Universal Shuffle Asymptotics, Part III: Continuous Alphabets and the Gaussian Mechanism

Part III of the Universal Shuffle Asymptotics series. Extends the canonical shuffle privacy theory from finite to arbitrary measurable output alphabets. The exact shuffled likelihood ratio is the sample mean of the one-message likelihood ratio on any output space; this is proved via a measure-theoretic sufficiency argument requiring no histogram structure. Under a finite third-moment condition: Berry–Esseen bound for the shuffle likelihood ratio (Theorem 4.1); GDP equivalence with explicit O(n−1/2) trade-off error (Theorem 5.3); LAN via direct logarithmic expansion with quantitative Le Cam distance (Theorem 6.1, Corollary 6.2); sharp JSD expansion χ²/(8n) + O(n⁻²) under finite fourth moment (Theorem 7.1). For the Gaussian mechanism W_x = N(x, σ₀² I_d): all canonical constants are explicit and dimension-free, with μ_n = √((e1/σ₀²−1)/n). Exact finite-n privacy curves reduce to the law of a scaled sum of iid log-normals (Theorem 8.1). Multi-message unbundled LR is an explicit symmetric polynomial with linear-term variance mχ²/n (Theorem 9.1). A new phenomenon for shrinking Gaussian noise: the standardized CLT breaks at σ₀⁻² ~ (1/2)log n, strictly before the finite-alphabet Poisson boundary (Theorem 11.1). Pure mathematics; no figures or code. Companion to Part I (arXiv:2602.09029) and Part II (DOI: 10.5281/zenodo.18841286).