OAI.ThreeTorus.actual_controlled_density_prediction: intermediate lemma from OpenAI, A smooth three-torus diffeomorphism with simple Lebesgue spectrum

Proved

OAI.ThreeTorus.actual_controlled_density_prediction

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.ThreeTorus.simple_lebesgue_spectrum, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A smooth three-torus diffeomorphism with simple Lebesgue spectrum, OpenAI Math Release, September 23, 2026, result family 144, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.ThreeTorus.actual_controlled_density_prediction of lean/OAI/Dynamics/ThreeTorus/ThreeTorus/ControlledBlocks.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ThreeTorus, with open Set Function Filter MeasureTheory, open scoped Topology BigOperators ComplexConjugate):

theorem actual_controlled_density_prediction {m : ℕ} (C : ColoredState (Fin m))
    (τR : Fin m → Fin m) (hτR : Involutive τR) (hrC : C.Reflection τR)
    (tags : Fin m → ℂ) (htags : ∀ r, ‖tags r‖ ≤ 1)
    (htrR : ∀ r, tags (τR r) = conj (tags r))
    {ε τ η κ : ℝ} (hε : 0 < ε) (hτ : 0 < τ) (hη : 0 < η) (hκ : 0 < κ)
    (hmass : (∫ z, blockTotal C.density z ∂phaseHaar) < 2) (Q : PhysicalJetData) (F₀ : PhysicalFrame) :
    ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus,
      ∃ Ψ : UnitAddCircle → ℂ, ∃ K : ℕ, ∃ A : UnitAddCircle → ℂ,
      0 < K ∧ ActualColoredBlocks C E K α A ∧ Ψ = (fun z => A z/(K:ℂ)) ∧
      (∀ r, physicalError (C.vector r) (E.vector r ∘ α) < η) ∧
      ((∫ z, Real.sqrt (blockTotal C.density z) ∂phaseHaar)-τ <
        ∫ z, Real.sqrt (blockTotal E.density z) ∂phaseHaar) ∧
      ((∑ r, ∫ z, E.density r z*‖Ψ z-tags r‖^2 ∂phaseHaar) < ε^2) ∧
      Continuous Ψ ∧
      ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => Ψ (x:UnitAddCircle)) ∧
      (∀ z, Ψ (-z) = conj (Ψ z)) ∧
      (∫ z, blockTotal E.density z ∂phaseHaar) < 2 ∧ E.Reflection τR ∧
      (∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ) ∧
      ∃ F' : PhysicalFrame, F'.chart = F₀.chart.trans α ∧ Q.Close η F₀ C.packets.V F' E.packets.V

Definitions. The statement is written with OAI.ThreeTorus.ActualColoredBlocks, OAI.ThreeTorus.ColoredState, OAI.ThreeTorus.PhysicalFrame, OAI.ThreeTorus.PhysicalJetData, OAI.ThreeTorus.TripleTorus, OAI.ThreeTorus.blockTotal, OAI.ThreeTorus.phaseHaar, OAI.ThreeTorus.physicalError, OAI.ThreeTorus.ColoredState.Reflection, OAI.ThreeTorus.ColoredState.density, OAI.ThreeTorus.ColoredState.vector, OAI.ThreeTorus.PassagePackets.GoodArc, OAI.ThreeTorus.PassagePackets and OAI.ThreeTorus.PhysicalJetData.Close (definitions bundle OAIQ13Tt000 of this split).

Used by. The proof of OAI.ThreeTorus.exists_predicted_whitening of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ13Tt000

Formal statement

section

attribute [-instance] _root_.OAI.ThreeTorus.tripleVolume_regular
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section

noncomputable section
open Set Function Filter MeasureTheory
open scoped Topology BigOperators ComplexConjugate
namespace ThreeTorus

theorem actual_controlled_density_prediction {m : ℕ} (C : ColoredState (Fin m))
    (τR : Fin m → Fin m) (hτR : Involutive τR) (hrC : C.Reflection τR)
    (tags : Fin m → ℂ) (htags : ∀ r, ‖tags r‖ ≤ 1)
    (htrR : ∀ r, tags (τR r) = conj (tags r))
    {ε τ η κ : ℝ} (hε : 0 < ε) (hτ : 0 < τ) (hη : 0 < η) (hκ : 0 < κ)
    (hmass : (∫ z, blockTotal C.density z ∂phaseHaar) < 2) (Q : PhysicalJetData) (F₀ : PhysicalFrame) :
    ∃ E : ColoredState (Fin m), ∃ α : Equiv.Perm TripleTorus,
      ∃ Ψ : UnitAddCircle → ℂ, ∃ K : ℕ, ∃ A : UnitAddCircle → ℂ,
      0 < K ∧ ActualColoredBlocks C E K α A ∧ Ψ = (fun z => A z/(K:ℂ)) ∧
      (∀ r, physicalError (C.vector r) (E.vector r ∘ α) < η) ∧
      ((∫ z, Real.sqrt (blockTotal C.density z) ∂phaseHaar)-τ <
        ∫ z, Real.sqrt (blockTotal E.density z) ∂phaseHaar) ∧
      ((∑ r, ∫ z, E.density r z*‖Ψ z-tags r‖^2 ∂phaseHaar) < ε^2) ∧
      Continuous Ψ ∧
      ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => Ψ (x:UnitAddCircle)) ∧
      (∀ z, Ψ (-z) = conj (Ψ z)) ∧
      (∫ z, blockTotal E.density z ∂phaseHaar) < 2 ∧ E.Reflection τR ∧
      (∃ δ : ℝ, 0 < δ ∧ δ < 1/8 ∧ δ < κ ∧ E.packets.GoodArc δ) ∧
      ∃ F' : PhysicalFrame, F'.chart = F₀.chart.trans α ∧ Q.Close η F₀ C.packets.V F' E.packets.V := by
  sorry

end ThreeTorus
end
end

end OAI
end

Source

OpenAI, A smooth three-torus diffeomorphism with simple Lebesgue spectrum, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 144; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/ThreeTorus/ThreeTorus/ControlledBlocks.lean (declaration OAI.ThreeTorus.actual_controlled_density_prediction)

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