 Local orthogonal nuisance reparametrization

For scalar interest parameter $\psi$, choose new nuisance coordinates $\eta$ and keep $\psi$ fixed. The transformed interest score function is $U_{\psi }+U_{\lambda }^{\top }{\partial }_{\psi }\lambda$, so the displayed differential equation makes its Fisher information cross-block with the nuisance score zero. Under smoothness and nonsingularity it has a local solution from initial coordinates on a fixed interest slice. Orthogonality is local expected-information decoupling; it implies first-order asymptotic independence of fitted blocks, not exact independence or nuisance-free conditional inference.