arXiv:2603.10073v2 Announce Type: replace-cross Abstract: We study estimation of a deterministic binary count from the shuffled outputs of a common single-message local channel. Privacy is required for every neighboring composition at fixed global parameters $(\varepsilon,\delta)$, and estimators may be biased. We obtain uniform critical Poisson and Skellam approximations and characterize the limiting minimax risk as an infimum over finite observed-mark Poisson experiments. Let $\Lambda$ be the noise intensity required by the boundary Poisson-shift profile and let $\lambda_{\min}$ be the intensity required by the full composition-maximized randomized-response profile. Binary randomized response is asymptotically minimax optimal if and only if $\Lambda=\lambda_{\min}$. When $\Lambda<\lambda_{\min}$, the minimax value lies strictly between $(\Lambda+\lambda_{\min})/2$ and $\lambda_{\min}$, and a randomized-response construction with observed rare marks strictly improves on binary randomized response. We give a finite-sample certified example. Binary randomized response is strictly suboptimal at $\varepsilon=0$ for every $0<\delta<1$, and this persists for sufficiently small positive $\varepsilon$ at each fixed target. For every fixed positive $\varepsilon$, strict suboptimality also occurs along a sequence of fixed privacy targets tending to zero. The variational characterization permits approximation by finite marked channels; it does not assert that a finite optimizer exists.