Theorem: The breaking functional, with every term displayed

On the field space $\mathfrak{F}$ of smooth multiplets with ghosts $c,\bar{c}$ compactly supported in ${M_T}^{\circ }$,

exactly as a formal series in $\kappa$, where

with ${T_0}^{\mu }{}_{\nu }={T_0}^{\mu \rho }{\eta }_{\rho \nu }$ and $C_2,M_2,J_3$ XII-C's finite coefficients. Equivalently ${\mathcal{N}}_{\nu }(e_{\mathrm{D}})=\kappa [E_{\varphi }{\partial }_{\nu }\varphi -{\mathcal{K}}_{\nu }]$.

Proof. By Lemma lem:decomp, $(S_{\mathrm{ext}},S_{\mathrm{ext}})=2s{S}_{\mathrm{D}}$. Write $\xi =c/2$, an odd compact vector field. Then $s\gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$ and $s\varphi =\kappa {\xi }^{\mu }{\partial }_{\mu }\varphi$, and $s{S}_{\mathrm{D}}=\int e_{\mathrm{D}}\cdot (K\xi +\kappa {\mathcal{L}}_{\xi }\gamma )+\kappa \int E_{\varphi }\xi \cdot \partial \varphi$ is the first variation of $S_{\mathrm{D}}$ along XII-C's scaled variation, evaluated coefficientwise in the Grassmann algebra. XII-C's proof of (eq:noether-input) computes this variation in two ways for an arbitrary compact interior parameter. Compact integration by parts gives $s{S}_{\mathrm{D}}=\int {\xi }^{\nu }[-{\mathcal{N}}_{\nu }(e_{\mathrm{D}})+\kappa E_{\varphi }{\partial }_{\nu }\varphi ]$. Direct variation gives: zero from the natural Einstein–Hilbert density; $\kappa \int {\xi }^{\nu }{\mathcal{B}}_{g,\nu }$ from the switched scalar action; and $-\kappa \int {\xi }^{\nu }[N_{0,\nu }({Z_{\sigma }}_{\epsilon })+\kappa Q_{\nu }({Z_{\sigma }}_{\epsilon })]$ from the added cubic functional, where XII-C's Lemma on the cubic divergence gives $N_{0,\nu }({Z_{\sigma }}_{\epsilon })=-w{Q}_{\nu }(A\gamma )+{\mathfrak{R}}_{{\sigma }_{\epsilon },\nu }$ with the apparent third ramp derivative cancelled. Hence $s{S}_{\mathrm{D}}=\kappa \int {\xi }^{\nu }{\mathcal{K}}_{\nu }$ with $\mathcal{K}$ as in (eq:Kden), and comparison of the two forms is (eq:noether-input). Substituting $\xi =c/2$ and doubling gives (eq:breaking). All steps are identities of local densities modulo total derivatives, integrated against the compact ghost.