Polyhedral degree-budget regularizer

Insert the mask directly before aggregation in a GNN or graph transformer. For each edge e=(u,v), compute a raw score q_e from the endpoint embeddings and set a_e=epsilon+(1-2epsilon)sigmoid(q_e), so every edge remains numerically available while the learned values become sparse. Add L_poly to the task loss. In every minibatch, evaluate the global mass constraint and all vertex degree constraints. Maintain an active subset family C for rank inequalities: initialize C with local neighborhoods and connected components, then every K optimization steps identify the most violated cuts using a min-cut or separation routine, add the five worst violations, and remove constraints that remain inactive for several epochs. Anneal alpha, beta, and gamma from zero during the first 10% of training to avoid early optimization collapse. The exact computed object is a penalty relaxation of Theta(G); the finite active set is an empirical approximation to all cut constraints. Test a 3-layer GraphSAGE and GAT on Cora, PubMed, and ogbn-arxiv against standard attention, L1 edge sparsity, and top-k pruning at matched average edge count. Record accuracy, peak GPU memory, effective edge count, connected components after thresholding, maximum mask degree, and training variance. The target is equal accuracy with n-1 to 2(n-1) effective edges and fewer disconnected supports than L1 or top-k baselines.