Two-Term Asymptotic Expansion of Jensen–Shannon Leakage for Subsampled Gaussian Sum Release

We consider subsampling at rate q among n users and a Gaussian sum-release with noise variance sigma^2. For neighboring datasets D0=(0,...,0) and D1=(mu,0,...,0), letting SNR=mu^2/sigma^2, we prove JSD(P0,P1) = (q*SNR)/(8n) + ((4-12q+7q^2)/64)*(SNR^2/n^2) + O(n^-3).The remainder is uniform for q in [eps,1] (any fixed eps>0) and bounded SNR. The proof combines an exact mixture representation, a local Edgeworth expansion, uniform sixth-derivative control, and a parity argument. Numerical quadrature of the exact binomial–Gaussian mixture corroborates the uniform n^-3 behavior.