Tree-Cone Distribution Head

Integrate this as a distribution head and auxiliary loss for a model whose output classes are arranged as a rooted tree. First store the parent index, children list, depth, and a postorder traversal of the tree. The network produces logits z in R to the power of |P|; use p=softmax(z) for a minimal implementation, or replace this with a hierarchical conditional-softmax head if reducing output computation is important. For each batch item, compute subtree masses by reverse traversal: initialize s_v=p_v and then add s_v to its parent, so after the traversal s_v=sum_{u in U_v} p_u. Given a training pair (x_a,x_b) with a known desired order a <= b, compute p=f_theta(x_a) and q=f_theta(x_b), then add lambda_dom times sum_v w_v relu(s_v(p)-s_v(q)) squared to the ordinary cross-entropy. If the direction of the pair is uncertain, use a binary target t and apply relu(s_v(p)-s_v(q)) for t=1 and relu(s_v(q)-s_v(p)) for t=0. The theorem supplies the exact reduction from all upper sets to rooted-subtree coordinates; the weights, lambda_dom, and whether to use logits or a hierarchical head are empirical choices. Start with a small MLP or Transformer on synthetic tree-ordered classification data and then test on a real taxonomy dataset. Compare flat softmax plus pairwise class penalties, standard hierarchical softmax, and this head at equal parameter count. Success means fewer stochastic-order violations on held-out pairs, improved hierarchical accuracy or calibration at equal cross-entropy, and O(|P|) loss computation rather than exponential enumeration of upper sets.