integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn

Plain-English statement

- Finite-box trace integrability for the explicit Gibbs-weighted Langevin trace display from component continuity. This is a convenience wrapper around `integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn`. It replaces the single scalar `ContinuousOn` hypothesis by separate continuity hypotheses for `V`, `Laplacian.laplacian f`, `gradient V`, and `gradient f`, all after the `WithLp.toLp 2` coordinate bridge. It still assumes the explicit Pi-space vector field has the supplied Frechet derivative on the closed box and that the pointwise differentiability hypotheses needed by the trace/display equality hold there. It does not derive those assumptions from a concrete test-function class, does not cancel face terms, and does not prove weighted IBP, generator domains, invariant laws, reversibility, stationarity, or KL/FI dissipation.

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 integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
    (F' : (Fin (n + 1) → ℝ) →
      (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
    (hF : ∀ x ∈ Set.Icc a b,
      HasFDerivAt
        (fun z : Fin (n + 1) → ℝ => fun i =>
          Real.exp (-V (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))) *
            fderiv ℝ f (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))
              (EuclideanSpace.single i (1 : ℝ))) (F' x) x)
    (hV : ∀ x ∈ Set.Icc a b,
      DifferentiableAt ℝ V
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
    (hfderiv : ∀ x ∈ Set.Icc a b,
      DifferentiableAt ℝ
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) => fderiv ℝ f y)
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
    (hf : ∀ x ∈ Set.Icc a b,
      DifferentiableAt ℝ f
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
    (hV_cont : ContinuousOn
      (fun x : Fin (n + 1) → ℝ =>
        V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      (Set.Icc a b))
    (hlap_cont : ContinuousOn
      (fun x : Fin (n + 1) → ℝ =>
        Laplacian.laplacian f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      (Set.Icc a b))
    (hgradV_cont : ContinuousOn
      (fun x : Fin (n + 1) → ℝ =>
        gradient V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      (Set.Icc a b))
    (hgradf_cont : ContinuousOn
      (fun x : Fin (n + 1) → ℝ =>
        gradient f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      (Set.Icc a b)) :
    IntegrableOn (fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
      (Set.Icc a b) volume := by
  have hcont := continuousOn_expNeg_langevinGenerator_rhs_of_components
    (a := a) (b := b) (V := V) (f := f)
    hV_cont hlap_cont hgradV_cont hgradf_cont
  exact integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn
    a b V f F' hF hV hfderiv hf hcont

/-- Finite-box trace integrability for the explicit Gibbs-weighted Langevin
trace display from global `C¹/C²` regularity, still assuming the explicit
Pi-space trace field has the supplied Frechet derivative on the box.

Compared with
`integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn`,
this wrapper derives the component continuity and pointwise differentiability
hypotheses from `ContDiff ℝ 1 V` and `ContDiff ℝ 2 f`.  The remaining
nontrivial regularity input is the explicit Pi-space field derivative `hF`.

It does not prove that field derivative, whole-space integrability, weighted
IBP, boundary cancellation, generator domains, invariant laws, reversibility,
or KL/FI dissipation. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:868published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: finite closed-box trace `IntegrableOn` handoff for the explicit Gibbs-weighted fderiv-coordinate field under component continuity plus the existing trace/display differentiability hypotheses

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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

Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.