Third and final part of the trilogy on asymptotic theory for neighboring shuffle experiments over finite alphabets. We treat general dominant blocks of arbitrary finite size, allow overlap between the dominant output sets under the two neighboring hypotheses, and prove a general Lévy–Khintchine limit theorem for shuffled histograms. After projecting onto the sum of the dominant tangent spaces, the dominant block yields a Gaussian factor; after quotienting by those same tangent spaces, the rare block yields a compound-Poisson jump field. The main theorem gives: (i) weak convergence of the full hybrid statistic to an independent Gaussian–compound-Poisson pair, (ii) O(n⁻¹) total-variation and Le Cam convergence for the projected jump experiment, and (iii) privacy-curve convergence for the full experiment in the interior, weak boundary, and regular strong-boundary regimes. A counterexample shows that no fully general strong-boundary privacy-curve theorem is possible without additional structural hypotheses. Two further results close the rate picture: the O(n⁻¹/²) comparison rate is shown to be sharp via an exact binomial-plus-Bernoulli calculation, and a boundary Berry–Esseen theorem establishes O(c) Le Cam proximity between the Poisson-shift and Gaussian shift experiments as the critical parameter c → 0. Together with Parts I (arXiv:2602.09029) and II, this yields a three-regime universality picture for finite-alphabet shuffle privacy.
Alex Shvets (Wed,) studied this question.