Minimal-Channel Harmonic Controller

Add a compact harmonic-state controller to a recurrent graph neural network, graph state-space model, or edge-feature adapter. Preprocessing takes the fixed incidence matrix B and computes an orthonormal basis H for ker(B): use sparse QR/SVD for small graphs, or obtain beta_1 cycle vectors with a spanning-tree fundamental-cycle construction followed by QR orthogonalization. At each time step, read the edge state F and compute harmonic coordinates Z=H^T F. Pass Z through a small recurrent cell or linear controller, then inject the result through A: U=Controller(Z), F_new=F+A U. For the simplest falsifiable version, set A=H and U=-eta Z, giving F_new=F-eta H H^T F; constrain eta to (0,1) with eta=sigmoid(raw_eta) so harmonic energy contracts monotonically. For a localized-device version, choose A as r selected physical edge columns, with r=beta_1 and optimize the selected edges or a learned sparse A. Before training, calculate the rank and condition number of H^T A; reject channel sets with rank deficiency. During training, log harmonic energy E_H=||H^T F||_F^2, cut energy E_C=||F-HH^T F||_F^2, and spillover ratio ||P_C A U||_F/||A U||_F. The mathematics determines the minimum rank and ideal contraction; the recurrent cell and channel selection are empirical additions. First test on temporal node/edge forecasting over ring, mesh, and IEEE-style graphs, comparing full edge-state recurrence, ordinary low-rank adapters, and the beta_1-channel controller at equal hidden size. Inject unobserved cycle perturbations at inference. Success is lower memory and parameter count with stable long-horizon predictions, harmonic energy decay at the prescribed rate, and no significant increase in cut-space forecasting error.