Theorem: F9a for the completed coefficient families

On ${\mathcal{D}}_{K_1}\times {\mathcal{D}}_{K_2}\times {\mathcal{D}}_{K_3}$, the three two-edge output families and the positive-order degree-two coefficient families are jointly continuous with values in $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$. The scalar family ${\mathcal{N}}_c(f,g,f^{\prime })$ is jointly continuous.

The degree-two order-zero families also have fixed-support test-family continuity.

Proof. The tensor test map $(f,g,f^{\prime })\mapsto f\otimes g\otimes f^{\prime }$ is jointly continuous into ${\mathcal{D}}_K(X)$. Lemma lem:strongcontinuity controls all compact coefficient seminorms.

For the scalar-only output every metric trace coordinate is zero. For either metric-quadratic output, its derivative trace is

Lemma lem:sourcecontinuity and the smooth Green-operator mapping property give continuity of this coordinate, including its smooth trace seminorms. The same proof applies after one fixed directional derivative. Later metric derivatives vanish by the finite-degree reduction. Thus every topology coordinate required by ${\tau }_{\Sigma }$ is controlled.

A distribution has a finite-order estimate on $K$, so

and the right-hand side is bounded by a product of finitely many seminorms of the three tests.

For order zero, the inherited test maps and separately continuous classical multiplication give separately continuous multilinear maps on the named Fréchet test spaces. The standard Fréchet-space uniform boundedness argument makes these finite multilinear maps jointly continuous. This is still a test-family conclusion.