Efficient-score projection identity
= Efficient-score projection identity
{title2=$P(\dot\ell\widetilde\ell)=P\widetilde\ell^2$}
The decomposition $\dot\ell=\Pi_{\mathcal N}\dot\ell+\widetilde\ell$ is orthogonal. Taking the <inner product> with the <efficient score> gives $P(\dot\ell\widetilde\ell)=\widetilde I$. Centering of the <efficient score> follows because the <nuisance tangent space> and the parametric <score function> lie in the closed <mean-zero L2 space>.