Granularity-Aware Feasible Routing

1. Integration point: apply this to a capacity-constrained mixture-of-experts router or an RL policy with a continuous gate allocation. For each input, let the router produce nonnegative expert loads $a_e$ whose sum is at most total token capacity $P$. Choose a token or capacity quantum $\delta$, and use the rounded allocation for dispatch. 2. Pseudocode: compute router scores; convert scores into nonnegative allocations $a_e$; optionally project them onto $\sum_e a_e\le P$; set $\hat a_e=\delta\lfloor a_e/\delta\rfloor$; dispatch at most $\hat a_e$ tokens to expert $e$; compute task loss and optionally penalize unused capacity $P-\sum_e\hat a_e$; backpropagate through $a_e$ with a straight-through estimator. During training, replace the hard floor in the backward pass by the identity or a clipped sigmoid surrogate. 3. Compute $\rho$, rounded capacity, and unused capacity exactly. Estimate $M$ empirically from validation perturbations as the largest loss reduction per additional assigned token or capacity unit. 4. First cheap experiment: compare continuous routing, nearest rounding, and downward rounding in a small Switch-style MoE on CIFAR-100 or a synthetic load-balancing task. Sweep $\rho\in\{0.01,0.02,0.05,0.1,0.2,0.5\}$ while scaling $P$ and $\delta$ together. The prediction is that normalized performance curves approximately collapse for equal $\rho$, downward rounding never violates capacity, and the task-loss gap remains below roughly $M\rho P$. Utilization should show sawtooth discontinuities when $1/\rho$ crosses an integer, with increased unused capacity near ratios just above $1/k$.