Dirichlet Follow-the-Leader Closes the Gap in Simultaneous Multiclass U-Calibration

Can one forecaster attain the optimal regret rate for every bounded proper loss and also adapt to every smooth proper loss? Recent work answered this up to a dimension gap. Its self-concordant perturbation gives roughly K 5/4 sqrt T worst-case regret and incurs an additional beta sqrt K K for beta-smooth losses. We close both gaps with a one-line forecaster. After observing class counts c t-1, draw the next prediction from Dir(c t-1), on the face of classes seen so far. This is a fresh Bayesian bootstrap of the outcomes. The analysis rests on an exact identity: averaging any bounded proper loss under Dir(alpha) equals a discrete derivative of its Dirichlet-averaged Bayes risk. The identity makes the be-the-perturbed-leader term telescope to a nonpositive Jensen gap. A one-count likelihood ratio then bounds stability by the inverse square root of that class's count. The resulting single, horizon-free algorithm satisfies E Reg at most 4 sqrt S T T at most 4 sqrt K T and E Reg at most 5 over 2 beta(1+ T) for every beta-smooth proper loss. Here S T is the number of observed classes. Known lower bounds show that both rates are optimal in their nontrivial regimes. The proof covers nondifferentiable losses and changes of the active simplex face.