Constraint-Free Skew Coupling

Use this at the coupling operation of a modular recurrent world model or graph neural network. Give every node or module \(i\) its own latent state \(\mathbf{e}_i\), positive metric \(\mathbf{M}_i=\mathbf{L}_i\mathbf{L}_i^{\top}+\epsilon\mathbf{I}\), internal skew operator \(\mathbf{J}_i=\mathbf{A}_i-\mathbf{A}_i^{\top}\), port projection \(\mathbf{B}_i\), and port output \(\mathbf{y}_i=\mathbf{B}_i^{\top}\mathbf{e}_i\). For each directed edge \((i,j)\), learn \(\mathbf{K}_{ij}\) and impose \(\mathbf{K}_{ji}=-\mathbf{K}_{ij}^{\top}\) rather than learning both directions independently. At each step compute all port outputs and set \(\mathbf{u}_i=\sum_j\mathbf{K}_{ij}\mathbf{y}_j\). Update each module with midpoint integration; either use explicit midpoint for a cheap baseline or solve one shared Newton system over the concatenated node states. Do not add equality penalties between node states. Compare against a standard message-passing GRU and a penalty-coupled latent Hamiltonian model. First test coupled oscillator trajectories and a small spring-mass graph, followed by a 10-50 node forecasting benchmark. Log total latent energy, rollout error, gradient norms, and wall-clock cost as graph size increases. Success means bounded energy drift, fewer exploding-gradient failures on long rollouts, and stable error as nodes are added, while matching one-step prediction without tuning a penalty coefficient.