Proposition: Linearized gauge identity

For a sufficiently differentiable covector field $\xi$, define

Then

At jet level the same identity holds for the induced second metric jet obtained from a third jet of $\xi$.

Proof. Substituting $h=\mathsf{K}\xi$ into the first-order connection gives

The first-order Riemann tensor is zero by commutation of third derivatives. Its Ricci and Einstein contractions are therefore zero. The calculation depends only on the relevant third derivative slots of $\xi$, so it also proves the jet assertion.