Exact Multi-Population Hyperplane Router

Integrate this at the MoE router immediately before dispatching tokens to two experts. Learn or freeze a projection W\in\mathbb R^{d imes D}, with d=2 or 3, and compute z=W h for hidden state h. For every population i, maintain a reservoir of 32--128 projected tokens and form a convex hull P_i; use an axis-aligned box as the first MVP. Set target fractions alpha_i, such as 0.5 for balanced routing. Enumerate candidate arrangement cells induced by hull edges or faces. For each cell, clip P_i by u^T x\le t, evaluate the resulting polygon or polyhedron volume, and solve C_{P_i}(u,t)=alpha_i vol(P_i) for all i plus u^T u=1. Reject roots violating the cell inequalities. If no exact root exists, minimize the normalized residual sum_i(C_{P_i}(u,t)/vol(P_i)-alpha_i)^2 and select the best candidate. Pseudocode: build P_i; generate cells; solve equations; filter feasible roots; dispatch using r(h)=1[ u^T W h>t]. The paper supplies the cap equations and semialgebraic cell constraints; hull maintenance and root solving are numerical approximations. First test on CIFAR-10 or TinyStories with a two-expert MLP, comparing Top-1 gating plus load loss against this router at equal active FLOPs. Record routing fractions, overflow rate, validation loss, and routing overhead. Success means below 1% population load imbalance without an auxiliary loss and equal-or-lower validation loss.