Proposition: Support and continuity of the new vertex

The additional action, its first variations, and the new local Euler term have support in $\mathrm{supp}a$. Their coefficient pairings are continuous polynomial maps in the compact-open smooth topology.

Bracketing a compact local functional with the new vertex produces a compact local functional. Finite multilocal products remain in ${\mathcal{F}}_{\mathrm{ml}}(U)$.

Proof. Every new variation contains a factor $a$ or one of its derivatives. Differentiation does not enlarge its support. On a compact set, the integral and its derivatives are bounded by finite products of smooth seminorms.

The variational bracket uses a finite number of Euler derivatives and integrations by parts. Smooth classical configurations have no undefined pointwise field products. The super-Leibniz rule proves the multilocal statement.