Inevitability of Encrypted Traffic Side-Channel Leakage in the Multi-Class Setting

The Side-Channel Existence Theorem proves I(X;Y)>0 in the binary, undefended setting, but is confined to pairwise arguments and ignores active defenses. We extend it to k classes via the per-class decomposition I(X;Y)= sum iπ i D KL(P YiP Y), with defense cost modelled by per-class Wasserstein-1 constraints x W 1(Q x D,P x) B. Three results follow: (1) a summation-form MI lower bound over all active classes; (2) a cascade critical cost theorem and a per-class budget corollary, nonzero where the uniform-budget bound vanishes; (3) an accuracy corollary Acc* 2 I 0/k>1/k. On a 95-class website fingerprinting dataset the measured MI has a strictly positive 95% confidence lower bound under every defense tested. Against the strongest pairwise baseline---a convex program over all 2 triangle constraints, also Θ(1) in k under the same non-vanishing-gap conditions---the summation form is only 1.45 times stronger, so the case for the per-class decomposition is structural: only it gives each class a critical cost and a cascade. FRONT's apparent 122 times gap is inflated mainly by threshold exclusion rather than the inequality chain: on the active classes it is 21 times, within 1.4 times of the 15 times measured undefended. Measuring the chain's two steps separately bounds the collapse onto one Lipschitz statistic below by 28 times, against a divergence step measured at 1.5 times. Undefended OVR distinguishability predicts post-defense per-class leakage at Spearman ρ=0.62 -- 0.77, the transfer the certification procedure relies on. The framework carries over unchanged to a 100-class QUIC/TCP pair.