integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2

Plain-English statement

- The normalized Gibbs measure annihilates the displayed Langevin operator on the compactly supported `C²` core. This is a normalized-measure corollary of the whole-space weighted-IBP theorem. It is a core-level infinitesimal stationarity statement, not semigroup invariance: extending it to a semigroup-stable generator domain requires an additional closure/core theorem or a separate martingale-problem argument.

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 integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2
    {n : ℕ}
    {V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : ContDiff ℝ 1 V)
    (hf : CompactlySupportedC2 f) :
    ∫ x, operator V f x ∂volume.withDensity
        (fun x =>
          (∫⁻ y, Measure.Gibbs.gibbsDensityENNReal V y ∂volume)⁻¹ *
            Measure.Gibbs.gibbsDensityENNReal V x) = 0 := by
  rw [TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul_of_neZero
    volume hV.continuous.measurable.aemeasurable]
  simp_rw [operator, smul_eq_mul, mul_assoc]
  rw [integral_const_mul]
  rw [Langevin.integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport
    hV hf.1 hf.2]
  simp

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:65published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Example 1.2.8 to Corollary 1.2.9: prove normalized Gibbs expectation of the displayed generator is zero on the C_c^2 test core

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`rw` rewrites by an established identity.
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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.
Almost-everywhere hypotheses depend on the stated measure and representative.
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.