The linear representatives

Let $\mathcal{D}(M)={C_c}^{\infty }(M;\mathbb{C})$ with its standard LF topology. On a fixed compact $K$, use the Fréchet space ${\mathcal{D}}_K(M)$ with all smooth seminorms. Backgrounds belong to the ambient smooth space

with convergence of all derivatives on compact sets. References to dependence on a closed causal set mean dependence on restrictions to its neighborhoods; no unmentioned topology on a nonsmooth closed set is imposed.

For $c\in \mathcal{D}(M)$, set

This off-shell insertion domain is identified with $\mathcal{D}(M)$.

Integration by parts in the bare expansion fixes the tests:

Thus

After free-field evaluation, the same notation denotes $\hat{\varphi }(c_0+\kappa c_1+{\kappa }^2{c}_2)$. The distinction between representative and class will always be made when applying $R_1$.