Geometric Binary Gate Solver

Integrate this as a discrete gating layer over N experts, prototype vectors, or binary residual branches. For each input h, let the network produce contribution vectors v_i(h) in R^d, such as v_i(h)=W_i h or a low-dimensional projection of expert outputs. The gate forms s(h)=sum_i b_i v_i and feeds s(h) to the next layer. For the exact two-dimensional variant, use a learned or fixed projection P in R^(2 by d). Compute z_i=P v_i and phi_i=atan2(z_i[1],z_i[0]). Construct all critical angles phi_i-pi/2 and phi_i+pi/2 modulo 2pi, sort them, and take one midpoint from every cyclic interval. At each midpoint theta, compute u=(cos(theta),sin(theta)), set b_i=sign(u dot z_i), evaluate the actual task score or the surrogate norm of the signed sum, and retain the best candidate. Compact pseudocode is: Z=P@V; critical=sort({phi_i-pi/2,phi_i+pi/2}); best=None; for interval in cyclic_intervals(critical): theta=midpoint(interval); b=sign(Z@u(theta)); score=task_loss_or_routing_score(b); best=argmin_or_argmax(best,score). For d greater than 2, sample K normalized directions u_k on the unit sphere, add the normalized current sum and gradient directions, generate b_i^(k)=sign(u_k dot v_i), and evaluate only these candidates. The geometry determines candidate generation; P, K, temperature, and the task-score definition are empirical choices. First test on a synthetic two-dimensional objective with N up to 16 and compare against exhaustive search to verify exact recovery. Then use a CIFAR-10 MLP with 32 binary residual branches or experts, comparing against independent random sign sampling and softmax top-1 routing. Record candidate evaluations, wall-clock routing cost, accuracy, and training stability. Success means exact agreement with exhaustive search in the two-dimensional objective and at least a fourfold reduction in routing evaluations at equal validation accuracy.