OAI.FiniteFactor.sourceSampling_second_pair_error: intermediate lemma from OpenAI, Backward intertwiners and a transitive commutant

Proved

OAI.FiniteFactor.sourceSampling_second_pair_error

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Hyperinvariant.main_theorem, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Backward intertwiners and a transitive commutant, OpenAI Math Release, September 27, 2026, result family 293, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.FiniteFactor.sourceSampling_second_pair_error of lean/OAI/Analysis/FiniteFactor/OrbitSampling/SecondCorrelation.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.FiniteFactor, with open MeasureTheory Filter, open scoped Classical):

theorem sourceSampling_second_pair_error (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
    (i : SourceCoordinate) (hi : i.1 ≤ stage) (n n' : ℤ)
    (hn : n ∈ sourceSamplingPowers stage) (hn' : n' ∈ sourceSamplingPowers stage) (hne : n ≠ n')
    (G : Set ℝ) (hG : MeasurableSet G) (a c s : ℝ) (hsx : s ∈ Set.Ico (0 : ℝ) 1) :
    |(∫ θ, samplingTestedSecondWeight (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
        (Set.Icc a c) G s (Int.fract ((n : ℝ)*θ i)) *
        samplingTestedSecondWeight (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
        (Set.Icc a c) G s (Int.fract ((n' : ℝ)*θ i)) ∂sourceAngleLaw) -
      samplingTestedSecondDensity (sourceSamplingWidth stage) (sourceSamplingCutoff stage) (Set.Icc a c) G s ^ 2| ≤
      104*(sourceBlockSize stage : ℝ)^(-74 : ℝ)

Definitions. The statement is written with OAI.FiniteFactor.SourceCoordinate, OAI.FiniteFactor.samplingTestedSecondDensity, OAI.FiniteFactor.samplingTestedSecondWeight, OAI.FiniteFactor.sourceAngleLaw, OAI.FiniteFactor.sourceBlockSize, OAI.FiniteFactor.sourceSamplingCutoff, OAI.FiniteFactor.sourceSamplingPowers and OAI.FiniteFactor.sourceSamplingWidth (definitions bundle OAIQ3Ff000 of this split).

Used by. The proof of OAI_FiniteFactor_sourceSampling_coordinate_pair_error of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ3Ff000

Formal statement

section

attribute [-instance] _root_.OAI.FiniteFactor.sourceBiasedRealMeasure_probability _root_.OAI.FiniteFactor.sourceSamplingLambda_finite _root_.OAI.FiniteFactor.sourceSamplingSummand_finite _root_.OAI.FiniteFactor.sourceSamplingTrimmed_finite _root_.OAI.FiniteFactor.sourceSamplingUntrimmed_finite _root_.OAI.FiniteFactor.sourceSamplingWeightedMeasure_finite _root_.OAI.FiniteFactor.sourceUnitCircle_probability
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical
noncomputable section

theorem sourceSampling_second_pair_error (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
    (i : SourceCoordinate) (hi : i.1 ≤ stage) (n n' : ℤ)
    (hn : n ∈ sourceSamplingPowers stage) (hn' : n' ∈ sourceSamplingPowers stage) (hne : n ≠ n')
    (G : Set ℝ) (hG : MeasurableSet G) (a c s : ℝ) (hsx : s ∈ Set.Ico (0 : ℝ) 1) :
    |(∫ θ, samplingTestedSecondWeight (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
        (Set.Icc a c) G s (Int.fract ((n : ℝ)*θ i)) *
        samplingTestedSecondWeight (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
        (Set.Icc a c) G s (Int.fract ((n' : ℝ)*θ i)) ∂sourceAngleLaw) -
      samplingTestedSecondDensity (sourceSamplingWidth stage) (sourceSamplingCutoff stage) (Set.Icc a c) G s ^ 2| ≤
      104*(sourceBlockSize stage : ℝ)^(-74 : ℝ) := by
  sorry

end
end FiniteFactor

end OAI
end

Source

OpenAI, Backward intertwiners and a transitive commutant, OpenAI Math Release, September 27, 2026, https://github.com/openai/math (result family 293; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/FiniteFactor/OrbitSampling/SecondCorrelation.lean (declaration OAI.FiniteFactor.sourceSampling_second_pair_error)

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