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

Proved

OAI.Percolation.EndpointProfile.eventually_neighborMoment_lower

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.EndpointProfile.eventually_neighborMoment_lower of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Kernel/DistinctMoments.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem eventually_neighborMoment_lower [Countable V] [Countable E] [Nonempty V]
    (hconn : Connected G) (hqt : QuasiTransitive G)
    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
    (i₀ : ι) (hw₀ : 0 < weight i₀) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) (hH : 1 < q.H) :
    ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
      b*(rootSusceptibility G root weight p)^(2-Real.log q.geometricScale/(q.H-1)) ≤
        rootNeighborMoment G hloc root weight p

Definitions. The statement is written with OAI.Percolation.Connected, OAI.Percolation.EndpointProfile, OAI.Percolation.IsAut, OAI.Percolation.LocallyFinite, OAI.Percolation.QuasiTransitive, OAI.Percolation.RootTransport, OAI.Percolation.rootSusceptibility, OAI.Percolation.vertexOrbit, OAI.Percolation.EndpointProfile.H and OAI.Percolation.EndpointProfile.geometricScale (definitions bundle OAIQ040C000 of this split); OAI.Percolation.rootNeighborMoment (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph and OAI.Percolation.pc (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.critical_root_power_integrable 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 Filter
section ProfileNeighborMoment
attribute [local instance] Classical.propDecidable Classical.decEq
namespace EndpointProfile
variable {G : BondGraph V E} {hloc : LocallyFinite G} {p₀ : unitInterval} {o : V} {s : ℝ}

theorem eventually_neighborMoment_lower [Countable V] [Countable E] [Nonempty V]
    (hconn : Connected G) (hqt : QuasiTransitive G)
    {ι : Type*} [Fintype ι] (root : ι → V) (weight : ι → ℝ) (Γ : Set (V ≃ V))
    (htr : RootTransport root weight Γ) (hΓ : ∀ a ∈ Γ, ∃ b, IsAut G a b)
    (hw : ∀ i, 0 ≤ weight i) (hw1 : ∑ i, weight i = 1)
    (i₀ : ι) (hw₀ : 0 < weight i₀) (hcover : ∀ x, ∃ i, x ∈ vertexOrbit G (root i))
    (c : ℝ) (hc : 0 ≤ c) (hmin : ∀ i, 1 ≤ c*weight i)
    (q : EndpointProfile G hloc p₀ (root i₀) s) (hp₀ : 0 < (p₀:ℝ)) (hH : 1 < q.H) :
    ∃ b : ℝ, 0 < b ∧ ∀ᶠ p in nhds p₀, p < pc G →
      b*(rootSusceptibility G root weight p)^(2-Real.log q.geometricScale/(q.H-1)) ≤
        rootNeighborMoment G hloc root weight p := by
  sorry

end EndpointProfile
end ProfileNeighborMoment
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/DistinctMoments.lean (declaration OAI.Percolation.EndpointProfile.eventually_neighborMoment_lower)

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