finite_klDiv_and_toReal_le_half_mul_information

Plain-English statement

- Applying a generator log-Sobolev inequality to the canonical density `d mu / d pi` gives finite canonical KL together with the KL--Fisher bound. The constant convention is the one used by `Generator.SatisfiesLogSobolev`: if a Gibbs law has curvature `lambda`, then its expected constant here is `C = lambda⁻¹`, so the conclusion reads `KL <= FI / (2 * lambda)`. The finiteness conjunct is essential: `ENNReal.toReal ∞ = 0`, so an inequality about `klDiv.toReal` alone would hide a failed entropy-domain obligation. Its proof comes from the entropy-integrability field of `hAdmissible`, not from LSI alone. Chewi and `Generator.SatisfiesLogSobolev` call the constant `C`; SALD and Lee--Shen--Tian use the reciprocal rate `c_LSI = lambda = C⁻¹`.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem finite_klDiv_and_toReal_le_half_mul_information
    (mu pi : Measure (EuclideanSpace ℝ ι)) [IsProbabilityMeasure mu]
    [SigmaFinite mu] [Measure.HaveLebesgueDecomposition mu pi]
    (generator :
      (EuclideanSpace ℝ ι → ℝ) →ₗ[ℝ] (EuclideanSpace ℝ ι → ℝ))
    {C : ℝ}
    (hLSI : Generator.SatisfiesLogSobolev pi generator C)
    (hAdmissible : Generator.LogSobolevAdmissible pi generator
      (InformationTheory.RNLogRatio.density mu pi))
    (hScore :
      InformationTheory.CanonicalRelativeFisher.SmoothFiniteScoreDomain mu pi)
    (hPair :
      InformationTheory.CanonicalDirichletFisher.DirichletPairDomain
        mu pi generator)
    (hGamma :
      InformationTheory.CanonicalDirichletFisher.HasCanonicalFisherGamma
        mu pi generator) :
    _root_.InformationTheory.klDiv mu pi ≠ ∞ ∧
      (_root_.InformationTheory.klDiv mu pi).toReal ≤
        (C / 2) *
          InformationTheory.CanonicalRelativeFisher.information mu pi hScore := by
  let _ : IsProbabilityMeasure pi := hLSI.1
  have hLogRatioIntegrable : Integrable (MeasureTheory.llr mu pi) mu := by
    rw [← MeasureTheory.integrable_rnDeriv_mul_log_iff
      hScore.absolutelyContinuous]
    simpa [InformationTheory.RNLogRatio.density] using
      hAdmissible.2.2.2.1
  have hFinite : _root_.InformationTheory.klDiv mu pi ≠ ∞ :=
    _root_.InformationTheory.klDiv_ne_top
      hScore.absolutelyContinuous hLogRatioIntegrable
  refine ⟨hFinite, ?_⟩
  have hApplied := hLSI.2.2
    (InformationTheory.RNLogRatio.density mu pi) hAdmissible
  have hEntropy :
      Generator.densityEntropy pi
          (InformationTheory.RNLogRatio.density mu pi) =
        (_root_.InformationTheory.klDiv mu pi).toReal := by
    unfold Generator.densityEntropy
    change
      (∫ x, (mu.rnDeriv pi x).toReal *
        Real.log (mu.rnDeriv pi x).toReal ∂pi) =
        (_root_.InformationTheory.klDiv mu pi).toReal
    rw [MeasureTheory.integral_rnDeriv_mul_log
      hScore.absolutelyContinuous]
    simpa [InformationTheory.RNLogRatio.logRatio] using
      (InformationTheory.RNLogRatio.toReal_klDiv_eq_integral_logRatio_of_probability
        mu pi hScore.absolutelyContinuous).symm
  have hDirichlet :
      Generator.dirichletForm pi generator
          (InformationTheory.RNLogRatio.density mu pi)
          (fun x => Real.log
            (InformationTheory.RNLogRatio.density mu pi x)) =
        InformationTheory.CanonicalRelativeFisher.information mu pi hScore := by
    rw [show
      (fun x => Real.log (InformationTheory.RNLogRatio.density mu pi x)) =
        InformationTheory.RNLogRatio.logRatio mu pi by
      funext x
      exact (InformationTheory.RNLogRatio.logRatio_apply mu pi x).symm]
    exact
      InformationTheory.CanonicalDirichletFisher.dirichletForm_density_logRatio_eq_information
        mu pi generator hScore hPair hGamma
  rw [hEntropy, hDirichlet] at hApplied
  exact hApplied

end

end CanonicalLogSobolev
end FunctionalInequalities
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/CanonicalLogSobolev.lean:44published source at ed709a7f320d

Proof architecture

turn an established generator LSI plus explicit canonical density/IBP/score domains into finite canonical KL and KL <= (C/2) FI

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.