Theorem: Raw admissibility on

For fixed $f,\varphi$, the following are equivalent:

When they hold, a representative is

Proof. Representability forces annihilation of every compact pure gauge $K\xi \in \mathcal{S}$, hence $\mathbf{H}=0$ by Proposition prop:threegauges. The decomposition (eq:H) gives the final equivalence. If $\mathbf{H}=0$, the compact current $J_{\mathrm{raw}}$ is conserved; Lemma lem:smoothprop and Theorem thm:smoothing supply the representative.