Proposition: The relative complex meets the free ghost shell only at zero

A ghost configuration $c\in {\mathfrak{g}}_W\otimes \Lambda$ with $\square c=0$ is zero. Consequently the restriction of the relative complex $({\mathfrak{F}}_W,s_{\mathrm{ext}})$ to XII-A's complete shell ${\mathcal{C}}_{\mathrm{sol}}$ has trivial ghost sector. The relative master equation induces no action on ${\mathcal{E}}_H$ beyond the inherited free differential $s_0$, and $s_0$ on ${\mathcal{E}}_H$ acts on unrestricted free ghosts, which are not elements of ${\mathfrak{g}}_W$.

Proof. A compactly supported solution of the homogeneous wave equation with support in $W\subset \{t>\epsilon \}$ has zero Cauchy data on a surface $\{t=t_0\}$, $\epsilon <t_0<{\min }_{\overline{W}}t$, and vanishes by uniqueness, coefficientwise XIB,BGP. The remaining assertions restate XII-A's construction: its carrier ghosts are free fields with arbitrary Cauchy data.