Lemma: The single-block weak insertion is defined

Equation (eq:blockC) is a continuous sesquilinear distribution on ${\mathcal{D}}_E\times {\mathcal{D}}_E$, and

This assertion is not an operator-domain claim for an arbitrary singularly smeared retarded insertion.

Proof. Between two finite-particle energy-smooth vectors, every fully normal-ordered multipoint kernel is smooth. Indeed, its Fourier integrals contain only the external vector wave functions and finitely many polynomial momentum factors. Arbitrarily high energy moments make these integrals absolutely convergent after Cauchy–Schwarz; the locally integrable mass-shell measure controls zero momentum. The same argument applies to every derivative.

Consequently each matrix element of (eq:blockC) is a finite sum of derivatives of $R(x-y)$ multiplied by smooth functions. It is a distribution, with finite-order bounds on compact sets. The bounds depend continuously on energy seminorms of the two vectors. Differentiating in $x$ and applying (eq:jetcontact) gives (eq:blockcontact).