Theorem: Complete contact ambiguity

Every permitted response has the form

Consequently, for every real test $F$ satisfying $F(0)=0$,

Proof. The difference $D=R-R_{\chi }$ is supported at the origin. A finite-order distribution supported at one point is a finite sum of derivatives of the delta distribution. This follows by subtracting a sufficiently high Taylor jet of a test at the point and using the finite-order continuity estimate.

In four dimensions, ${\partial }^{\alpha }{\delta }_0$ has scaling degree $4+\text{abs}\alpha$. The common scaling-degree restriction therefore permits only ${\delta }_0$. Reality makes its coefficient real. Since $\hslash >0$, write that coefficient as $c\hslash$. Evaluation on $F$ gives the last assertion.