Lemma: Smooth propagation for the currents at hand

The causal propagation of $J_{\mathrm{raw}}$ and of $\mathbf{H}$ is smooth. If $J_{\sec }$ is compact, its causal propagation is smooth as well.

Proof. Each current is a finite sum of compact slice distributions with smooth spatial coefficients and piecewise smooth densities on a bounded time interval. Spatial derivatives of those densities have uniform bounds on each time piece, and their spatial supports have a common compact bound.

For a density $c(s)$, causal propagation is

All output-time derivatives can be moved onto the sine propagator. On the bounded source-time interval these propagators act continuously on smooth compact spatial data. Spatial derivatives commute with them. The resulting function is smooth. A source ${\delta }^{(j)}(t-t_{\ast })c_{\ast }(x)$ gives a time derivative of $S_{\mathrm{w}}(t-t_{\ast })c_{\ast }$, also smooth. Finite propagation gives spacelike compactness.

The required decompositions follow from (eq:Jraw), (eq:H), and (eq:Aexplicit); the surface measures cancel in $J_{\sec }$.