Introduction: three constructions that must remain distinct

The present problem has three different maps:

The first can descend even when the second fails to intertwine the proposed representative identification. The third can record that failure without repairing it.

Manuscript VI identifies this distinction for a second-order Einstein–scalar expansion CheungVI. We reprove the required classical statements on a supported local domain. No proof of a companion manuscript is an unprinted premise.

Relational perturbation theory and higher-order invariant observables have substantial antecedents BFHPR,Frob. In particular, neither relational observables nor gauge cancellations in previously constructed quantum correlation functions are claimed as new here. The question addressed here is compatibility of one fixed finite-time prescription with one fixed compact switch, followed by a precisely limited finite operational interpretation.