The propagated defect

Define

The massless kernel gives

It is smooth and spacelike compact, with data

Thus $\zeta =0$ is exactly two compact smooth Cauchy-data equations, not an unspecified global condition.

The advanced potential is

Its jump is $A(0+)-A(0-)=-b$. The resulting Dirac coefficient in $D^{\ast }A$ is

which equals $j_{\Sigma }$, because $b^{\nu }=u_0{\partial }^{\nu }\varphi {|}_0$. The section current has no remaining Dirac measure: it is piecewise smooth and locally bounded.