Theorem: Free-shell ghost sector of

1. A ghost $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$ with $\square c=0$ is zero. An antighost $\bar{c}\in {C_c}^{\infty }({M_T}^{\circ })\otimes {\Lambda }^{\mathrm{odd}}$ with $\square \bar{c}=0$ is zero.

2. Consequently

   $${\mathfrak{F}}_W\cap {\mathcal{C}}_{\mathrm{sol}}=\{(\gamma ,\varphi ,0,0,b):(\gamma ,\varphi ,b)\in {{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}\},$$

   the bosonic free shell with zero ghost sector.

3. At every configuration in (eq:intersection) the kinematic rule (eq:kin) takes the values

   $$s\gamma =0,s\varphi =0,sc=0,s\bar{c}=ib,sb=0.$$

   Every $\kappa$-dependent term of $s$ vanishes there, and the only nonzero component is the frozen free rule $s_0\bar{c}=ib$. For an antifield-independent $F$, $(s_{\mathrm{ext}}F)(u)=⟨{F^{\prime }}_{\bar{c}}(u),ib⟩$ on the intersection; for $F$ without antighost argument, in particular for $\mathcal{O}$, $(s_{\mathrm{ext}}F)(u)=0$ there.

Proof. (i) $\mathrm{supp}c\subset \overline{W}$, compact in $\{t>\epsilon \}$, so $t_1:={\min }_{\overline{W}}t>\epsilon >0$ and Lemma lem:uniq applies; $\mathrm{supp}\bar{c}$ is compact in the open slab, so $t_1>0$ and the lemma applies again. (ii) A configuration of ${\mathfrak{F}}_W$ lies in ${\mathcal{C}}_{\mathrm{sol}}$ exactly when its bosonic part lies in ${{\mathcal{C}}_{\mathrm{sol}}}^{\mathrm{bos}}$ and its ghosts solve the free ghost equations; by (i) the latter forces $c=\bar{c}=0$. (iii) Each $\kappa$-dependent term of (eq:kin) is linear in $c$, as is $\mathcal{R}c$; the antighost rule is $c$-independent. The last assertion is (eq:sF) evaluated at $c=0$.