Abstract

We examine the compatibility of a finite-time harmonic relational prescription with compactly switched Einstein–scalar perturbation theory. For a specified second-order Einstein–scalar solution-jet space, we construct harmonic coordinate coefficients with fixed Cauchy data and prove that the resulting scalar functional descends under formal diffeomorphisms that are trivial near the initial surface. The metric is an unknown of the coefficient equations; an explicit sourced family establishes nonvacuity.

The same geometric dressing does not automatically descend after the scalar correction is replaced by the retarded response of a compactly switched vertex. We derive the exact first-order switch defect and give two representatives with a strictly nonzero smeared gap, even though the switch equals one near the observation smearing. No correction beginning at second order can remove this first-order defect while that representative identification and lower-order prescription are retained.

We distinguish the classical second-order coefficient from the four-term expansion of a renormalized interacting composite. We also state and prove the finite-dimensional certificate criterion, including its catalogue-completeness and finite-sampling requirements. No renormalized second-order relational product, complete finite-renormalization response space, or radiative response matrix is constructed here. The quantum certification problem is therefore -open. The closed result is the classical descent theorem together with the compact-switch obstruction, not a quantum renormalization separator.