Cross-Order Hodge Stability Scaling

1. Integration point: implement a simplicial or Hodge neural layer whose hidden states are feature-valued cochains x_k on oriented k-simplices. Replace its explicit diffusion or residual propagation line with x_k <- x_k - alpha_k Delta_k^+ x_k, optionally followed by a learned feature transform, nonlinearity, and normalization. Construct signed incidence matrices from the simplex list and compute the up-Laplacian action without materializing a dense matrix: Delta_k^+ x = B_{k+1}(B_{k+1}^T x), where B_{k+1} is the boundary matrix between (k+1)- and k-simplices. 2. Pseudocode: choose a reference order r=k-1; initialize a random Gaussian vector q and normalize it; repeat J times q <- Delta_r^+ q / ||Delta_r^+ q||; set lambda_hat <- q^T Delta_r^+ q; set alpha <- eta / max(lambda_hat, epsilon), with eta initially 1.0; compute y <- B_{k+1}(B_{k+1}^T x_k); update x_k <- x_k - alpha*y; then apply the learned channel map. 3. The mathematical input is the cross-order inequality lambda_k^+ <= lambda_{k-1}^+; power iteration, epsilon, eta, and refresh frequency are empirical choices. Compare shared lower-order scaling against per-order power-iteration scaling, unconstrained learned step sizes, and a standard simplicial GNN. Use a small 2- or 3-order network on synthetic complexes, citation complexes, or mesh data for node and edge classification, plus a diffusion-denoising task. Measure maximum stable depth, activation norms, loss at equal FLOPs, power-iteration overhead, and validation accuracy. Success means substantially deeper stable networks or lower spectral-estimation cost without reducing accuracy.