We study one-step neighboring shuffle experiments for ε₀-locally differentially private channels on growing input alphabets d → ∞. On the privacy side, we prove an exact likelihood-ratio quotient compression theorem, a universal extremal χ² bound sharp at two-point laws, construct explicit obstruction families for which the shuffled privacy curve is identical to binary randomized response for all d, and establish a sharp diluting/persistent dichotomy governed by the pairwise χ² divergence. On the estimation side, we prove a universal minimax lower bound of order (d−1)/(nχ*(W)) for frequency estimation and show that permutation-equivariant channels are without loss of generality for the uniform-point Fisher criterion. On the mechanism design side, we show that calibrated GRR is not shuffle-optimal; the optimal low-budget mechanism is an augmented GRR channel that concentrates an aggressive local signal on a random fraction of users. In the regime 0 < C ≤ C*(d), this construction is optimal among all permutation-equivariant channels with arbitrary finite output alphabet, up to conditionally identical output refinements. We further prove that GRR is the unique matched-budget optimizer within the entire subset-selection family.
Alex Shvets (Wed,) studied this question.