Risk-budgeted MoE capacity reservations

Integrate this at the MoE router's capacity-management operation, not in the expert MLP weights. For each training step or serving window, record the number of tokens routed to expert e, X_e, before capacity truncation. Maintain exponentially weighted statistics: mu_e <- beta mu_e + (1-beta)X_e and v_e <- beta v_e + (1-beta)(X_e-mu_e)^2, with sigma_e=sqrt(v_e+epsilon). Choose a menu T={0.01,0.02,0.05,0.10}. For every alpha in T compute the candidate capacity c_e(alpha)=ceil(mu_e+sigma_e*sqrt((1-alpha)/alpha)), then select alpha_e values under the available total device-token budget. For small expert counts, enumerate menu combinations; otherwise initialize every expert at alpha=0.10 and repeatedly upgrade the expert with the largest estimated reduction in overflow per additional reliability cost, until sum_e -log(1-alpha_e) reaches B. During routing, use c_e(alpha_e) as the expert capacity; overflow tokens use the existing second-choice or residual fallback. The mathematical quantities are the Cantelli reservation, menu costs, and aggregate budget; mu_e, sigma_e, B, and the Wasserstein radius rho are estimated or tuned empirically. First test an 8-expert top-1 MoE on synthetic bursty routing plus CIFAR-100 or a small language-model benchmark, against a fixed global capacity factor and mean-plus-fixed-buffer capacities. Measure dropped-token fraction, padding waste, step latency, load imbalance, and validation loss at equal total capacity. Success means lower overflow at equal capacity, or at least 20% lower reserved capacity at the same overflow rate without validation-loss degradation.