trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt

Plain-English statement

- Trace-summand display for the explicit Pi-space vector field `x ↦ exp (-V x) * fderiv f x eᵢ`. This is the pointwise bridge from Mathlib's finite-box divergence-theorem trace integrand to the Langevin display `exp (-V) * (Δ f - <∇V, ∇f>)`. It uses the supplied Frechet derivative of the explicit Pi-space vector field, plus pointwise differentiability assumptions needed by the compiled coordinate-divergence display. It does not prove that the supplied derivative exists on a box, prove continuity or integrability, invoke a divergence theorem, cancel boundary terms, establish weighted integration by parts, define a generator domain, or prove invariant/reversible Gibbs laws.

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 trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt
    {n : ℕ}
    {V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    {x : Fin (n + 1) → ℝ}
    {F' : (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ)}
    (hF : 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)
    (hV : DifferentiableAt ℝ V
      (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
    (hfderiv : DifferentiableAt ℝ
      (fun y : EuclideanSpace ℝ (Fin (n + 1)) => fderiv ℝ f y)
      (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
    (hf : DifferentiableAt ℝ f
      (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) :
    (∑ i, F' (Pi.single i (1 : ℝ)) i) =
      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))))) := by
  have htrace :=
    _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt
      (ι := Fin (n + 1)) hF
  have hdisplay :=
    coordinateDivergence_expNeg_fderivCoordinateField_langevinGenerator_display_of_differentiableAt
      (V := V) (f := f)
      (x := (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) hV hfderiv hf
  rw [← htrace]
  simpa using hdisplay

/-- Continuity of the scalar Langevin display on a finite Pi-box from
component continuity.

The hypotheses keep the analytic regularity inputs explicit: continuity of the
potential, the Mathlib Laplacian display, and the two gradient fields after the
`WithLp.toLp 2` coordinate bridge.  The theorem only assembles these component
facts into continuity of
`exp (-V) * (Δ f - <∇V, ∇f>)`.

It does not prove that the components are continuous from a `ContDiff` or
test-function class, does not prove differentiability of the explicit vector
field, and does not prove trace integrability, IBP, boundary cancellation,
generator domains, invariant laws, reversibility, or KL/FI dissipation. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:582published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: pointwise handoff from Mathlib's Pi-space finite-box trace summand for the explicit field `exp(-V) * fderiv f eᵢ` to the scalar display `exp(-V) * (Delta f - <grad V, grad f>)`

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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`rw` rewrites by an established identity.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

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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.