The relative differential of the observable is the pairing

Proposition prop:relO is a statement about the off-shell ghost $c\in {\mathfrak{g}}_W$ evaluated on a bosonic shell configuration. It is not an evaluation in ${\mathcal{E}}_H$: by Proposition prop:carrier, a nonzero relative ghost is not a free-shell configuration, and XIII-A's display of $s_0\mathcal{O}$ for unrestricted free ghosts retains the coordinate ramp and scalar switching terms, which vanish identically for $c\in {\mathfrak{g}}_W$. The two displays are different evaluations of the same functional $⟨{{\mathcal{O}}^{\prime }}_{\gamma },\mathcal{R}c⟩$.