Theorem: Balanced tree composition

The edge product is defined by the ordinary wavefront product criterion. Its internal-vertex pushforward is compactly supported. Its external wavefront satisfies

Thus it excludes configurations in which all external covectors are nonzero-compatible future causal covectors or all are past causal covectors. For $n$ external field slots this gives a microcausal kernel.

Proof. A zero sum of all incident edge covectors at every vertex would, at a leaf, force the unique edge covector to vanish. Removing that edge and repeating proves that every edge covector would vanish, which is excluded in the product criterion. Zero covectors may occur in individual factors; the same induction applies to any proposed nontrivial cancellation.

Compact graph support makes the internal projection proper. In the pushforward wavefront condition all internal covectors are zero. Summing balanced edge covectors leaves the stated external sum. A sum of causal covectors with one common time orientation cannot vanish unless every covector vanishes. Differential operators and smooth weights preserve the bound Hormander.