The geometric second-order term is necessary

The expression

is not the second-order relational coefficient. It lacks both the second-order inverse-coordinate displacement and the quadratic Taylor term.

Likewise, replacing the inverse map by $y-\kappa w-{\kappa }^{2}v$ is incorrect. The term ${\kappa }^2{w}^{\nu }{\partial }_{\nu }{w}^{\mu }$ in (eq:Xinverse) is forced by formal inversion.