One proposed construction, two different maps

A relational scalar functional can be defined geometrically before an interaction switch is introduced. A retarded interacting field can also be defined perturbatively after an interaction is multiplied by a compact test function. These operations do not commute merely because both use retarded inverses.

The distinction is between

and

The first map has a natural coefficient gauge action inherited from diffeomorphisms. The second requires an intertwining statement between that action and the switched dynamics. We prove the first statement through second order and exhibit failure of the second for a specified naive identification.

This obstruction must be resolved before a finite-renormalization separator is interpreted as a gauge-invariant response.

Figure fig:VI-two-maps separates the geometric map from the proposed switched realization and records the required logical order.