hasFDerivAt_smul_vectorField_off_countable

Plain-English statement

- Open-box/off-countable Frechet derivative wrapper for a scalar cutoff times a Pi-space vector field. This derives the `Hd` shape required by the finite-box divergence-theorem handoffs from separate derivative hypotheses for the scalar cutoff and the vector field on the same open-box minus exceptional set. It still does not prove trace integrability or any no-boundary conclusion.

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_smul_vectorField_off_countable
    {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))
    (s : Set (Fin (n + 1) → ℝ))
    (hχ : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt χ (χ' x) x)
    (hG : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt G (G' x) x) :
    ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt (fun y => χ y • G y)
        (χ x • G' x + (χ' x).smulRight (G x)) x := by
  intro x hx
  exact hasFDerivAt_smul_vectorField_of_hasFDerivAt χ (χ' x) G (G' x) x
    (hχ x hx) (hG x hx)

/-- Closed-box continuity of the cutoff-smul product-rule trace from only the
coordinate component continuity needed by the trace summand.

The expanded summand is
`χ x * (G' x eᵢ)ᵢ + (χ' x eᵢ) * (G x)ᵢ`.  This leaf therefore assumes
continuity of exactly these component functions.  It does not prove that `χ'`
or `G'` are derivative fields, does not construct cutoffs, and does not prove
boundary cancellation or weighted integration by parts. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1095published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff route: derive the divergence-theorem off-countable derivative hypothesis for a cutoff-smul field from separate cutoff and vector-field derivative hypotheses

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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

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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.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.