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

Proved

OAI.FiniteFactor.sourceProjection_biased_prefix_pairs

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.sourceProjection_biased_prefix_pairs of lean/OAI/Analysis/FiniteFactor/BiasedPairs/PairIdentity.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 ENNReal NNReal Topology):

theorem sourceProjection_biased_prefix_pairs (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ)
    (hs : SourceOrderedTorusSampling θ)
    {p : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hp : p ∈ sourceAlgebra θ) (hproj : IsStarProjection p)
    (hinv : (1-p)*sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg*p = 0)
    (j n : ℕ) :
    ∀ᵐ z ∂(sourceBiasedTorusMeasure j (sourceBiasedPrefixMask n) : Measure SourceSamplingTorus),
      z ∈ sourceCappedPairZeroSet θ p

Definitions. The statement is written with OAI.FiniteFactor.SourceAngles, OAI.FiniteFactor.SourceCoordinate, OAI.FiniteFactor.SourceHilbert, OAI.FiniteFactor.SourceOrderedTorusSampling, OAI.FiniteFactor.SourceSamplingTorus, OAI.FiniteFactor.SourceTorus, OAI.FiniteFactor.sourceAlgebra, OAI.FiniteFactor.sourceBiasedPrefixMask, OAI.FiniteFactor.sourceBiasedTorusMeasure, OAI.FiniteFactor.sourceBlockSize, OAI.FiniteFactor.sourceCappedPairZeroSet, OAI.FiniteFactor.sourceLogWeight, OAI.FiniteFactor.sourceMeasure and OAI.FiniteFactor.SourceAngles.nonneg (definitions bundle OAIQ3Ff000 of this split); OAI.FiniteFactor.sourceLogWeight_nonneg and OAI.FiniteFactor.sourceOperator (definitions bundle OAIQ3Ff002 of this split).

Used by. The proof of OAI.FiniteFactor.sourceProjection_biased_ratio_augmented of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ3Ff002

Formal statement

section

attribute [-instance] _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
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 ENNReal NNReal Topology
noncomputable section

theorem sourceProjection_biased_prefix_pairs (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ)
    (hs : SourceOrderedTorusSampling θ)
    {p : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hp : p ∈ sourceAlgebra θ) (hproj : IsStarProjection p)
    (hinv : (1-p)*sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg*p = 0)
    (j n : ℕ) :
    ∀ᵐ z ∂(sourceBiasedTorusMeasure j (sourceBiasedPrefixMask n) : Measure SourceSamplingTorus),
      z ∈ sourceCappedPairZeroSet θ p := 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/BiasedPairs/PairIdentity.lean (declaration OAI.FiniteFactor.sourceProjection_biased_prefix_pairs)

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