isInvariantOn_normalizedGibbs_on_compactlySupportedC2

Plain-English statement

- A semigroup satisfying the integrated-generator contract on the compactly supported `C²` core preserves normalized Gibbs expectations on that core. This theorem composes the concrete Gibbs integration-by-parts identity with the abstract semigroup-to-invariance bridge. The semigroup contract remains an explicit hypothesis: no Langevin SDE, Markov semigroup, core closure, or measure-determining extension is constructed here.

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 isInvariantOn_normalizedGibbs_on_compactlySupportedC2
    {n : ℕ}
    {V : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    {P : ℝ →
      (EuclideanSpace ℝ (Fin (n + 1)) → ℝ) →
        EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    {generator :
      (EuclideanSpace ℝ (Fin (n + 1)) → ℝ) →
        EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : ContDiff ℝ 1 V)
    (hcore : CoreContract V generator (Set.ofPred CompactlySupportedC2))
    (hsemigroup : WeakGenerator.IntegratedSemigroupGeneratorContract
      P generator (Set.ofPred CompactlySupportedC2)
      (volume.withDensity
        (fun x =>
          (∫⁻ y, Measure.Gibbs.gibbsDensityENNReal V y ∂volume)⁻¹ *
            Measure.Gibbs.gibbsDensityENNReal V x))) :
    WeakGenerator.IsInvariantOn P
      (volume.withDensity
        (fun x =>
          (∫⁻ y, Measure.Gibbs.gibbsDensityENNReal V y ∂volume)⁻¹ *
            Measure.Gibbs.gibbsDensityENNReal V x))
      (Set.ofPred CompactlySupportedC2) := by
  refine WeakGenerator.isInvariantOn_of_integral_generator_eq_zero hsemigroup ?_
  intro f hf
  rw [hcore.generator_eq_operator_on_core f hf]
  exact integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2 hV hf

end LangevinGenerator
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:90published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Corollary 1.2.9 route: compose Gibbs core annihilation with an explicit integrated-semigroup contract on C_c^2

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Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Pointwise support, topological support, and compact support retain distinct meanings.
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.