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

Proved

OAI.Percolation.raw_derivative_template_row_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.raw_derivative_template_row_lower of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Deformation/AveragedDecrease.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 raw_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (d : ℕ) (hd : ∀ x, degree G x ≤ d)
    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
    (k : ℕ) (w : RawCorridor G k)
    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
        if e ∈ r.retained then r.weight P p else 0) ≤ c)
    (t : ℝ) (ht : 0 ≤ t) :
    Real.exp (-t*c) * (w.probability P *
      (retainedSusceptibilityProduct G P p t w -
        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re)) ≤
      ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2

Definitions. The statement is written with OAI.Percolation.LocallyFinite, OAI.Percolation.QuasiTransitive, OAI.Percolation.RawCorridor, OAI.Percolation.matrixCoeff, OAI.Percolation.RawCorridor.base, OAI.Percolation.RawCorridor.endpoint, OAI.Percolation.RawCorridor.probability, OAI.Percolation.RawCorridor.retained, OAI.Percolation.RawCorridor.retained_finite and OAI.Percolation.RawCorridor.weight (definitions bundle OAIQ040C000 of this split); OAI.Percolation.corridorDerivativeContribution, OAI.Percolation.degree, OAI.Percolation.maxSusceptibility and OAI.Percolation.retainedSusceptibilityProduct (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_uniform_averaged_decrease 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 AveragedCalibratedDecrease
attribute [local instance] Classical.propDecidable Classical.decEq
variable [Countable V] [Countable E] [Nonempty V]
variable {ι : Type*} [Fintype ι]

theorem raw_derivative_template_row_lower (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (d : ℕ) (hd : ∀ x, degree G x ≤ d)
    (P : V → V → ℝ) (hP : ∀ x y, 0 ≤ P x y) (hpcrit : 0 < (pc G:ℝ))
    (p : unitInterval) (hp : p < pc G) (hpnear : (pc G:ℝ)/2 ≤ (p:ℝ))
    (T : Hilbert V →L[ℂ] Hilbert V) (hT : HasKernel G p T)
    (k : ℕ) (w : RawCorridor G k)
    (c : ℝ) (hc : (∑ e ∈ (w.retained_finite hG).toFinset, ∑' r : RawCorridor G k,
        if e ∈ r.retained then r.weight P p else 0) ≤ c)
    (t : ℝ) (ht : 0 ≤ t) :
    Real.exp (-t*c) * (w.probability P *
      (retainedSusceptibilityProduct G P p t w -
        (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2*(matrixCoeff (T^3) w.base w.endpoint).re)) ≤
      ∑' xy : V × V, corridorDerivativeContribution G P p t w xy.1 xy.2 := by
  sorry

end AveragedCalibratedDecrease
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/AveragedDecrease.lean (declaration OAI.Percolation.raw_derivative_template_row_lower)

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