Input statement: XIV-A: the breaking functional and the two spaces

On the frozen sheet (XII-C's cubic-only $S_{\mathrm{D}}$, inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ with coefficient $-1$, inert $g$ and ${\sigma }_{\epsilon }$), with the kinematic rule

($\mathcal{R}=K/2$, $c=2\xi$ recovers $K\xi$), and the extended action

one has, on the space $\mathfrak{F}$ of smooth multiplets with ghosts compactly supported in ${M_T}^{\circ }$,

with $w={\sigma }_{\epsilon }-1$ and the displayed XII-C densities. Every term of ${\mathcal{K}}_{\nu }$ carries $1-g$, $\partial g$, $w$, $\partial {\sigma }_{\epsilon }$, or ${\partial }^2{\sigma }_{\epsilon }$; hence ${\mathcal{K}}_{\nu }=0$ on $P_g\cap \{t>\epsilon \}$. On the unrestricted space the late-switch value is nonzero:

and every extension in the kinematic class ${\mathfrak{S}}_{\mathrm{kin}}$ has this antifield-degree-zero component; no $\hslash$ correction removes it. On the relative space ${\mathfrak{F}}_W$ (ghost $c\in {\mathfrak{g}}_W\otimes {\Lambda }^{\mathrm{odd}}$),

and XI-B's fermion $\Psi =-i\int {\bar{c}}_{\mu }(2D^{\mu }\gamma -b^{\mu })$ preserves both statements: $S_{\Psi }[\Phi ,{\Phi }^{\ast }]:=S_{\mathrm{ext}}[\Phi ,{\Phi }^{\ast }+\delta \Psi /\delta \Phi ]$ has $(S_{\Psi },S_{\Psi })=\mathfrak{K}$ on $\mathfrak{F}$, $0$ on ${\mathfrak{F}}_W$, and

The relative complex meets the free ghost shell only at zero: ${\mathfrak{g}}_W\cap \{\square c=0\}=\{0\}$ XIVA.