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

Proved

OAI.Percolation.rooted_selected_label_mass_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.rooted_selected_label_mass_lower of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Escape/PathSelection.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 rooted_selected_label_mass_lower [Countable V] [Countable E] [Nonempty V]
    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
    (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)
    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
    (p : unitInterval) (hA : HasKernel G p A)
    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J))
    (x : V) (hx : x ∈ vertexOrbit G (γ (segmentCenter J))) (hρ : ‖P‖ < 1)
    (hsep : ∀ s ∈ segmentOrbit G γ,
      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))))
    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (t : ℝ) (ht : 0 < t) :
    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m)/t +
      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L) ≤
    ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0

Definitions. The statement is written with OAI.Percolation.EscapeLabel, OAI.Percolation.LocallyFinite, OAI.Percolation.PositiveRealMatrix, OAI.Percolation.RootTransport, OAI.Percolation.matrixCoeff, OAI.Percolation.openSimpleGraph, OAI.Percolation.reachBall, OAI.Percolation.segmentOrbit, OAI.Percolation.vertexOrbit and OAI.Percolation.RawCorridor.Selected (definitions bundle OAIQ040C000 of this split); OAI.Percolation.earlyCollisionCost, OAI.Percolation.segmentCenter, OAI.Percolation.segmentNegativeIndex, OAI.Percolation.segmentPositiveIndex and OAI.Percolation.suffixCollisionCost (definitions bundle OAIQ040C001 of this split); OAI.Percolation.degree, OAI.Percolation.escapeLabelProbability, OAI.Percolation.labelCorridor, OAI.Percolation.vertexAut and OAI.Percolation.walkKernel (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph, OAI.Percolation.HasKernel and OAI.Percolation.Hilbert (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.exists_selected_path_mass 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}
section SelectedLabelMass
attribute [local instance] Classical.propDecidable Classical.decEq
variable {ι : Type*} [Fintype ι]

theorem rooted_selected_label_mass_lower [Countable V] [Countable E] [Nonempty V]
    (G : BondGraph V E) (hloc : LocallyFinite G) (a : ℝ)
    (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)
    (c : ℝ) (hc : 0 < c) (hmin : ∀ r, 1 ≤ c*weight r)
    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hsym : IsSelfAdjoint P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hdiag : ∀ x, 1/2 ≤ (matrixCoeff P x x).re)
    (p : unitInterval) (hA : HasKernel G p A)
    (J L m : ℕ) (γ : Fin (J+J+1) → V) (hinj : Function.Injective γ)
    (hγ : ∀ i : Fin (J+J), (openSimpleGraph G Set.univ).Adj (γ i.castSucc) (γ i.succ))
    (hdiam : ∀ i j, γ j ∈ reachBall G (γ i) (J+J))
    (x : V) (hx : x ∈ vertexOrbit G (γ (segmentCenter J))) (hρ : ‖P‖ < 1)
    (hsep : ∀ s ∈ segmentOrbit G γ,
      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (segmentPositiveIndex J i))) ∧
      (∀ i, J ≤ (openSimpleGraph G Set.univ).dist (s (Fin.last (J+J))) (s (segmentNegativeIndex J i))) ∧
      2*J ≤ (openSimpleGraph G Set.univ).dist (s 0) (s (Fin.last (J+J))))
    (d : ℕ) (hd : ∀ z, degree G z ≤ d) (t : ℝ) (ht : 0 < t) :
    1 - (4*c*(earlyCollisionCost ‖P‖ J + suffixCollisionCost ‖P‖ c J*‖P‖^L) +
      2*((3*(d:ℝ)+3)*((((J+L)+J):ℕ)+1)*Real.sqrt c*‖P‖^m)/t +
      2*c*(((d+1)^2:ℕ):ℝ)*Real.sqrt c*‖P‖^L) ≤
    ∑' ξ : {ξ : EscapeLabel G γ (Fin.last (J+J)) 0 L m // ξ.first (segmentCenter J) = x},
      if (labelCorridor G J L m γ hγ ξ.1).Selected hloc p (J+m) t then
        escapeLabelProbability G hloc a γ (J+J) hdiam (segmentCenter J) (Fin.last (J+J)) 0 L m ξ.1 else 0 := by
  sorry

end SelectedLabelMass
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/Escape/PathSelection.lean (declaration OAI.Percolation.rooted_selected_label_mass_lower)

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