Efficient-score projection identity ($P(\dot{\ell }\tilde{\ell })=P{\tilde{\ell }}^2$)

The decomposition $\dot{\ell }={\Pi }_{\mathcal{N}}\dot{\ell }+\tilde{\ell }$ is orthogonal. Taking the inner product with the efficient score gives $P(\dot{\ell }\tilde{\ell })=\tilde{I}$. Centering of the efficient score follows because the nuisance tangent space and the parametric score function lie in the closed mean-zero L2 space.