Cumulative-Fair MoE Capacity Envelopes

Integrate the allocator immediately before top-k MoE routing. At the start of every window of B minibatches, estimate unconstrained expert demand d[e] from router probabilities and compute a technical envelope q_tech[e] using current expert memory and latency limits: q_tech[e]=min(capacity_e, floor(C*available_compute_e/sum_e available_compute_e)). Maintain cumulative demand D[e] and accepted tokens S[e] across windows. Solve the small quadratic projection problem for q_fair with projected gradient descent or a differentiable bisection multiplier: initialize q=q_tech; repeatedly update q toward the Jain or disparity objective; clip q to [0,q_tech]; project onto sum(q)<=C; and reject updates violating the efficiency budget Delta. During routing, assign at most q_fair[e] tokens to expert e; overflow tokens go to their next-best expert. After routing, update S using accepted assignments and measure routing loss, overflow, expert latency, and total FLOPs. The source-derived quantities are the Jain index, cumulative ratios, and two-stage feasibility/fairness construction. Estimate q_tech, the loss functions ell, and Delta empirically from expert latency, memory, overflow, and validation loss. First test a small Switch-Transformer or Mixtral-style model on C4, comparing standard top-k plus auxiliary load-balancing loss against this allocator at equal total FLOPs. Sweep rho and Delta. The quantitative prediction is that J_t rises toward one as rho is relaxed or Delta increases, while validation-loss or throughput degradation stays below Delta. When Delta is reduced below the minimum cost of equalization, the fairness constraint becomes active and produces a visible kink in J versus Delta. The allocator should make no change when the technical ratios already satisfy the disparity bound.