= Score factorization implies parameter orthogonality
{title2=$\ell_\psi=\lambda t_\psi\quad\Longrightarrow\quad I_{\psi\lambda}=0$}
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 $\lambda\ne0$. 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.
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