Simplicial Ky-Fan Spectral Budget

Integrate this into a simplicial or hypergraph message-passing layer whose propagation matrix is P=\widetilde B_r\widetilde B_r^*, acting on features X\in\mathbb R^{|F_{r-1}| imes c}. Start with the exact signed incidence matrix B_r, optionally multiply each incidence column or nonzero entry by a learned positive gate g, and define \widetilde B_r=B_r\operatorname{diag}(g) or a symmetrically normalized variant. Precompute every codimension-one-face degree d(\sigma), construct the sorted conjugate sequence d^ op, and select a small set \mathcal K such as {1,2,4,8}. During each training step, estimate the largest k eigenvalues of P using k-step Lanczos or simultaneous power iteration; do not form P densely, and implement matrix-vector products as v\mapsto\widetilde B_r(\widetilde B_r^*v). Add \alpha\mathcal L_{\mathrm{budget}} to the task loss. Pseudocode: compute X_next=activation(XW+\beta PX); for k in \mathcal K, run Lanczos(P,k) to obtain \widehat\lambda_1,\ldots,\widehat\lambda_k; compute budget_k=\sum_{j=1}^k d_j^ op; add relu(\sum_i \widehat\lambda_i-budget_k)^2. The mathematical budget is computed exactly from the input complex; eigenvalues are estimated numerically, and the theorem is a certificate only when g=1. First test on a small synthetic simplicial classification benchmark, comparing a 4-layer simplicial GNN against the same model without the penalty. Measure validation accuracy, feature Dirichlet energy, largest propagation eigenvalue, gradient norm, and oversmoothing as depth increases. Success is a lower top-eigenvalue tail and stable accuracy at 8-16 layers without increasing per-epoch cost by more than the Lanczos overhead.