Pullback-Commuting 3-Axis Network

1. Exact integration point: implement a three-axis block for data with time, spatial, and group axes, such as a tensor h[t,x,g]. Each axis has a gated residual transition T_i(h,u_i), with shared latent width and separate axis embeddings. Add the commutation loss to the ordinary task loss. 2. Pseudocode: sample a latent state h and local labels u1,u2,u3; compute both paths for each pair, d12 = norm(T2(T1(h,u1),u2) - T1(T2(h,u2),u1)) and analogously d13,d23; enumerate the six triple compositions, compute their maximum pairwise distance d123; optimize task_loss + lambda_pair*(d12²+d13²+d23²) + lambda_assoc*d123². Use gradient clipping and optionally normalize defects by the latent-state norm. 3. Taken directly from the paper's mechanism: colored path interchange, pullback uniqueness, and triple consistency implied by compatible local diagrams. Estimated empirically: the transition Lipschitz constant L using Jacobian-vector power iteration, epsilon as the validation mean pairwise defect, and collision rate by testing whether equal boundary conditions produce materially different latent outputs. 4. First cheap experiment: train a three-axis gated SSM on Moving-MNIST or a synthetic 3D advection dataset. Compare fixed-order scanning, random scan-order scanning, and the commuting model at equal parameter count. Measure task error, rollout error at horizons 1, 10, 50, and the disagreement across all six scan orders. The quantitative prediction is that six-order disagreement will scale approximately as 3*epsilon*(1+L+L²), within 30 percent, and that rollout disagreement will stop growing when measured L is below one. Sweep lambda_pair and identify whether a sharp reduction occurs once pairwise defects fall below the task-loss noise scale.