Proposition: Dependence support

Let $K_f=\mathrm{supp}f$, and let

This is compact. The functional ${\mathcal{O}}_f$, regarded as a functional of smooth $h,{\varphi }_0$ before restricting ${\varphi }_0$ to the free shell, has dependence support contained in ${\mathcal{D}}_f$.

Proof. The retarded scalar response observed in $K_f$ uses sources only in $J_{\eta }^{-}(K_f)$. Its source is restricted by $\chi$ to $0<t<T$. The coordinate response in $K_f$ uses $C(h)$ only between ${\Sigma }_0$ and $K_f$, again inside $J_{\eta }^{-}(K_f)$. The direct scalar factor in the coordinate term is evaluated in $K_f$. Differentiation does not enlarge these supports.

A finite-time Minkowski causal past of a compact set has bounded spatial extent on the closed slab $0\le t\le T$, proving compactness. The conclusion follows from finite propagation.