Orthogonal Latin Expert Router

Integrate this at the MoE router immediately before top-1 or top-2 expert selection. Choose E=n experts and process each microbatch in chunks of n squared tokens; map token position t to grid coordinates (i,j). Precompute h mutually orthogonal Latin squares L_s, or use L_a(i,j)=((i-1)+a(j-1)) modulo n plus 1 for distinct slopes when n is prime. For token t=(i,j), form a candidate set C_t from the labels L_s(i,j) over a small subset S_t of routing families. The router network emits logits z_t,u for candidate experts only. Maintain an exponential moving average qhat_u of the number of tokens sent to expert u; select argmax of z_t,u minus tau times log(qhat_u plus epsilon), then dispatch by expert. For top-2 routing, choose the best two distinct candidates and renormalize their softmax weights. The Latin tables and their balance and orthogonality guarantees are exact; qhat_u, logits, and temperature tau are empirical. First test a 4-expert or 8-expert Switch-style MLP on a small language-model benchmark such as WikiText-2, comparing standard noisy top-1 routing, random hash routing, and this candidate-constrained router at equal expert FLOPs. Record dropped-token fraction, coefficient of variation of expert loads, auxiliary load-balancing loss, training loss at fixed steps, and all-reduce or dispatch time. Success means near-zero overflow or dropped tokens, lower load variance, and equal or better perplexity without increasing router FLOPs; a useful systems signal is reduced padding in expert GEMMs and faster step time at the same parameter count.