Birkhoff Modular Mask Regularizer

Integrate the construction into a discrete structured-mask or hierarchical MoE-routing module. First define a prerequisite poset P over binary structural decisions: for example, a fine-grained token may be retained only if its coarse group is retained, or expert e may be activated only if its parent expert is active. A routing state is an order ideal D subseteq P, represented by a binary vector m satisfying m_child<=m_parent. Choose Q as the desired family of legal states, such as all masks with one of several legal compute budgets, and verify decomposition closure by enumerating states for a small P and checking every pair of endpoint states and every decomposition. Construct g using a linear program: solve for g and gamma subject to sum_{p in D_q}g(p)=0 for every q in Q, sum_{p in D_s}g(p)>=gamma for every s outside Q, gamma>=0.1, and |g(p)|<=G. At each training step, produce relaxed Bernoulli or Gumbel probabilities for the decisions, sample or enumerate an order ideal, and add lambda times F_g(s) to the task loss. For large P, enforce validity by a topological projection that repeatedly removes every selected node whose prerequisite is absent. Use a straight-through estimator for the projected mask. The modular score is computed exactly from the selected ideal; only g is estimated once by the LP. The first experiment should use a small Transformer with eight hierarchical experts or four-level token pruning, comparing this regularizer with independent mask penalties at equal average FLOPs. Record invalid-state rate, validation loss, expert utilization, routing entropy, and gradient variance. Success means substantially fewer illegal configurations and equal or lower validation loss at the same compute budget, with negligible routing overhead.