What this comparison does not prove

A classical observable $B_1$ on ${\mathrm{Sol}}_1$ can be transported to ${\mathrm{Sol}}_2$ by

However, this is not generally the assertion that the same evaluation formula defines the transported observable. For example, if

then

which need not equal $B_2(\psi )$.

This distinction is the classical counterpart of the need to specify an identification between cutoff-dependent descriptions rather than assert equality in an unchanged realization.

In perturbative algebraic quantum field theory, algebraic cutoff comparison is distinct from a direct numerical or operator removal of the switch BFR (Section 4). The present classical map does not establish the corresponding quantum isomorphism for the nonlocal harmonic insertion. Such a result would require the product domain and localization hypotheses to be proved.