From Pointwise Queries to Population Probabilities: A Sharp Preservation Threshold

A Boolean question can report either its truth value at an unknown state or its probability under an unknown population law. We compare the optimal guarantees of these two interfaces when a decision maker may choose one question at random from a fixed finite menu. Let e be the minimax error for predicting a Boolean target from a pointwise answer, and let R be the minimax excess zero-one risk from an exact population answer, relative to a decision maker who knows the population law. We prove e ≤ R ≤ 2e and identify the exact universal preservation range: R=e whenever e=0 or e ≥ 1/3, whereas every positive attainable error below 1/3 admits a finite interface with R>e. Writing G(η) for the supremum of R over interfaces with e=η, we give an explicit lower bound proving G(η)/η → 2 as η ↓ 0 through all positive rational values. In contrast, the minimax absolute error for recovering the target probability is exactly e for every interface. The proofs use affine extension, the classical plug-in risk comparison, and explicit adversarial populations. A small example has a complete rational finite-game certificate. The results concern exact, restricted query interfaces and provide no model-performance guarantee.