Second-order Jensen–Shannon Divergence Expansion in the Subsampled Gaussian Sum-Release Model

We derive a two-term asymptotic expansion for the Jensen–Shannon divergence between neighboring distributions in the subsampled Gaussian sum-release model, a fundamental primitive in differential privacy. For n users with participation probability q and Gaussian noise N(0,σ²), we prove: JSD(P₀,P₁) = (q·SNR)/(8n) + c₂(q)·SNR²/n² + O(SNR²/n³) where SNR = μ²/σ² is the signal-to-noise ratio and c₂(q) = (4 - 12q + 7q²)/64. Key results:- Explicit second-order coefficient with phase transition at q* ≈ 0.453- Uniform bounds over q ∈ [ε,1], SNR ∈ (0,1], n ≥ 2- Rigorous treatment of the common atom at zero- Exact cancellation identities for cross-term control Version 2.0: Complete rewrite with corrected proofs addressing remainder bounds, tail estimates, and domain specifications.