Different right inverses need not give the same observable

An actual section also requires $C(h)$ to belong to the declared gauge group and its transformed configuration to remain admissible. The algebraic condition $CK=\mathrm{id}$ alone does not establish either fact.

Two right inverses with different boundary, initial-data, or support prescriptions need not define the same observable even when they invert the same formal differential operator.

On a common domain, $-w_{\mathrm{P}}$ has the right-inverse sign of $w_{\mathrm{R}}$, but may have tails or past behavior incompatible with ${\mathcal{G}}_{\mathrm{R}}$. This does not identify the two sections.

A coupled Poisson coefficient would be

The unchanged-product expression tested earlier is different because it does not carry these complete coefficient data.