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+Δ 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
nΔ2→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=nKΔ2K−2. The matching lower bounds hold on every fixed polynomial window
nΔ2≤n−κ. 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
Δ=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.