Mean-preserving error tangent space ($\{\gamma \in L^2(f):E_f\gamma =E_f(\epsilon \gamma )=0\}$)

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 $0<E_f{\epsilon }^2<\infty$, 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 $\epsilon \psi (X)$ with $\psi \in L^2(v)$. The same constraints apply to other paths only under regularity permitting differentiation of the first moment.