Lemma: Support and tree-order properties

Suppose the parameters in Definition def:contacttemplate are chosen so that the seed in (eq:contacttemplate) is supported in the existing compact graph closure ${\mathcal{K}}_{\mathrm{gr}}$. Then

A well-defined pushforward has support in the image of that set. If this value is used as a contact change at order ${\hslash }^2$ in the time-ordered product, it does not alter the selected tree coefficient.

Proof. The tensor factor ${\partial }^{\alpha }\delta (r_I)$ is supported on $r_I=0$. Multiplication by a smooth function and application of a finite differential operator do not enlarge distributional support. The support hypothesis supplies the compact graph closure. A pushforward, when defined, has support contained in the image of the original support.

For the last assertion, a change $\delta {\mathsf{T}}_2={\hslash }^{2}C$ changes the retarded expression $(i/\hslash )({\mathsf{T}}_2-{\mathcal{V}}_g{\star }_H\mathcal{O})$ by $i\hslash C$. It has no coefficient at ${\hslash }^0$. The inherited tree datum is consequently unchanged.