= Mean-preserving error tangent space
{title2=$\{\gamma\in L^2(f):E_f\gamma=E_f(\varepsilon\gamma)=0\}$}
For independent-error regression with a zero-mean error, bounded density paths $f_t=f(1+t\gamma)$ must preserve both normalization and the first <moment>. Their <score functions> therefore satisfy two constraints. When $0<E_f\varepsilon^2<\infty$, truncation followed by two small bounded moment corrections shows these scores are dense in the displayed <closed subspace of a Hilbert space>. By <independence>, they are orthogonal to every $\varepsilon\psi(X)$ with $\psi\in L^2(v)$. The same constraints apply to other paths only under regularity permitting differentiation of the first <moment>.
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