The task and its limits

The analytical source is Cheung's X-C, version 3 XC. The supplied X-A-ST and X-B-ST manuscripts fix the supported classical prescription and its first scalar response XAST,XBST. The first response is recalled as an inherited input, not reconstructed as a new achievement of this file.

Quadratic scalar perturbations admit particularly direct linear-field constructions. Their relation to perturbative agreement has a broader established literature HW,DHP. That context does not supply the particular regulated coordinate prescription or the supported host contract used here. Nor is a general renormalized product family imported from that literature.

All coupling expansions below are finite formal expansions over

with complexification for tests and field coefficients. An equality modulo ${\kappa }^3$ means equality of the three displayed coefficient slots. It is not a numerical error estimate at a nonzero real coupling.