Coset-aware MoE routing repair

Integrate this immediately after the ordinary router chooses top-1 or top-2 experts and before dispatch. Use microbatches of B tokens and let c[j] be the number assigned to expert j. Start with a practical lattice: partition the E experts into groups G_1,...,G_g and require each group total to be divisible by r, or require each expert count to lie in a target residue class modulo m; encode these linear congruences as A c = b (mod m). This is the finite-coset version of c belonging to L+a. Compute residues from the current c. If the target coset is violated, select at most K low-confidence tokens whose second-best experts are available, and greedily apply swaps token t: expert i to expert j, updating c by e_j-e_i. Choose the swap with largest routing-margin loss per unit residue correction, where the margin is the difference between the chosen and alternative gate logits. Stop when A c=b mod m or K swaps are exhausted. In pseudocode: logits=g(x); choices=topk(logits); c=count(choices); while residue(Ac-b)!=0 and swaps<K: enumerate feasible alternatives; choose argmin(delta_margin / residue_progress); swap; update c. The paper supplies the index-vector, lattice, and quotient formulation; the engineer estimates the appropriate lattice empirically by measuring recurring count residues and selecting m or group constraints that correlate with overflow. The first experiment should use a 12-expert Switch-Transformer-style layer on CIFAR-100 or a small language-model corpus, comparing standard load-balancing loss against the same router plus repair at equal capacity factor. Measure overflow/drop rate, coefficient of variation of expert loads, auxiliary-loss contribution, training stability, and validation accuracy. A successful signal is substantially fewer dropped tokens and less batch-to-batch load variance with no meaningful increase in routing cost or degradation in perplexity.