Lemma: Smooth propagation of the present sources

The causal propagations of $J_{\mathrm{raw}}$ and $H$ are smooth. If $E_{1}H=0$, the causal propagation of $J_{\mathrm{raw}}-D^{\ast }{G}_{1,A}H$ is smooth as well.

Proof. The sources are finite sums of slice distributions with smooth compact spatial coefficients and piecewise smooth densities on a bounded time interval. For a density $c(s)$, propagation is

All output-time derivatives act on the smooth Cauchy propagator; spatial derivatives commute with it. On compact time intervals, Cauchy estimates control every derivative. A slice derivative gives a derivative of $S_w(t-s_{\ast })c_{\ast }$, also smooth.

If $E_{1}H=0$, its advanced potential is compact by (eq:compactprimitive). It is piecewise smooth with smooth spatial coefficients, so the same argument applies after $D^{\ast }$.