integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn

Plain-English statement

- Finite-box trace integrability for the explicit Gibbs-weighted Langevin trace display, assuming the displayed scalar RHS is continuous on the box. This closes the integrability handoff only under explicit regularity data: the Pi-space vector field has the supplied derivative on the closed box, the pointwise differentiability hypotheses needed by the Langevin display hold on the closed box, and the scalar display `exp (-V) * (Δ f - <∇V, ∇f>)` is continuous on that box. The theorem does not prove those regularity hypotheses, does not prove whole-space integrability, does not cancel finite-box face terms, and does not prove weighted IBP, invariant law, 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_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))))
    (hcont : ContinuousOn
      (fun x : Fin (n + 1) → ℝ =>
        Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) *
          (Laplacian.laplacian f
              (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) -
            inner ℝ
              (gradient V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
              (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 h_exp : IntegrableOn
      (fun x : Fin (n + 1) → ℝ =>
        Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) *
          (Laplacian.laplacian f
              (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) -
            inner ℝ
              (gradient V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
              (gradient f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))))
      (Set.Icc a b) volume :=
    hcont.integrableOn_compact isCompact_Icc
  refine (integrableOn_congr_fun ?_ measurableSet_Icc).mp h_exp
  intro x hx
  exact (trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt
    (V := V) (f := f) (x := x) (F' := F' x)
    (hF x hx) (hV x hx) (hfderiv x hx) (hf x hx)).symm

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:804published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: finite closed-box handoff proving Mathlib trace-summand `IntegrableOn` for the explicit Gibbs-weighted fderiv-coordinate field when the scalar Langevin display is already continuous on the box

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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.
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.