Compact localization of the regularized graphs

Gaussian smoothing creates tails, so finite-$\tau$ support is not the original retarded support. This must be handled explicitly.

Let $K_{\Gamma }$ be the copied compact closed support set determined by the graph weights and inherited internal retarded lines. Choose

It is an auxiliary localization, not a contact normalization.

Let $z_1,z_2$ be the two residual field variables, attached to vertices $v_{a_1},v_{a_2}$. Before pushforward define

The fixed operator $D_{\Gamma }$ includes the prescribed residual field derivatives. The attachments are separately labelled even when they occur at the same vertex.

With the inherited Wick coefficients $c_{\Gamma }$, define the regularized time-ordered coefficient kernel

No new combinatorial multiplier is introduced.