Hermite-fiber MoE router initialization

1. Integration point: apply this during initialization of a top-1 or top-2 MoE router, or before fitting a Gaussian-mixture density head, using a calibration buffer of hidden states h in R^d from a pretrained encoder. Standardize the buffer by subtracting its mean and whitening with a regularized covariance estimate so that the approximate unit-covariance assumption is reasonable. The projected router can replace the full d-dimensional distance computation; retain a residual linear correction only if accuracy drops. 2. Pseudocode: collect n hidden states x_i; compute M2=(1/n) sum_i (x_i x_i^T-I); obtain the top r=min(k,d) eigenvectors v_a by randomized Lanczos; for every a,b compute b_hat_ab=(1/n) sum_i[((v_a^T x_i)(v_b^T x_i))x_i-(v_a^T x_i)v_b-(v_b^T x_i)v_a-(v_a^T v_b)x_i]; form W by QR orthogonalization of all v_a and b_hat_ab; set z_i=W^T x_i; fit k weighted centroids (nu_j,pi_j) with EM or k-means-plus-weighted refinement; initialize router logits as log pi_j-||W^T h-nu_j||^2/(2 sigma^2). 3. Computed from the mathematics: the H2/H3 estimators, eigenspace, contraction range, and projected logits. Estimated empirically: r, sigma, whitening regularization, and whether the residual adapter is needed. 4. First experiment: use a 4- or 8-expert Transformer MoE on a small language-model or CIFAR-100 encoder benchmark, comparing random router initialization, k-means initialization, and this method at identical training FLOPs. Measure early loss reduction, router load balance, expert utilization, final validation accuracy or perplexity, and projection dimension dim(W). A useful success signal is 20-30% lower loss after the first 1-5% of training with no worse final loss, or a 2x reduction in router projection FLOPs at matched validation quality. Use a held-out calibration split to detect overfitting of moment estimates.