Corollary: No interacting ghost differential on

No nonzero relative ghost is a configuration of XII-A's shell. The restriction of the relative complex to ${\mathfrak{F}}_W\cap {\mathcal{C}}_{\mathrm{sol}}$ carries the inherited free rule $s_0$ evaluated at $c=0$ and nothing else. Hence $({\mathfrak{A}}_W,s_{\mathrm{ext}})$ supplies no differential on ${\mathcal{E}}_H$ beyond the frozen $s_0$, and in particular no interacting ghost differential ${\hat{s}}_{\mathrm{int}}$.

Proof. Theorem thm:ghost(i)–(iii). ${\mathcal{E}}_H$'s ghosts are free fields with arbitrary Cauchy data (Input statement input:XIIA); by (i) none of them, except zero, lies in ${\mathfrak{g}}_W\otimes \Lambda$.