Spectrally Safe Phase Propagation

1. Integration point: modify the propagation matrix in a directed GNN, graph preprocessing step, or graph diffusion block. For each edge use A_uv=e^{iθ_uv}, multiply by inverse out-degree D_+^{-1}, and retain a real-valued output projection after the complex propagation. 2. Pseudocode: form P=(1−α)I+αD_inv@A_gain; draw a fixed random complex probe x with unit norm; run K=5–10 power iterations x←P@x/norm(P@x), recording rho_hat=norm(P@x); if rho_hat>r_target, replace P by (r_target/(rho_hat+ε))*P. Apply h_next=equivariant_activation(P@h@W). Compute the monitor every several hundred optimizer steps, with gradients stopped through rho_hat, or use a soft penalty max(0,rho_hat−r_target)^2. 3. The exact comparison theorem is used for the unnormalized gain adjacency; rho_hat is an empirical estimate for the normalized residual operator, and separately estimate rho(A(D)) using the same power iteration. 4. First cheap experiment: train 2- to 6-layer directed GCNs on Cora, PubMed, and a synthetic long-chain graph, comparing ordinary real adjacency propagation, unconstrained complex gains, and this rescaled version. Record exploding activations, gradient norms, validation accuracy, and feature oversmoothing versus depth. The target signal is fewer exploding runs and stable training at depths where the unconstrained phase model diverges, without reducing accuracy or requiring smaller learning rates.