Proposition: Graphwise properness

For each supplied rooted graph of $\mathcal{O}$, attach the compactly switched vertex and either one or two cross-lines. On the closed support set determined by the existing graph weights and retarded internal lines, projection over all internal vertices is proper. A support-preserving extension does not spoil this property.

Proof. XII-D and XII-E provide compact support sets for their rooted graphs. Internal retarded vertices are bounded in time below by the ramp or compact switch, above by the compact observation, and spatially by the corresponding causal regions.

The new local interaction vertex lies in $\mathrm{supp}g$, a compact set. Adding cross-lines restricts the product support inside the product of these existing compact graph sets with $\mathrm{supp}g$. Its closure is compact. Projection from this set is proper. An extension supported in the closure of the same graph support, including allowed total-diagonal terms, preserves compactness.