Lemma: A nonzero surface coefficient is not microcausal

If $b$ is smooth near $\Sigma$ and $b{|}_{\Sigma }\ne 0$, then $C^{\ast }(\theta b)$ has a nonzero timelike wavefront covector.

Proof. In the fixed component convention the surface map in (eq:sharpadjoint) has

It is injective. Some component has local form

Choose compact localizers whose spatial pairing with $\beta$ is nonzero. Its Fourier transform along $(\tau ,0)$ is a nonzero constant plus $O(|\tau {|}^{-1})$; the latter bound follows by one-sided integration by parts on the smooth Heaviside amplitude. It is not rapidly decreasing. If no timelike wavefront covector occurred on that compact patch, a finite microlocal partition would imply rapid decay in a cone about the normal rays, a contradiction.