Corollary: Exact scope of the local relational interpretation

On the common-flow family of Theorem thm:localmatching, the encoded phases both acquire $t/{\sigma }_{\mathrm{p}}$. The complete declared phase instrument is therefore invariant under that comparison.

It is not whole-${\mathfrak{g}}_U$ invariant on the two-coefficient family.

Proof. The common-flow assertion follows from (eq:commonflow) and (eq:phasecovariance). Under ${\xi }_2$, only $b_2$ changes, so the relative phase and, generically, its cosine change.