No quantum products have been used

All products in the proofs above are products of smooth classical functions. A Green operator is applied to a smooth source; nested applications are compositions of well-defined Cauchy solution maps. This does not establish products of the associated distributional quantum kernels.

In particular, no microcausal functional domain, time-ordered extension, or renormalized retarded product for the harmonic insertion has been constructed.

The distinction matters because the coordinate prescription is part of the observable. Fröb's one-loop calculation of a relational scalar two-point function includes cancellations between ordinary self-energy contributions and coordinate corrections Frob. That calculation specializes the coordinate Green function to a Feynman prescription Frob (Sections 2–3). Its cancellation cannot be imported as a result for the present zero-Cauchy-data prescription.