Corollary: No equivalent compact current when the criterion fails

If ${\mathsf{E}}_1\mathbf{H}\ne 0$, none of the three functionals in Theorem thm:harmonic has a smooth compact conserved representative on its stated domain. This excludes all representatives modified by on-shell annihilators, including field-equation terms.

Proof. By Proposition prop:threegauges, a residual wave $\xi$ has $B_{\mathrm{raw}}(K\xi )\ne 0$. An annihilator cannot change that value on a solution. Every smooth compact conserved tensor gives zero by (eq:gaugeannihilation).