Universal Shuffle Asymptotics, Part III: Dominant-Block Quotient Geometry and Hybrid Gaussian--Compound-Poisson Limits in Finite-Alphabet Shuffle Privacy

Abstract

Part I of this series (arXiv:2602.09029) establishes a sharp Gaussian (LAN/GDP) limit theory for neighboring shuffle experiments in the fixed full-support regime. Part II (arXiv:2603.10073) identifies the first universality-breaking frontier: critical Poisson, Skellam, and multivariate compound-Poisson regimes. The present paper completes the finite-alphabet weak-limit theory by identifying the dominant-block quotient geometry that governs neighboring shuffle experiments. We treat dominant blocks of arbitrary finite size, allow overlap between the dominant output sets under the two neighboring hypotheses, and show that the limiting experiment decomposes according to this geometry: projecting onto the sum of the dominant tangent spaces yields a Gaussian factor, while quotienting by those same tangent spaces isolates a compound-Poisson jump field in the rare block. We also identify the regimes in which this quotient description determines the full privacy-curve, as well as the obstruction that appears when projected jump limits alone do not suffice. Two further sections sharpen the rate picture and the boundary interface: we show that the O(n^{-1/2}) rate for the full hybrid experiment is sharp in general, identify a compatibility condition restoring the O(n^{-1}) rate, and prove a boundary Berry--Esseen theorem giving O(c) Le Cam proximity between the critical Poisson-shift and Gaussian shift experiments as c tends to 0. Together with Parts I--II, this yields a three-regime universality picture and a precise finite-alphabet Levy--Khintchine layer for shuffle privacy.