Universal Shuffle Asymptotics: Sharp Privacy Analysis in the Gaussian and Non-Gaussian Regimes (Parts I–II)

A two-part series developing a complete sharp privacy theory for the shuffle model of differential privacy across all asymptotic regimes. **Part I (Gaussian Regime)** establishes exact likelihood-ratio identities for shuffled histograms, sharp Jensen–Shannon divergence expansions through O(n⁻³), asymptotic equivalence to Gaussian Differential Privacy (GDP) with Berry–Esseen bounds, Local Asymptotic Normality (LAN) and quantitative Le Cam equivalence to a one-dimensional Gaussian shift, exact finite-n privacy curve formulas, and an application to frequency estimation where exact accounting yields strictly less noise than amplification bounds. **Part II (Non-Gaussian Regimes)** characterizes the critical frontier where local privacy ε₀(n) grows with population size n and classical Lindeberg conditions fail. For binary randomized response with eᵋ⁰⁽ⁿ⁾/n → c², the neighboring shuffle experiment converges in Le Cam distance to a Poisson-shift limit (canonical pair) or a Skellam-shift limit (proportional compositions), with explicit TV bounds and limiting privacy curves. For general finite alphabets in a sparse-error critical regime, a multivariate Poisson point process / compound-Poisson limit is established. Results are unified into a three-regime phase diagram: sub-critical (Gaussian/GDP), critical (Poisson/Skellam/PPP), and super-critical (no privacy). Both papers are self-contained. Full analytical proofs, explicit constants, numerical illustrations, and reproducible Python code are provided.