Corollary: Strict admission and injectivity

For every nonzero real $f$,

The linear map $f\mapsto B_f$ is injective, also after complexification.

Proof. At ${\Psi }_f=(0,{\varphi }_f)$, the metric derivative is $-q_a$. Lemmas lem:positive and lem:noncancellation give a nonzero timelike wavefront component, forbidden for a bulk linear metric descendant. The derivative is nonzero, so $B_f\ne 0$. For a complex source, separate real and imaginary parts on real configurations.