Fixed-Composition Shuffle Asymptotics in the Full-Support Gaussian Regime

Experiment-level asymptotics for privacy amplification by shuffling, in the fixed full-support Gaussian regime: a fixed finite-output local randomizer with full support and neighboring binary datasets differing in one user. We give exact likelihood-ratio identities for shuffled histograms and a conditional-expectation linearization with explicit typical-set control. The sharp objects are information-theoretic constants of the fixed-composition experiment, not the privacy curve itself: (1) the correct fixed-composition Fisher constant Iπ = vᵀΣπ⁺v, replacing the optimistic i.i.d.-mixture proxy, with Σπ = (1−π)Σ0 + πΣ1; (2) a uniform second-order Jensen–Shannon expansion JSD(Tn,k‖Tn,k+1) = Iπ/(8n) + O(n⁻²) over all proportional compositions k = Θ(n), and the corresponding leading constant for smooth f-divergences; (3) a Gaussian shift / GDP limit on the local scale ε = tμn (μn = √(Iπ/n)), where the privacy curve admits the sharp local expansion δ(tμn) = μn{φ(t) − tΦ(−t)} + O(n⁻¹), with the error genuinely below the leading term; (4) Local Asymptotic Normality with quantitative Le Cam distance to the Gaussian shift; (5) exact finite-n positive-part privacy curves, given separately as an accounting layer and deliberately not used as an asymptotic certificate; (6) a comparison of bundled vs. unbundled multi-message shuffling at the level of the leading Gaussian parameter. The global trade-off approximation has error of the same order O(n⁻¹/²) as the signal and is explicitly flagged as not a small-δ finite-n certificate. A numerical companion study illustrates exact calibration against the Feldman–McMillan–Talwar amplification bound at moderate population sizes. Scope: the results are confined to binary inputs, a fixed finite output alphabet, full support, and a fixed local randomizer; growing-alphabet and growing-ε regimes (where Gaussian limits break) are treated separately.