Relative derivatives and punctured products

For completeness, the use of the relative-space wavefront relation can be justified from the two-point relation: the two-point kernel is the pullback by $(x,y)\mapsto x-y$. Its wavefront covectors have the shape $(k,-k)$. Restriction to $(x,y)=(z,0)$ is permitted because the restriction conormal has shape $(0,k^{\prime })$, which cannot meet that set for nonzero $k$. The quoted off-diagonal orientation therefore transfers to $\mathcal{S}(z)$.