Score factorization implies parameter orthogonality (${\ell }_{\psi }=\lambda t_{\psi }⟹I_{\psi \lambda }=0$)

In a regular common-support statistical model with probability density function $a(\lambda ,y)e^{\lambda t(y;\psi )}$, the score function in $\psi$ is $\lambda t_{\psi }$. Its zero expectation gives $\mathbb{E}{t}_{\psi }=0$ for $\lambda \ne 0$. The mixed log-likelihood derivative is $t_{\psi }$, so the cross Fisher information matrix block vanishes. At zero the same conclusion follows by continuity when the regular derivatives extend there. This proves parameter orthogonality; it does not establish exact independence of fitted parameters.