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

Proved

OAI.Percolation.deformed_root_sum_retained

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.deformed_root_sum_retained of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Deformation/FiniteScale.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 deformed_root_sum_retained (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (a : ℝ) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hA : HasKernel G p A) (k : ℕ) (t B : ℝ) (ht : 0 ≤ t)
    (htB : t*‖A‖ ≤ B) (hB : B*‖P‖^k ≤ 1/4) :
    ‖A‖/(2*c) ≤ averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t

Definitions. The statement is written with OAI.Percolation.LocallyFinite, OAI.Percolation.PositiveRealMatrix, OAI.Percolation.QuasiTransitive, OAI.Percolation.averagedDeformedSusceptibility and OAI.Percolation.matrixCoeff (definitions bundle OAIQ040C000 of this split); OAI.Percolation.vertexAut and OAI.Percolation.walkKernel (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph, OAI.Percolation.HasKernel, OAI.Percolation.Hilbert and OAI.Percolation.pc (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.exists_finite_scale_obstruction 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 AveragedRetention
attribute [local instance] Classical.propDecidable Classical.decEq
variable [Countable V] [Countable E] [Nonempty V]
variable {ι : Type*} [Fintype ι]

theorem deformed_root_sum_retained (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (a : ℝ) (p : unitInterval) (hp0 : 0 < (p:ℝ)) (hp : p < pc G)
    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
    (hcov : ∀ x, ∃ i, ∃ g ∈ vertexAut G, g (root i) = x)
    (c : ℝ) (hc : 0 < c) (hmin : ∀ i, 1 ≤ c*weight i)
    (P A : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P) (hPs : IsSelfAdjoint P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hA : HasKernel G p A) (k : ℕ) (t B : ℝ) (ht : 0 ≤ t)
    (htB : t*‖A‖ ≤ B) (hB : B*‖P‖^k ≤ 1/4) :
    ‖A‖/(2*c) ≤ averagedDeformedSusceptibility G (walkKernel G hloc a) p k root weight t := by
  sorry

end AveragedRetention
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/Deformation/FiniteScale.lean (declaration OAI.Percolation.deformed_root_sum_retained)

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