Plateau invariance and the causal qualification

For a ghost test supported inside $P_g$, the two terms in (eq:sSfirst) vanish. The restricted symmetry therefore survives.

Under a simultaneous background pullback supported in $U$, the metric, fluctuation, scalar, ghost, antighost, and auxiliary fields transform in their tensor types. The fermion and ghost density are natural, $g\circ F^{-1}=g$, and the flat scalar term is unchanged where its coefficient $1-g$ is nonzero. Thus the integrated reference action is exactly invariant under this simultaneous pullback.

This background covariance is different from quantum-gauge invariance with $\bar{q}$ held fixed.

If an observation point belongs to $P_g$, its vertical past curve crosses a region where $g$ changes from zero. Hence $\mathrm{supp}\mathrm{d}g$ meets its causal past. Without an observation-plateau or comparable causal hypothesis, compactness alone does not prove the asserted intersection.

Even a nonempty causal intersection does not prove a nonzero ghost contraction with a particular observable. Propagator support, index contractions, and smearings still have to be evaluated.