Shifted-Complex Spectral Defect Regularizer

Integrate the method into a hypergraph neural network or sparse MoE router whose current batch constructs hard top-k hyperedges. Let N tokens or nodes produce router scores s_{i,e}; select each hyperedge e as a set of k+1 nodes using top-k selection, and use the resulting binary incidence matrix B for the forward message-passing or expert-routing operation. Every T training steps, compute the regularizer on the hard graph: (1) enumerate every selected facet and all of its k-element ridges; (2) assign a consistent orientation to each facet and build sparse B; (3) compute vertex degrees d_i; (4) sort d_i nonincreasingly and form dT[r]=count_i(d_i>=r); (5) calculate C=B^T B with sparse matrix multiplication; (6) calculate q=sum_{a,b} C[a,b]^2; (7) calculate D=(sum_r dT[r]^2-q)/max(1,sum_r dT[r]^2); and (8) add mu*D to the task loss. Do not eigendecompose the Laplacian. Since hard top-k selection and sorting are nondifferentiable, use a straight-through estimator for facet membership, or apply the structural term during periodic rewiring: resample candidate facets, rank them by router score minus mu times their estimated contribution to D, and retain the best fixed-size set. A smoother but theorem-free variant can use soft B_theta=softmax(scores/temperature) for backpropagation while always reporting exact hard D for validation. Sweep mu over {1e-4,1e-3,1e-2} and compare against random routing, fixed degree balancing, and a spectral-norm penalty. First run a 2-layer k=2 hypergraph GNN on synthetic node classification with a fixed number of facets, then test a sparse transformer router at equal expert assignments. Success means lower validation loss or higher accuracy at equal edge count, fewer routing rewiring oscillations, and lower batch-to-batch message-passing variance. Failure is increased task loss or no improvement over simple degree balancing.