Independence of auxiliary cutoffs

In the smooth-current construction, two temporal cutoffs ${\chi }_1,{\chi }_2$ differ only in a bounded slab. For a spacelike-compact $q$,

is smooth and compact, and the representative difference is $2{\left(L_{\mathrm{E}}r\right)}^{♯♯}$. It pairs to zero with every vacuum solution by formal self-adjointness. Thus auxiliary cutoff independence is on shell.

By contrast, the physical interaction switch $g$ changes the retarded observable. It cannot be discarded using this auxiliary-cutoff argument. Equation (eq:remainder) is the explicit comparison required for that change.

The auxiliary cutoff in Figure fig:smoothing-classification changes the current only by an on-shell annihilator. It is distinct from the physical interaction switch in Figure fig:switched-remainder.