chewi_theorem_1_2_22_forward

Plain-English statement

The same Poincare energy argument yields exponential chi-square decay once its density evolution supplies the required dissipation curve.

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 χ²'(t)=-2D(t) and χ²(t)≤CD(t), then χ²(t)≤χ²(0)exp(-2t/C).

Intuition

At the scalar level, variance and chi-square use the same coercivity-and-dissipation calculation; their distinction belongs to the concrete density representation.

Conditions

C is positive
the chi-square curve has the exact -2 dissipation derivative
Poincare coercivity is available along the curve

Why these conditions cannot be dropped

the scalar theorem cannot manufacture the Radon-Nikodym density or its evolution
the exact rate depends on the factor 2 in the dissipation identity

Proof route

reuse the variance-style Poincare decay theorem
leave density identification to the downstream semigroup layer

Lean interface notes

this is not yet a theorem about a particular pair of measures
the name records the exact source theorem and direction

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Lean statement

theorem chewi_theorem_1_2_22_forward
    {C : ℝ} (hC : 0 < C) (curve : DissipationCurve 2)
    (hPI : ∀ s : ℝ,
      curve.energy s ≤ C * curve.dissipation s)
    {t : ℝ} (ht : 0 ≤ t) :
    curve.energy t ≤
      curve.energy 0 * Real.exp (-(2 / C) * t) :=
  chewi_theorem_1_2_21_forward hC curve hPI ht

/-- Backward scalar direction of Chewi, Theorem 1.2.22. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:278published source at 7bcd37294df1