Proposition: Linear admission

For fixed classical backgrounds, every $c_j$ in (eq:tests) is smooth and supported in $K=\mathrm{supp}f$. The maps $(h,k)\mapsto c_j\in {\mathcal{D}}_K(M)$ are continuous polynomial maps at fixed $\epsilon$. Each $F_{c_j}$ belongs to ${\mathcal{L}}_{\mathrm{off}}$, and its class $\hat{\varphi }(c_j)$ is well defined.

Proof. The coordinate coefficients are smooth and continuous by the handoff. Formula (eq:tests) uses smooth products and derivatives multiplied by $f$, so its support is contained in $K$, with continuous dependence in all compact-test seminorms. As a functional of the only quantum configuration, $F_{c_j}$ has smooth first derivative and all higher derivatives zero.