Yang–Baxter Current-Conserving Neural Flow

First, integrate the mechanism into a local sequence model with feature dimension d, such as a neural ODE, recurrent state-space model, or learned cellular automaton. For the minimum viable prototype, use one shared learned real matrix h of size d² by d² and construct the global vector field from translated pair interactions. If real-valued stable dynamics are desired, parameterize the generator as A = K - K transpose; if complex unitary dynamics are desired, use the generator minus iH. Second, use this training procedure: sample a minibatch; construct h_12 = h tensor I_d and h_23 = I_d tensor h; compute c = h_12 h_23 - h_23 h_12; compute r = (h_12 + h_23)c - c(h_12 + h_23); add lambda times ||r||_F² divided by (||h||_F² + epsilon)² to the task loss; update h and the predictor parameters. During rollout, integrate the local vector field with RK4 or a matrix-exponential method. The paper supplies the commutator certificate, local-current construction, and implication from zero residual to current conservation. The implementation must estimate residual norms, finite-chain commutators, rollout errors, and current drift numerically. Start with d = 2 and chain length L = 16, training on trajectories from a known integrable nearest-neighbour spin chain. Compare an unconstrained local model, the residual-regularized model, and a GRU with matched parameter count on one-step and 128-step prediction. Measure relative current drift delta_J(T) = ||J(T) - J(0)|| divided by ||J(0)||. The falsifiable prediction is that delta_J(T) grows approximately linearly at short and intermediate horizons, with slope proportional to ||r(h)||_F. Reducing the residual by ten times should reduce the drift slope by approximately ten times. At residual zero, remaining drift should be at the numerical integration and floating-point error floor.