Remark: Tests versus configurations

A compact real vector field $v\in {\mathfrak{g}}_W$ is a legal smearing test for the ghost field of ${\mathcal{E}}_H$: $c(v)=\int v_{\mu }{c}^{\mu }$ is an element of the carrier, with $s_{0}c(v)=0$. That element is a functional, not a configuration, and $v$ is not thereby promoted to a free ghost solution. Likewise a compact ghost that is not a free solution is a legal element of ${\mathfrak{F}}_W$ and a legal argument of the breaking functional; it is not an element of ${\mathcal{C}}_{\mathrm{sol}}$.