Introduction: the equation space precedes descent

A proposed relational expression must define a function on the equivalence classes actually declared. This question precedes any subsequent representation or measurement construction:

Only the first object and its classical descent are studied here.

Two failures must be distinguished. An expression can assign different values to equivalent representatives. Alternatively, its proposed transformation can fail to preserve the equation space, so that the intended quotient action is not defined. An off-shell cancellation excludes neither failure.

Relational and metric-dependent coordinate constructions have a substantial literature; see, for example, Brunetti, Fredenhagen, and Rejzner BFR. Explicit gravitational dressings and their nonlocal kernels are developed by Donnelly and Giddings DG. Our question is narrower: for the displayed first-order functionals, which equations, identifications, and support prescriptions make descent valid?