Theorem: Native classical insertion

Equation (eq:Fclass) is a continuous polynomial on $\mathcal{X}\times \mathcal{X}$. As $f$ varies it defines a distribution-valued polynomial section, without spatial smoothing or a replacement of the original Newton inverse.

Proof. Every local term uses finitely many derivatives and products of Schwartz functions. Lemma lem:Newton controls the sole nonlocal factor. On the compact support of $f$, all these operations are continuous in finitely many seminorms. The dependence on $f$ is continuous in the ordinary test topology.