= Solution
The map $h\mapsto Ph$ is a <bounded linear functional> on <L2 space>, since $|Ph|\leq\|h\|_2$ by the <Cauchy-Schwarz inequality>. Its kernel, the <mean-zero L2 space>, is therefore closed. Every nuisance <score function> and the parametric <score function> $\dot\ell$ are centered, so both $\Pi_{\mathcal N}\dot\ell$ and $\widetilde\ell$ are centered. Hence
$$
\boxed{P\widetilde\ell=0.}
$$
Moreover, the <efficient score> belongs to the <orthogonal complement> of the <nuisance tangent space>. Writing $\dot\ell=\widetilde\ell+\Pi_{\mathcal N}\dot\ell$ gives
$$
P(\dot\ell\widetilde\ell)
=P\widetilde\ell^2+P\bigl((\Pi_{\mathcal N}\dot\ell)\widetilde\ell\bigr)
=\boxed{\widetilde I}.
$$
This is the <efficient-score projection identity>; it remains valid when the <efficient information> is zero.
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