Theorem: First-order background identity

For every smooth classical $h$ and every $\xi \in {\mathfrak{g}}_p$,

Equivalently,

is background invariant modulo ${\kappa }^2$.

Proof. The coordinate term in (eq:variation) vanishes by the supplied preflight identity (eq:preflight). The switch operator vanishes because $g=1$ and $\mathrm{d}g=0$ on a neighbourhood of $\mathrm{supp}\xi$. The remaining difference is

whose field class is zero.