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

Proved

OAI.FiniteFactor.sourceBiasedPrefix_domination

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.sourceBiasedPrefix_domination of lean/OAI/Analysis/FiniteFactor/BiasedPairs/PrefixDomination.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.FiniteFactor, with open MeasureTheory Filter ProbabilityTheory CouplingParameters Finset, open scoped ENNReal):

theorem sourceBiasedPrefix_domination (j n : ℕ) :
    (sourceBiasedTorusMeasure j (sourceBiasedPrefixMask n) : Measure SourceSamplingTorus) ≪
      ∑ I ∈ (blockPrefix sourceBlockSize n).powerset,
        (sourceSamplingLambdaTorus I : Measure SourceSamplingTorus)

Definitions. The statement is written with OAI.FiniteFactor.SourceCoordinate, OAI.FiniteFactor.SourceSamplingTorus, OAI.FiniteFactor.sourceBiasedPrefixMask, OAI.FiniteFactor.sourceBiasedTorusMeasure, OAI.FiniteFactor.sourceBlockSize, OAI.FiniteFactor.sourceSamplingLambdaTorus, OAI.FiniteFactor.CouplingParameters.BlockCoordinate and OAI.FiniteFactor.CouplingParameters.blockPrefix (definitions bundle OAIQ3Ff000 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ3Ff000

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

namespace FiniteFactor
open MeasureTheory Filter ProbabilityTheory CouplingParameters Finset
open scoped ENNReal
noncomputable section

theorem sourceBiasedPrefix_domination (j n : ℕ) :
    (sourceBiasedTorusMeasure j (sourceBiasedPrefixMask n) : Measure SourceSamplingTorus) ≪
      ∑ I ∈ (blockPrefix sourceBlockSize n).powerset,
        (sourceSamplingLambdaTorus I : Measure SourceSamplingTorus) := 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/PrefixDomination.lean (declaration OAI.FiniteFactor.sourceBiasedPrefix_domination)

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