Lemma: Compact balanced-tree composition

Let a finite tree have internal and external vertices, one balanced line on each edge, and smooth vertex weights. Suppose the joint support of the weighted lines is compact. Then their product is defined, its pushforward over internal vertices is defined, and an $n$-slot output kernel has wavefront contained in $Z_n$.

Differentiation and multiplication by fixed smooth coefficients preserve this conclusion.

Proof. Pull each line back to the full vertex configuration space. Its covector is $p$ at one endpoint, $-p$ at the other, and zero elsewhere.

A forbidden product cancellation would assign line covectors whose signed sum vanishes at every vertex. In a forest a leaf forces the covector on its only incident edge to be zero. Removing that edge and repeating forces every line covector to be zero. This contradicts the nonzero covector required for a forbidden cancellation. Adding lines successively therefore defines the product.

The same leaf recursion bounds every line covector by a fixed multiple of the sum of the vertex covector norms. Thus the covector-sum upper bound is closed; no circulation with unbounded internal covectors is hidden in it.

Compact support makes projection over the internal variables proper. A pushforward wavefront point has zero covectors at all internal vertices. Summing the remaining vertex covectors cancels every line pair, so the external sum is zero. The distributional product, pullback, and pushforward criteria are the standard ones Hormander,BDH.