Lemma: Admission of the displayed retarded forests

Consider a finite forest of vertex positions with:

1. a retarded Klein–Gordon kernel or its derivatives on each edge;

2. smooth multipliers at vertices;

3. a compact retained integration region;

4. external field copies attached to specified vertices, allowing several copies at one vertex.

The distribution obtained by integrating internal vertices and inserting the external field copies exists. Its external wavefront is contained in $Z_n$. It therefore avoids ${\overline{V}}_{+}^n\cup {\overline{V}}_{-}^n$.

This applies to the finitely many bare-coordinate and scalar-response graphs in (eq:completed), at fixed width.

Proof. Choose a root in each component. Root positions and oriented edge differences are independent linear coordinates for a forest. In those coordinates the unweighted edge distribution is a tensor product of the existing retarded distributions, with constant distributions in the root positions. It therefore exists without multiplying dependent singular factors.

Its wavefront annihilates simultaneous translation of all vertices of each component. Equivalently, the sum of its vertex covectors is zero. Differentiation and smooth multiplication preserve this bound Hormander.

Place every external copy at its assigned vertex. The resulting map from vertex positions to the list of external positions is smooth. Pushforward is defined after the compact localization. Its wavefront formula, including its possible normal covectors, still imposes zero total external covector: the transpose differential assigns each external covector to its vertex, and preserves their total sum. Internal integrations set the internal covectors to zero. This proves the $Z_n$ bound.

A collection of co-oriented causal covectors can have zero sum only when all are zero, which has already been excluded.

For the application, expand (eq:coordinates) and (eq:completed) by the finite Leibniz rule. The graphs are the single-coordinate line, the two coordinate branches, the nested second-coordinate line, the mixed coordinate–scalar branches, the local quadratic metric insertion, and the iterated scalar line. They are forests, not closed graphs. Finite propagation, the initial cutoff $\sigma$, the compact switch, and the final observation test place every retained vertex in a compact subset of $D_f$. A smooth compact localization equal to one near that retained set justifies the pushforwards.