Lemma: Two-edge multiplication and single-vertex smoothing

Pull back two endpoint kernels satisfying Lemma lem:endpoints to the three-vertex variables of a two-edge tree. Their distributional product exists.

After multiplication by a compact smooth joint coefficient, pushforward integrating all vertices except any one vertex is smooth.

Proof. Each of the two edges has a leaf belonging to no other edge. A nonzero covector from that edge has a nonzero component at that leaf, whereas the other edge has zero component there. Thus the two pulled-back wavefront covectors cannot sum to zero. The distributional product criterion applies.

For the second assertion, the pushforward wavefront bound requires zero covectors in all integrated slots. Remove an integrated leaf different from the retained vertex. Its covector forces its incident edge to be inactive, meaning that the edge contributes the zero covector in the augmented wavefront bound. Repeat on the remaining edge. Both edges must be inactive, so no nonzero covector remains at the retained vertex.

If the retained vertex is the central vertex, remove the two integrated leaves immediately. If it is a leaf, remove the other leaf first and then the integrated central vertex. The endpoint norm comparison justifies the same reasoning for closed conic limits. Compact multiplication makes the pushforward proper. Its wavefront set is empty, so it is smooth.