Proof: Verification of eq:Xinverse

Proof. Substitution into ${{\mathsf{X}}_{\mathbf{g}}}^{\mu }(x)=x^{\mu }+\kappa w^{\mu }(x)+{\kappa }^2{v}^{\mu }(x)$ cancels the first-order terms. At second order, Taylor expansion contributes $-w^{\nu }{\partial }_{\nu }{w}^{\mu }$, which is cancelled by the displayed inverse coefficient.