Corollary: Singular support and the smooth tail

The singular support is the projection to the base of the relation in (eq:main). In particular:

1. off the diagonal, geometric singularities occur at the regular or limiting geometric travel times $T=\pm R_k$;

2. in nonquotient sectors, the additional singular front is $T=\pm L_{\mathrm{ap}}$;

3. the open region $T^2>{L_{\mathrm{ap}}}^2$ is smooth.

The third statement does not say that the covariance or causal kernel vanishes there.

Proof. The first two statements follow from the exact wavefront equality. For the third, $Q<-1$, where the summed boundary expression is smooth by the first step of Proposition prop:upper.