Compact-source Green tails are available

For compact $g$, the retarded and advanced responses ${G_{\mathcal{B}}}^{\mathrm{ret},\mathrm{adv}}g$ are smooth and have uniformly compact spatial support on $J⋐I$. Indeed their support lies in the causal future or past of $\mathrm{supp}g$, and the coordinate propagation radius is bounded by

on compact time intervals. Their rod sources meet Lemma lem:Newton.

This proves a Green-response tail. It is not a proof of a state-covariance tail.