Spherical Height-Routed Multiscale Network

1. Integration point: construct a hierarchical MLP, Transformer feed-forward block, or MoE layer with levels m=0,...,M. Each level contains q child experts or graph modules; a router produces logits over levels or leaves, and each leaf has height h=2m. The selected path determines computation depth, feature resolution, or expert identity. For an inexpensive prototype use q=2 or 4, M=4, shared parent projections, and separate child residual adapters. 2. Pseudocode: compute router logits z_b(2m), set p_b=softmax(z_b), compute the task loss, average p_b over the minibatch to obtain p_B, evaluate rho[0]=(q-1)/q and rho[2m]=(q*q-1)*q^(-2m-1), add tau times the KL term to the task loss, and backpropagate through router and experts. Use dense soft routing initially, then switch to top-1 or top-2 routing after warmup. 3. Computed from the paper: the target shell law and q-ary descendant growth. Estimated empirically: p_B, total-variation discrepancy D_M, per-level gradient norms, activation variance, routing load, and effective FLOPs. 4. First cheap experiment: compare a six-layer residual MLP or small ViT on CIFAR-10 against an equally sized dense baseline and an unconstrained hierarchical router. Sweep tau in {0, 1e-3, 1e-2, 1e-1}. The falsifiable prediction is that constrained-router discrepancy D_M decreases during training and remains near the target law, whereas unconstrained routing collapses toward m=0 or maximum depth. For q=2, the predicted tail ratio is rho(2(m+1))/rho(2m)=1/4 for m>=1; measured tail ratios should approach 0.25. Also test inference compute: expected selected height should remain finite and approximately equal to the truncated target mean rather than growing with M. A useful failure criterion is no improvement in routing stability or a measured tail ratio differing from 0.25 by more than 20 percent after convergence.