Corollary: Exact sharp insertion is not bulk

For every nonzero real $f$,

The map $f\mapsto B_f$ is injective.

Proof. At $(0,{\varphi }_f)$, the metric derivative is $-q_a$. Lemmas lem:witness and lem:boundary give a nonzero timelike wavefront component, forbidden for a bulk first derivative. It is also a nonzero derivative, so $B_f\ne 0$.

For complex $f$, reality of $L$ and separation into real and imaginary parts show that $B_f=0$ forces both parts to vanish.