How many predictions per example are necessary to learn several classification tasks simultaneously? For a class $\mathcal H$, let $K(\mathcal H)$ be the least list size that makes realizable PAC learning possible. Hanneke, Moran, and Waknine asked for $K(\mathcal H_1\otimes\mathcal H_2)$, leaving a gap between $(K(\mathcal H_1)-1)(K(\mathcal H_2)-1)$ and $K(\mathcal H_1)K(\mathcal H_2)$. We close the gap and determine a stronger risk spectrum. For nonempty classes with finite $k_j=K(\mathcal H_j)$, set $M=\prod_j k_j$. If a learner may output $L$ tuple labels, then its optimal worst-case expected realizable error converges to $\left(1-\frac{L}{M}\right)_+.$ Consequently, $K(\bigotimes_j\mathcal H_j)=\prod_jK(\mathcal H_j)$. The result is quantitative at every sample size. Its lower bound is a strong direct-product inequality in terms of the factor learning curves. The proof turns infinite list-DS pseudo-cubes into explicit finite Bayes experiments. On an unseen coordinate, the posterior contains $k_j$ competing labels. A difference of adjacent Bayes list risks lower bounds the $k_j$-th posterior mass. Under independent products, these masses multiply, while a list of size $L$ omits at least $M-L$ cells of the top-posterior grid. This posterior order-statistic argument also shows that optimal weak multiclass accuracies multiply. We complement the list theorem with an exact fixed-marginal tensorization under standard minimax regularity and dimension-dependent rates for uniform and agnostic product learning.
Keywords: list learning, direct products, product classes, DS dimension, multiclass learning, risk spectrum