Proposition: Local extension interface

These component distributions are defined off the origin and have scaling degree at most six in four dimensions. They admit extensions with that bound. Two such extensions differ by

with smaller ambiguity for components with lower scaling degree. No extension is selected here.

Proof. Off the origin, the singular Feynman factors at a null separation have the same covector orientation, so their product meets the wavefront product test. The scalar Feynman singularity has scaling degree two; at most two derivatives occur in the displayed component. The flat short-distance expansion gives the upper bound six. The finite-scaling-degree extension theorem then gives an extension without raising that bound. A difference is supported at one point, and $\mathrm{sd}({\partial }^{\gamma }{\delta }_0)=4+|\gamma |$, which gives the ambiguity bound BF.