Scale-Local Information Governs Sample Synthesis from Noisy Smoothed Scores

Statistically equivalent targets can have asymptotically different sampling costs under the same score interface. We establish this through an oracle that returns the full normalized smoothed-score vector at one shared scale, with independent Gaussian observation noise and a precision cap. For geometrically spaced Fourier and spatial families, constant target capacity coexists with an optimal cost of $$\begin{gathered} Theta( \\ sqrtd) \end{gathered}$$ queries at precision $B=d$ and fixed, sufficiently small average total variation error. Preserving the entire statistical experiment, we then construct Blackwell-equivalent spatial targets with a diverging ratio of optimal costs at the same dimension, precision, and accuracy. Matching bounds across all precision levels explain this separation at fixed relative TV accuracy in a local perturbation regime under a uniform prior. Precision determines the feasible coordinate block length; at high precision, the optimal gain over coordinate thinning is of the order of its square root. A blockwise converse quantifies the dependence required by an accurate joint output. Shared activation attains the bound by matching coordinate marginals, cancelling single-coordinate error terms, and controlling the remaining interactions. Nonadaptive samplers match lower bounds allowing adaptive queries and random stopping.