Proposition: Exact basic admissions

The domain contains ${\mathcal{P}}_{\mathrm{mc}}$, all $J(a)$, and all $B_f$. For $\eta =(k,\sigma )$,

Third derivatives vanish. The mixed and linear kernels are compact distributions.

Proof. Bulk admission is Lemma lem:bulk. The first metric derivative of $J(a)$ is $q_a$; its higher derivatives vanish. Differentiate (eq:B) and integrate the scalar variation by parts to obtain (eq:descendants). For a fixed smooth $k$, $Lk$ is smooth on $V$, so the first term in $D_{\eta }{B}_f$ is a regular scalar linear functional. Every source in the remaining metric derivatives is smooth and compactly supported in $V$.

For compact distribution kernels, the bilinear expression in (eq:B) is continuous in smooth fields restricted to a fixed compact neighbourhood of $K_{\ast }$. The kernel theorem gives its compact mixed kernel. Alternatively, multiply the retarded kernel by $\Theta (z^0)$ and transfer the finite $C$ and scalar derivatives to it. The product exists because a nonzero wavefront pair of the wave kernel has nonzero observation-slot covector, whereas a wavefront covector of the step factor has zero observation-slot component. The operations are therefore legitimate BDHIntro.