Characterizing finite posets whose probabilistic powerdomain are RB-domains

What the math gives to ML

The paper identifies a precise coordinate structure for stochastic order on distributions over a finite rooted tree: dominance can be checked through cumulative probabilities of rooted subtrees rather than exponentially many upper sets. This gives a practical structured-output module for neural networks whose labels form a hierarchy, taxonomy, or causal tree. The transferable asset is the simplicial tree cone, which turns a global partial-order constraint into nonnegative local cumulative-flow constraints. The most direct experiment is to replace flat categorical outputs with tree-flow probabilities and train with a subtree-dominance loss, measuring constraint violations, calibration, and compute against flat softmax and hierarchical-softmax baselines.