The weakest sufficient physical certificate

A full homogeneous germ decomposition is unnecessary if one can directly prove

with evaluated complete coefficients and

For the inherited family, neither the coefficients nor this remainder have been calculated.

If both routes establish (eq:desired-certificate), their complete sums agree by (eq:translation-bridge). Individual fibers need not correspond to individual sectors. After scalar absolute continuity and an anchor are used to integrate the certificate, uniqueness of finite real power–logarithm expansions identifies the singular coefficients and finite part; see Appendix app:decisions.