Theorem: Continuous admission of test families

The maps

are continuous and linear. For finitely many fixed compact test supports, every fixed finite pointwise word in these generators depends jointly continuously on those tests in their smooth fixed-support topologies.

Proof. For coordinate functionals, smooth finite-time Cauchy estimates make $a\mapsto {\mathsf{G}}_{\mathrm{A}}a$, its derivatives, and its initial trace continuous on each fixed compact source stage. Formula (eq:qa) then gives strong continuity of $q_a$.

For $B_f$, the compact mixed kernel is a finite derivative of the fixed retarded-step distribution multiplied by $f(x)$. On a common compact support stage, strong seminorms of this kernel are bounded by finitely many smooth seminorms of $f$. This follows by applying its finite distributional order to the bounded family of joint tests multiplied by $f$.

The only nonzero types of boundary coordinates are

Higher direction lists give zero. Each map in (eq:Btrace) is continuous on fixed test supports. The test-function LF topology is final over those stages, proving the linear continuity assertions.

For a word with fixed supports, its coefficient kernel is an exterior tensor product of fixed distributions multiplied by products of the varying smooth test functions. Finite-order estimates on the common compact region prove joint continuity of this restricted parameterization. Its boundary coordinates follow from the finite Leibniz formula. This restricted statement does not establish joint continuity of multiplication on all of ${\mathcal{P}}_{\Sigma }$.