chewi_theorem_1_2_22_scalar_equivalence

Plain-English statement

Under its exact dissipation identity, Poincare coercivity is equivalent to shifted exponential chi-square 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

χ²(s)≤CD(s) for every s iff χ²(s+t)≤χ²(s)exp(-2t/C), assuming (χ²)'=-2D.

Intuition

The scalar differential argument is identical to variance decay; the distinction is in how the concrete density curve and dissipation are constructed.

Conditions

C>0
the chi-square curve has exact right derivative -2D
shifted decay or coercivity holds at every time

Why these conditions cannot be dropped

the scalar layer cannot manufacture the Radon-Nikodym density
the converse needs decay after arbitrary restarts

Proof route

reuse the generic shifted Gronwall theorem
reuse the generic one-sided converse
package the two implications as an iff

Lean interface notes

the source-facing name preserves the distinction from the variance theorem
concrete chi-square evolution remains an explicit downstream node

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

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Lean statement

theorem chewi_theorem_1_2_22_scalar_equivalence
    {C : ℝ} (hC : 0 < C) (curve : DissipationCurve 2) :
    (∀ s : ℝ,
      curve.energy s ≤ C * curve.dissipation s) ↔
    (∀ s t : ℝ, 0 ≤ t →
      curve.energy (s + t) ≤
        curve.energy s * Real.exp (-(2 / C) * t)) :=
  chewi_theorem_1_2_21_scalar_equivalence hC curve

/-- Forward direction of Chewi, Theorem 1.2.26, between arbitrary times. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:298published source at 7bcd37294df1