Exact direct-plus-reflected subtraction

Thus the correctly named pieces are

The direct contribution is not a pure function of $\tau$, and is not a pure Pauli–Jordan contribution. Distinct Fourier phases do not prevent their integrated distributions from cancelling on an open region.

The substitutions $v=\tau$ and $v=u$ are distributionally unambiguous. For example, $(t,s)\mapsto (\tau ,s)$ is an invertible linear coordinate change, after which $D(\tau ,\mathbf{z})$ is $D$ tensored with the constant distribution in $s$. The same argument applies to $u$. No time-ordering prescription or wavefront extension is hidden in these substitutions.