Lemma: Two-edge leaf criterion and one-vertex smoothing

Pull back two such endpoint kernels to a three-vertex two-edge tree. Their product exists. After multiplication by a compact smooth joint coefficient, integration over all vertices except any one gives a smooth function at the remaining vertex.

Proof. Each edge has a leaf not belonging to the other edge. At that slot, only its own nonzero edge covector occurs. The two wavefront covectors therefore cannot sum to zero, which verifies the distributional product criterion.

For pushforward to one vertex, every integrated slot must have zero covector. Remove an integrated leaf different from the retained vertex. Its incident edge must contribute zero; endpoint norm equality forces its other endpoint contribution to vanish as well. Repeat for the remaining edge. No nonzero covector survives at the retained vertex. The same norm comparison excludes one-slot directions in closed conic limits. Compact multiplication makes the pushforward proper, so its wavefront set is empty.