Lemma: Smeared sharp metric response

For $a\in \mathcal{D}(V;T^{\ast }U)$, smearing a massless metric covariance against $q_a$ in one slot gives a smooth function of the other slot. Reverse ordering has the same property.

Proof. The kernel $q_a$ has only timelike conormal wavefront covectors. A nonzero covariance wavefront covector is null in each slot and nonzero in both. The distributional composition criterion therefore has no opposing pair in the contracted variable. Its output wavefront can contain a covector only if such a matching pair survives, which is impossible. Compactness of $q_a$ supplies properness for the integration. The output is smooth BDHIntro.