) (hh : 0 < hV G)
    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
    (η : ℝ) (hη : 0 < η) :
    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
      rootTriangle G root weight (p:ℝ) ≤ K*((pc G:ℝ)-(p:ℝ))^(-η)

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, OAI.Percolation.rootTriangle and OAI.Percolation.vertexAut (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph, OAI.Percolation.cluster, OAI.Percolation.law and OAI.Percolation.pc (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.critical_bounded_of_transport_finiteness 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
universe u v
variable {V : Type u} {E : Type v}
open MeasureTheory
section SubpowerTrace
attribute [local instance] Classical.propDecidable Classical.decEq

theorem critical_rootTriangle_subpower
    [Countable V] [Countable E] [Nonempty V] [Nonempty E]
    (G : BondGraph V E) (hloc : LocallyFinite G) (hconn : Connected G)
    (hqt : QuasiTransitive G) (hh : 0 < hV G)
    {ι : Type*} [Fintype ι] [Nonempty ι] (root : ι → V) (weight : ι → ℝ)
    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
    (hcov : ∀ x, ∃ i, ∃ e ∈ Γ, e (root i) = x)
    (hw : ∀ i, 0 < weight i) (hw1 : ∑ i, weight i = 1)
    (hp₀ : 0 < (pc G:ℝ)) (hp₁ : (pc G:ℝ) < 1)
    (hfin : ∀ i, ∀ᵐ ω ∂law (pc G), (cluster G ω (root i)).Finite)
    (η : ℝ) (hη : 0 < η) :
    ∃ K : ℝ, 0 < K ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
      rootTriangle G root weight (p:ℝ) ≤ K*((pc G:ℝ)-(p:ℝ))^(-η) := by
  sorry

end SubpowerTrace
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/Kernel/TraceBounds.lean (declaration OAI.Percolation.critical_rootTriangle_subpower)

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