Proposition: Dependence and fixed-width continuity

The coefficients ${{\mathcal{O}}^{(j)}}_{f,g,\epsilon }$, $j=0,1,2$, depend on the configuration only through its restriction to every neighbourhood of $D_f$. The same holds for the dynamical evaluation, with its fixed free input.

At fixed $\epsilon >0$, $h\mapsto w[h]$ is continuous linear and $(h,k)\mapsto v[h,k]$ continuous polynomial in the smooth compact-open topologies. The smeared observable coefficients are continuous polynomial functionals, and depend continuously linearly on $f$ at a fixed configuration.

Proof. The local coefficient operators do not enlarge source dependence. Every retarded inverse at a point $x$ uses only $J_{\eta }^{-}(x)\cap M_{T_{\ast }}$. If an intermediate point lies in that past, its own truncated past lies there as well. Applying this successively to (eq:coordinatesolutions) and (eq:scalarsolution), then to (eq:relationalcoeff), proves dependence on $D_f$.

The Cauchy estimates in Lemma lem:cauchy, continuity of multiplication and differentiation in $C^{\infty }$, and compact smearing give all stated continuity assertions. The constants may depend on $\epsilon$; derivatives of the ramp scale as ${\epsilon }^{-r}$.