Gauge fermion and free action

All action identities are bulk variational identities, modulo divergences and tested with compact interior variations. No BRST boundary charge or boundary variational principle is asserted.

The Fierz–Pauli part is invariant under its linear gauge map, and the scalar is unchanged by $s_0$. Consequently $s_0{S}_0=0$ as a free bulk variational functional, since ${s_0}^2\Psi =0$. This does not contradict the XI-A breaking: the bosonic gauge-fixing term alone is not gauge invariant. The independent auxiliary and ghost terms produce the free BRST-invariant completion.