Grassmannian Tropical Router

Integrate this module at the router-logit stage of a small top-1 or top-2 mixture-of-experts Transformer. Given token representation h in R^d, use three learned scalar projections x(h)=u_x^T h, y(h)=u_y^T h, and t(h)=u_t^T h, optionally normalized with LayerNorm. Choose M ordered trainable parameters psi_1<...<psi_M by parameterizing psi_j=psi_min+softplus(cumsum_j(delta)) and clipping them to an interval shorter than pi/2 to avoid degeneracies. Enumerate all N-subsets K of {1,...,M}. Precompute c_K=sum_{k in K} sin(psi_k), d_K=sum_{k in K} cos(psi_k), and e_K=sum_{k in K} sin(2 psi_k), then compute s_K=2*x*c_K+2*y*d_K-2*t*e_K+b_K, where b_K is a learned log routing weight. During training, use softmax(s/tau) or noisy softmax for load balancing; anneal tau from 1.0 to 0.1, and use argmax/top-k at inference. For a strict Grassmannian prior, initialize b_K=0 and use a Vandermonde positive matrix A_{rj}=lambda_j^(r-1), with 0<lambda_1<...<lambda_M, then set b_K=alpha*log(det(A[:,K])+epsilon), where det(A[:,K]) is the Plucker minor. The mathematical quantities are the subset phase coefficients and boundary geometry; empirical quantities are the projections, psi values, and expert losses. Add an auxiliary loss penalizing router load imbalance and, optionally, a locality penalty that samples token pairs near a predicted phase boundary and encourages only subsets differing by one index to compete. First test on a 4-expert, 2-layer Transformer trained on WikiText-103 or CIFAR-10 sequence modeling, comparing a standard linear top-1 router at matched parameters and FLOPs. Measure validation perplexity or accuracy, expert-load entropy, number of disconnected routing components, and loss variance. Success means equal-or-better validation quality with lower routing collapse and fewer expert switches for neighboring token embeddings; a stronger result is improved quality at the same active-expert FLOPs.