OAI.FiniteFactor.sourceSamplingSquaredError_ae: intermediate lemma from OpenAI, Invariant-projection counterexamples for every irrational rotation

Proved

OAI.FiniteFactor.sourceSamplingSquaredError_ae

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.FiniteFactor.main_theorem, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Invariant-projection counterexamples for every irrational rotation, 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.sourceSamplingSquaredError_ae of lean/OAI/Analysis/FiniteFactor/OrbitSampling/ErrorEvent.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 Topology):

theorem sourceSamplingSquaredError_ae (r : Bool) (b : SourceCoordinate → Bool)
    (a c a' c' : SourceCoordinate → ℝ) :
    ∀ᵐ θ ∂sourceAngleLaw,
      Summable (fun stage => sourceSamplingSquaredError stage r b
        (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) ∧
      Tendsto (fun stage => sourceSamplingSquaredError stage r b
        (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) atTop (𝓝 0)

Definitions. The statement is written with OAI.FiniteFactor.SourceCoordinate and OAI.FiniteFactor.sourceAngleLaw (definitions bundle OAIQ3Ff000 of this split); OAI.FiniteFactor.sourceSamplingSquaredError (definitions bundle OAIQ3Ff002 of this split).

Used by. The proofs of OAI.FiniteFactor.sourceSamplingLambda_rectangle_limit_event and OAI.FiniteFactor.sourceSamplingTrimmed_marginals_event of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ3Ff002

Formal statement

section

attribute [-instance] _root_.OAI.FiniteFactor.sourceBiasedRealMeasure_probability _root_.OAI.FiniteFactor.sourceFiniteReferenceKernel_markov _root_.OAI.FiniteFactor.sourceHilbert_separable _root_.OAI.FiniteFactor.sourceNormalFunctionalSpace_complete _root_.OAI.FiniteFactor.sourceNormalPredualSpace_complete _root_.OAI.FiniteFactor.sourceNormalPredualSpace_normedGroup _root_.OAI.FiniteFactor.sourceNormalPredualSpace_normedSpace _root_.OAI.FiniteFactor.sourceResamplingKernel_markov _root_.OAI.FiniteFactor.sourceSamplingLambda_finite _root_.OAI.FiniteFactor.sourceSamplingSummand_finite _root_.OAI.FiniteFactor.sourceSamplingTorus_borel _root_.OAI.FiniteFactor.sourceSamplingTorus_compact _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 Topology
noncomputable section

theorem sourceSamplingSquaredError_ae (r : Bool) (b : SourceCoordinate → Bool)
    (a c a' c' : SourceCoordinate → ℝ) :
    ∀ᵐ θ ∂sourceAngleLaw,
      Summable (fun stage => sourceSamplingSquaredError stage r b
        (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) ∧
      Tendsto (fun stage => sourceSamplingSquaredError stage r b
        (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) atTop (𝓝 0) := by
  sorry

end
end FiniteFactor

end OAI
end

Source

OpenAI, Invariant-projection counterexamples for every irrational rotation, OpenAI Math Release, September 27, 2026, https://github.com/openai/math (result family 293; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/FiniteFactor/OrbitSampling/ErrorEvent.lean (declaration OAI.FiniteFactor.sourceSamplingSquaredError_ae)

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