Proposition: Exact retained infinitesimal defects

For every smooth $(h,\varphi )$ and $\xi \in {\mathfrak{g}}_U$,

In particular, first-order covariance would require $a=\xi$ in the coordinate identity. That requirement is not satisfied.

Proof. Lemma lem:preflight gives

Linearity of $W{T}_{g,\epsilon }$ gives

and polarization gives

Hence

At first order,

At second order,

Direct differentiation of (eq:Jw) gives

Therefore the second-order integrand is

Using (eq:Rtwo) yields (eq:infdefect).