= Solution
Fix $P=P_{\theta,\eta}$. Let $\mathcal N$ be the <nuisance tangent space>: the closed linear span in $L^2(P)$ of <score functions> of <statistical paths> that vary only the <nuisance parameter> $\eta$. By the preceding argument, $\mathcal N$ is contained in the <mean-zero L2 space> $L^2_0(P)$.
Let $\Pi_{\mathcal N}$ denote <orthogonal projection> onto this <closed subspace of a Hilbert space>. The <efficient score> and scalar <efficient information> are
$$
\boxed{\widetilde\ell_{\theta,\eta}=\dot\ell_{\theta,\eta}-\Pi_{\mathcal N}\dot\ell_{\theta,\eta},
\qquad
\widetilde I_{\theta,\eta}=P_{\theta,\eta}\widetilde\ell_{\theta,\eta}^{\,2}.}
$$
The <efficient score> is the component of the parametric <score function> that cannot be reproduced by changing the <nuisance parameter>. The <efficient information> is its squared <L2 norm>; it can be zero, so positivity must not be assumed in the definition.
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