Extreme classification: beating chance with one training example from each class

We study a minimal classification problem: Given independent labeled observations $X\sim P$ and $Z\sim Q$ from two unknown distributions $P,Q$, and given an independent target $Y$ drawn with equal probability from $P$ or $Q$, can one classify $Y$ strictly better than chance whenever $P\neq Q$? The one-nearest-neighbor rule succeeds for every pair of multivariate Gaussian distributions with distinct means and a common positive-definite covariance matrix but can perform strictly worse than chance even for smooth densities on the real line. We construct a fixed randomized kernel rule whose expected accuracy is exactly $1/2+\operatorname{MMD}_k^2(P,Q)/4$, and obtain characteristic kernels on countably generated measurable spaces from countable families of measurable binary questions. We also prove that a deterministic order rule on $\mathbb R$ beats chance for every pair of distinct Borel probability measures. A measurable encoding then gives a deterministic distribution-free rule which beats chance on every countably generated measurable space, in particular every separable metric space. Finally, we show that no rule works for every distinct pair of distributions and every unknown unbalanced class prior; under adaptive target-class selection, every rule other than a fair coin is strictly worse than chance for some finitely supported pair.