Intrinsic-Volume Router Regularizer

Integrate the regularizer into a bias-free top-1 or top-2 MoE router. Let the router produce logits r_e(x)=w_e^T x for feature x in R^n; expert e receives inputs in the polyhedral cone C_e={x:(w_e-w_j)^T x >= 0 for every j}. During training, periodically estimate the intrinsic-volume vector of each C_e from Gaussian probes. Sample g_s~N(0,I_n), compute the winning expert e=argmax_j w_j^T g_s, and assign the probe to that expert. For each assigned probe, determine its exposed face by evaluating the router's near-tied constraints; use the rank of the active constraint matrix A_active to estimate the face dimension k=n-rank(A_active), then increment histogram h_e[k]. Normalize with \hat v_{e,k}=(h_e[k]+alpha)/(sum_q h_e[q]+(n+1)alpha), using alpha=1 or 5 for smoothing. Every T=100 to 500 optimizer steps, add lambda times the sum over experts of L_IV(C_e) to the task and load-balancing losses. If differentiable projection is unavailable, use stop-gradient through the histogram and update the router with a moving-average penalty; a more advanced version can use a differentiable quadratic-program projection. Compute rho_k exactly using log omega_j=(j/2)log(pi)-lgamma(j/2+1). The theorem supplies the target inequality; the empirical ingredients are the Gaussian-projection estimator and active-face rank estimate. First test on a 4- or 8-expert MLP-MoE on CIFAR-100 or a small language-model routing benchmark, comparing standard load-balancing loss with load-balancing plus this penalty at equal FLOPs. Track expert utilization entropy, dead-expert fraction, router gradient variance, validation accuracy, and loss spikes. Success means fewer dead or oscillating experts and equal-or-better validation loss without increased dispatch imbalance; failure means unchanged routing behavior or degraded accuracy at matched utilization.