Non-Abelian KPZ State-Space Coupling

1. Integration point: implement the mechanism inside a residual SSM or RNN. Let the hidden tensor be h with shape batch by sequence length by channel count, and add a scalar phase channel phi. Use periodic or reflective finite differences along the sequence axis: D h at position l equals one half of h at l plus one minus h at l minus one, while D2 h equals h at l plus one minus two h at l plus h at l minus one. Add the residual drift h <- h + delta_t times [nu D2 h + lambda times (D h) elementwise squared divided by two + K F(D phi, D h)]. 2. Pseudocode: compute D h and D2 h; compute the nonlinear drift; update h; pass the result through the normal SSM transition; every 100 optimizer steps estimate latent covariance, variance, and effective dimension; rescale lambda according to s(d) = 1 for d less than or equal to 2 and exp(-beta times (d minus 2)) otherwise. Enforce an explicit stability limit delta_t times nu times the spectral norm of the finite-difference Laplacian below one half. 3. Taken from the paper: the relevance law and the critical dimension d equal to 2. Estimated in code: nu, lambda, K, latent variance, effective dimension, and c_N. Estimate d using the participation ratio of the local latent covariance, d_eff = (trace C)^2 divided by trace(C squared). 4. First experiment: compare a GRU or small SSM with and without this drift on Copy Memory and sequential MNIST, using equal parameter budgets. Sweep initial g and latent width. Coarse-grain hidden trajectories by averaging adjacent positions at scales b equal to 2, 4, 8, and 16, then fit g(b). The prediction is g(b) proportional to b raised to the power 2 minus d_eff below dimension two, logarithmic growth near dimension two, and decay above dimension two. Require the measured log-log slope to agree with 2 minus d_eff within 20 percent.