Lemma: Bulk derivatives

Every smooth directional derivative of a member of ${\mathcal{P}}_{\mathrm{mc}}$ is again in ${\mathcal{P}}_{\mathrm{mc}}$. At every smooth configuration, its first metric derivative is a compact tensor distribution with no nonzero nonspacelike wavefront covector.

Proof. Directional differentiation contracts kernel slots with smooth directions. Evaluation at a smooth configuration contracts further slots with smooth fields.

The wavefront bound for these contractions allows zero covectors in all contracted slots. A forbidden nonspacelike covector in the remaining first-derivative slot would therefore produce a forbidden tuple in (eq:MC). The same argument applies to every remaining number of slots.