Orthogonal anti-Latin hash routing

Integrate the construction at the routing decision in an MoE layer whose tokens correspond to a d by d image-patch grid or a sequence reshaped into that grid. Prepare K mutually orthogonal anti-Latin squares offline and store them as an integer tensor A with shape [K,d,d]. For token position (i,j), select a channel r and compute q = A[r,i,j]. In a fixed top-1 router, map q directly to expert q when the number of experts is d. For two-channel routing, compute q1 = A[r,i,j] and q2 = A[s,i,j], then use the pair as a structured code; with d squared experts use expert index q1 times d plus q2, while with fewer experts hash the pair but retain both labels as auxiliary features. Rotate r and s across layers or batches. A practical hybrid is score_e = alpha times fixed_score_e plus (1-alpha) times learned_gate_e(x), with alpha initially near one and decayed if learned routing improves accuracy. The exact load count is computed from the identity sum over i,j of indicator[A[r,i,j] = q] = d, so no stochastic estimate or load-balancing loss is needed for the fixed component. Measure expert counts, overflow, token dropping, pairwise co-routing collisions, router entropy, and training variance. First test a Switch-style Transformer or ViT-Tiny on CIFAR-100 with d = 8, 64 tokens, and 8 experts. Compare anti-Latin routing against random balanced hashing and learned top-1 routing with an auxiliary load loss at equal FLOPs. Success is zero or near-zero overflow, lower batch-to-batch expert-load variance, and equal or better validation accuracy without the auxiliary balancing term.