chewi_theorem_1_2_26_forward

Plain-English statement

Log-Sobolev coercivity turns entropy dissipation into exponential KL decay.

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.

Mathematical statement

If KL'(t)=-FI(t) and KL(t)≤(C/2)FI(t), then KL(t)≤KL(0)exp(-2t/C).

Intuition

LSI lower-bounds the available Fisher dissipation by a multiple of the current KL divergence, so entropy cannot decay slower than the corresponding exponential curve.

Conditions

C is positive
the entropy curve has derivative minus Fisher information
the LSI bound holds along the curve

Why these conditions cannot be dropped

density regularity and entropy differentiation are genuine analytic obligations
the factor C/2 determines the rate 2/C

Proof route

multiply KL≤(C/2)FI by 2/C
obtain (2/C)KL≤FI
invoke the generic scaled-dissipation decay theorem

Lean interface notes

the existing KL/FI scalar bookkeeping can later instantiate this curve
the converse LSI criterion remains a separate theorem

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Lean statement

theorem chewi_theorem_1_2_26_forward
    {C : ℝ} (hC : 0 < C) (curve : DissipationCurve 1)
    (hLSI : ∀ s : ℝ,
      curve.energy s ≤ (C / 2) * curve.dissipation s)
    {t : ℝ} (ht : 0 ≤ t) :
    curve.energy t ≤
      curve.energy 0 * Real.exp (-(2 / C) * t) := by
  simpa [sub_zero] using
    chewi_theorem_1_2_26_forward_from hC curve hLSI ht

/-- Backward scalar direction of Chewi, Theorem 1.2.26. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:327published source at 7bcd37294df1