Degree-Capped Simplicial Residual Step

Modify the residual coefficient in a simplicial GNN, hypergraph network, or higher-order diffusion block. Given the complex K, compute D_{\max}=\max_{\sigma\in F_{r-1}}d(\sigma) directly from the incidence list. For an unnormalized up-propagation P=B_rB_r^T, set \eta=\rho/(D_{\max}+\varepsilon), with \rho initially 0.1-0.5 and \varepsilon=10^{-6}; use X_{\mathrm{next}}=\operatorname{activation}(XW+\eta PX) or X_{\mathrm{next}}=X+\eta PXW. If the model uses a learned weighted incidence matrix, estimate \lambda_{\max}(P) every M steps by 5-10 power iterations: initialize a random unit vector v, repeatedly compute v\leftarrow Pv/\|Pv\|_2, and then estimate \widehat\lambda=v^TPv; use an exponential moving average of this estimate and set \eta=\rho/(\widehat\lambda+\varepsilon). For batched complexes, compute one bound per graph and apply a graph-specific \eta, or use the largest bound in the batch. The theorem-derived part is the unweighted certificate \lambda_{\max}\le D_{\max}; power iteration is only needed after learning weights. Also compute the Brouwer quantity B_\ell=((r+1)/2)f_r+(f_{r-2}/r)\binom{\ell+1}{2} and compare measured top-\ell eigenvalue sums against it as a diagnostic. First experiment on synthetic 2-complexes with controlled maximum triangle degree and on a small simplicial face-classification task. Compare fixed tuned residual scales, symmetric normalization, and this degree-capped rule at depths 4, 8, and 16. Record activation norms, gradient explosions, training loss at equal epochs, and accuracy. Success means fewer unstable runs and comparable or faster loss descent without a validation sweep over layer-wise residual coefficients.