Conservative Parallel-Edge Decomposition

1. Integration point: use this as the message-construction layer of a graph forecaster, neural ODE, or battery and energy-system world model. For every physical edge, create K parallel channel MLPs, one for each input type such as electrical, thermal, mechanical, or learned residual flow. Do not concatenate all inputs into one unconstrained edge MLP except as an ablation. 2. Pseudocode: for every edge j and channel k, compute p[j,k] = EdgeMLP_k(x[tail[j]], x[head[j]], u[j,k]); set p[j] = sum_k p[j,k]; aggregate q = incidence @ p; add q_ext = incidence_ext @ p_ext; update node states using q + q_ext. If the target obeys exact conservation, use incidence aggregation directly; otherwise add the displayed residual penalty. 3. Computed from the paper: parallel decomposition and conservation through vertex aggregation. Estimated empirically: channel functions, channel count K, and any constitutive correction in the node update. Measure conservation residual independently from prediction error. 4. First cheap experiment: train on a synthetic multi-domain graph with known electrical and thermal flows, then test on a small battery-temperature forecasting dataset or coupled RC network. Compare an unconstrained composite-edge GNN against K-channel decomposition with matched parameter count. The predicted signature is machine-precision internal conservation, with norm of 1^T B P below 1e-6 apart from floating-point error, while the unconstrained baseline has a nonzero residual that grows with rollout length. If the node dynamics are conservative, long-horizon energy drift should scale with external-flow and integration error rather than with the number of internal message-passing steps.