Proposition: Off-collision construction on the admissible set

On ${\mathcal{U}}_{\Gamma }$, the specified graph product exists. Differentiation and multiplication by its smooth weights preserve existence. On compactly supported portions its pushforward is proper. The grouped difference agrees there with (eq:formal-two-line).

Proof. This is the distribution product criterion applied to the complete graph variables. The derivative kernels have the prescribed sum-zero bounds, and the line kernels are balanced. Properness is the graphwise support statement inherited from XIII-B. Restriction to an open set does not enlarge the original compact support control XIIIB,Hormander.