Corollary: Smooth spectator rule

Suppose a selected graph coefficient has the form $a[h,\varphi ]D{T}_{\epsilon }^{\rho }$ after fixing smooth configurations, where $a[h,\varphi ]$ is smooth on $X$. If these multipliers depend continuously on the configurations in the compact smooth seminorms required by Theorem thm:descendants, its numerical finite part is

Convergence is uniform on configuration sets for which those multiplier seminorms are bounded.

Proof. Use Theorem thm:descendants with $a=a[h,\varphi ]$ and its uniform coefficient bounds.