Lemma: Admission of the finite scalar-completion forests

At fixed width the kernels of

and the scalar-slot preimages obtained by polarizing (eq:scalaridentity) are admitted compact polynomial kernels. Their metric variables can range over $D_f$; their scalar variable lies in $C$.

Proof. Expand (eq:coordinates) and (eq:completed) by the finite Leibniz rule. The propagator graphs are the single coordinate line, the two coordinate branches, the nested coordinate response, the mixed coordinate–scalar branches, a local quadratic metric insertion, and the iterated scalar line. Each is a forest.

Root positions and oriented edge differences are independent coordinates for a forest. Its unweighted propagator distribution is a tensor product of existing retarded kernels in those differences. It is constant in simultaneous root translations, so its wavefront has zero total vertex covector. Smooth multipliers and derivatives preserve that bound. Attaching several external field copies to one vertex is a diagonal pushforward; its additional normal covectors also have zero total sum. Integrating internal vertices sets their covectors to zero. Thus each external wavefront lies in $Z_n$.

All contributing vertices are in the retained finite-time causal past $D_f$. Smooth compact localizations equal to one near that compact region make the pushforwards proper. They do not change the retained kernels. The same argument applies with an extra advanced transposition and compact generator coefficients to the preimages in (eq:scalaridentity). Standard tensor-product and pushforward wavefront rules justify these operations Hormander.