Theorem: Complete first-order ambiguity

The class in Definition def:prescriptions is nonempty. For every fixed reference scale ${\mu }_0>0$, all its members are exactly

Every real $c_{\mathsf{P}}$ is allowed. Consequently,

For the joint kernel, the difference is

Proof. Lemma lem:reference supplies a member. The difference between any member and the reference is supported at the origin. A point-supported distribution is a finite sum of derivatives of ${\delta }_0$. The scaling degree of ${\partial }^{\alpha }{\delta }_0$ in dimension four is $4+|\alpha |$. The order bound therefore permits only $c{\delta }_0$. Reality makes $c$ real.

Conversely, adding a real multiple of $\hslash {\delta }_0$ preserves every condition in Definition def:prescriptions. Thus the catalogue is exhaustive, not merely an upper bound. Evaluation of ${\delta }_0$ proves the annihilator formula, and the joint statement follows from the difference-map pullback.