Simultaneous Reconstruction of Shuffled Histograms from a Retained Statistic: Exact leading coefficients for finite hypothesis families

Final part of the trilogy. We prove a general Lévy–Khintchine limit for neighboring shuffle experiments: a common Gaussian factor on the dominant tangent space, a compound-Poisson jump field on the quotient carrying the full neighboring shift Δ, O(n⁻¹) projected Le Cam rate, privacy-curve convergence in regular regimes, a strong-boundary obstruction, sharp O(n⁻¹/²) hybrid rate with a compatibility condition restoring O(n⁻¹), and boundary Berry–Esseen giving O(c) Poisson-to-Gaussian proximity. A numerical companion demonstrates that dominant-block overlap, cross/native split-law compatibility, and boundary regularity are the three geometric levers controlling privacy level, certification speed, and asymptotic safety. Completes the series with Parts I–II. arxiv: Part I Part II Part III