= Local orthogonal nuisance reparametrization
{title2=$\partial_\psi\lambda(\psi,\eta)=-I_{\lambda\lambda}^{-1}I_{\lambda\psi}$}
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.
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