Strang-Split Anisotropic Kernel Layer

Integrate this as a propagation block in a graph neural operator, point-cloud encoder, or latent world model. Replace one unrestricted attention layer by three sparse propagation sublayers: direction 1 half-step, direction 2 full-step, and direction 1 half-step. Each direction has a scalar MLP $s_r(x)$, order coefficient $\beta_r$, and bandwidth $\varepsilon_r$; value projections and feed-forward layers may be shared to keep parameter counts comparable. For each direction compute $\ell^{(r)}_{ij}=-\|x_j-x_i\|^2/(4\varepsilon_r)+\beta_r(s_r(x_j)-s_r(x_i))$, apply a neighbor-wise softmax to obtain $P_r$, and update with $h\leftarrow h+\alpha_r(P_rV(h)-h)$. Use $\alpha_1=\delta/2$, then $\alpha_2=\delta$, then repeat direction 1 with $\alpha_1$. For a direct semigroup interpretation, keep $\delta$ small; for ordinary deep-network training, tune residual scales and use normalization after the three-step block. The paper determines the symmetric composition and its $O(\delta^3)$ local error; training estimates $s_r$, $\beta_r$, and bandwidths. Measure noncommutativity through $\|P_1P_2-P_2P_1\|_F$ and clip logits before softmax. First test on a 2D advection-diffusion field with diffusion matrix $A=R(\phi)\operatorname{diag}(0.05,0.5)R(\phi)^T$, training a small point-cloud next-step predictor. Compare one isotropic ordered layer, sequential composition, and Strang composition at matched FLOPs. Evaluate one-step MSE, 50-step rollout error, and feature-norm growth. The expected signal is lower rollout error and reduced depth sensitivity for Strang splitting, especially when the principal diffusion direction changes spatially.