Proposition: Gauge fixing is not plateau invariant

The fixed term (eq:gf) is not invariant under ${\mathcal{G}}_U$. At the linear level its breaking is $-\square D^{\nu }$, as in (eq:egf).

Proof. Take $\xi =f{\partial }_1$, with nonzero $f\in {C_c}^{\infty }(U)$. Directly,

A nonzero compact interior $f$ cannot satisfy $\square f=0$, by Cauchy uniqueness with zero past data. Starting from $\gamma =0$, a genuine pure-gauge metric has ${\gamma }^{\prime }=K\xi +O(\kappa )$, so the nonzero displayed term is the leading gauge-fixing action. The Einstein action on that flat pullback is unchanged. Hence the discrepancy is gauge fixing, not a switching defect.