Proposition: Smooth scalar-linear admission

For fixed $(h,k)$, all tests in (eq:tests) lie in ${\mathcal{D}}_{K_f}$. Their dependence on backgrounds is continuous and polynomial at fixed $\epsilon$: $c_1$ is linear in $h$, and $c_2$ is quadratic in $h$ plus linear in $k$.

Thus each $F_{c_j}$ lies in the specified off-shell linear insertion domain, and $\mathbf{\varphi }(c_j)$ is a well-defined element of ${{\mathfrak{A}}_0}^{(1)}$.

Proof. The supplied coordinate maps are smooth and continuous in the background topology at fixed width. Equations (eq:tests) use their finite products and derivatives multiplied by $f$. This proves smoothness, support, polynomial dependence, and continuity in every ${\mathcal{D}}_{K_f}$ seminorm.

As a functional of the only quantum configuration $\psi$, $F_{c_j}$ has smooth first derivative $c_j$ and vanishing higher derivatives.