Proposition: Observation-law perturbation allowance

Let $P$ be the ideal joint law of the rule, points, and admitted auxiliary observations in the uncued task. Let $P^{\prime }$ be another law on the same spaces with $d_{\mathrm{TV}}(P,P^{\prime })\le e$, and retain the target $Y=f_R(X)$. Every predictor has accuracy under $P^{\prime }$ at most

truncated at one.

More generally, for a conditional rule posterior $\pi (r\mid O)$ and a reference posterior ${\pi }_0(r\mid O)$, the difference between their optimal current-label probabilities is at most

when the same point and label functions are used.

Proof. Mapping the joint law to target and prediction cannot increase total variation. The expectation of a correctness indicator changes by at most $e$. For the conditional statement, each candidate-label probability is the posterior mass of a subset of rule IDs. Its change is at most total variation; taking a maximum preserves that allowance.