Rarity Determines Minimax Rates of Semantic Entropy From Weak Pairwise Judgments With a Known Channel

The semantic entropy of a language model is the entropy of the meanings of its sampled answers. The meanings are never observed, and the entropy is estimated from the verdicts of a pairwise equivalence judge. Given the meanings of n answers, a known channel declares each pair equivalent independently, with probability q+Delta within a meaning and q across meanings. At most K meanings occur, and q is bounded away from 0 and 1. For a weak judge, with nDelta2→0, the global minimax rate is set by distributions near a vertex of the simplex and overstates the difficulty elsewhere. We show that the difficulty is set by how many meanings can be rare. Over distributions with at most s meanings below a fixed probability τ<1/(K-s), the minimax squared risk is of order N-1/s for s≥2, N-1log2(e+N) for s=1 and N-1 for s=0, capped at one, where N=nKDelta2K-2. The matching lower bounds hold on every fixed polynomial window nDelta2≤ n-kappa. One estimator attains these rates without knowing s or τ: it averages products of centered verdicts along all spanning trees of every k-set of answers, projects the resulting power sums onto the valid moment set and inverts them. The case s=K-1 gives the global rate N-1/(K-1) and the consistency threshold: with Delta=n-γ the entropy is uniformly consistently estimable exactly when γ<K/(2K-2). On polynomial windows, the same estimator attains the sharp local minimax constant on shrinking neighborhoods of interior points with distinct probabilities. Restricting the average to stars multiplies the constant by KK-3.