 Mean-zero score identity under quadratic-mean differentiability

For a differentiable-in-quadratic-mean path, normalization gives $⟨(\sqrt{p_t}-\sqrt{p})/t,\sqrt{p_t}+\sqrt{p}⟩=0$. The two factors converge in L2 space to $g\sqrt{p}/2$ and $2\sqrt{p}$. Continuity of the inner product therefore gives $Pg=0$. This proves score function centering without differentiating the density integral.