hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff

Plain-English statement

- Global `C¹/C²` version of `hasFDerivAt_expNeg_fderivCoordinateField_of_differentiableAt`.

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 hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff
    {n : ℕ}
    {V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    {x : Fin (n + 1) → ℝ}
    (hV : ContDiff ℝ 1 V)
    (hf : ContDiff ℝ 2 f) :
    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)))
      (fderiv ℝ
        (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))) x) x := by
  exact hasFDerivAt_expNeg_fderivCoordinateField_of_differentiableAt
    (V := V) (f := f) (x := x)
    (hV.differentiable one_ne_zero _)
    ((hf.fderiv_right (m := 1) (by norm_num)).differentiable one_ne_zero _)

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:770published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: discharge the explicit Pi-space field differentiability input for `z ↦ exp(-V(toLp z)) * fderiv f (toLp z) eᵢ` from global `C¹/C²` regularity

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
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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.