integrableOn_smul_vectorField_trace_of_continuousOn

Plain-English statement

- Closed-box integrability for the trace of the cutoff-smul product-rule derivative, assuming that trace expression is continuous on the closed box. This is a compact-box integrability handoff only. It does not prove continuity of the trace from component assumptions, construct a smooth cutoff, prove tail decay, or pass from finite boxes to whole-space weighted integration by parts.

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_smul_vectorField_trace_of_continuousOn
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (χ : (Fin (n + 1) → ℝ) → ℝ)
    (χ' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] ℝ)
    (G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (G' : (Fin (n + 1) → ℝ) →
      (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
    (htrace : ContinuousOn
      (fun x => ∑ i, ((χ x • G' x + (χ' x).smulRight (G x))
        (Pi.single i (1 : ℝ))) i)
      (Set.Icc a b)) :
    IntegrableOn
      (fun x => ∑ i, ((χ x • G' x + (χ' x).smulRight (G x))
        (Pi.single i (1 : ℝ))) i)
      (Set.Icc a b) volume :=
  htrace.integrableOn_compact isCompact_Icc

/-- Scalar cutoff vanishing outside the open Pi-box implies Mathlib's finite-box
signed face-term sum is zero for the cutoff-smul vector field.

Regularity of the cutoff-smul field is not addressed here; this is only the
finite-box support-to-face producer. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1189published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 finite-box cutoff route: discharge compact-box integrability of the cutoff-smul product-rule trace from closed-box trace continuity

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Hidden assumptions and non-claims

Integrability is an input or proved output; a displayed integral alone does not supply it.
Pointwise support, topological support, and compact support retain distinct meanings.
A formal generator display does not establish a closed operator domain.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.