The natural switched master residual

On the completion, retain

Use the same kinematical transformation and the natural gauge fermion, but no post-variation source switching.

For compact ghost coefficients, direct variation gives

Here

To verify the formula, use

and ${\partial }_{\mu }{T_0}^{\mu }{}_{\nu }=(P_m\varphi ){\partial }_{\nu }\varphi$. All switch derivatives remain XG,XGST.

Let ${\mathcal{S}}_{\mathrm{sw}}$ be this natural reference action plus its signed antifield lift. The argument of Theorem thm:BV now gives

The result has ghost number one and odd parity.

For example, with

one has $P_m{\varphi }_m=0$ and ${T_0}^{00}=1/2$. For a nonzero compact switch with a plateau, choose $\beta \ge 0$, nonzero and compactly supported where ${\partial }_{t}g>0$, and put $c=\theta \beta {\partial }_t$. The leading coefficient of (eq:swCME) is

Such a transition exists on the completion along a vertical line reaching the plateau. Calling this detector a retained laboratory test additionally requires its support to lie in $\mathcal{L}$, as qualified in X-G-ST.

The full off-shell breaking is not confined to $\mathrm{supp}\mathrm{d}g$, because of the term containing $1-g$. On $U$, all terms in (eq:breaking) vanish.