Passivity-Certified Softmax Optimizer

1. Integration point: use this module wherever a network currently maintains simplex weights, especially MoE router probabilities, attention mixtures, differentiable architecture choices, or learned convex combinations of experts. Maintain one filter state z per routed token and route; the forward pass uses x=softmax(v), while the backward signal produces g=-gradient with respect to x. 2. Pseudocode: initialize v and z; at each minibatch compute x=softmax(v), loss L, and g=-stop_gradient(grad_x(L)); update z <- z + dt*(A z + B g), then update v <- v + dt*(C z + D g), while optimizing ordinary network weights with Adam. Parameterize A as A=-(R R^T+alpha I) to make it Hurwitz, then solve or check the KYP LMI for P,Q,epsilon. Reject filter parameters if the LMI is infeasible. For a cheap scalar or diagonal implementation use H_i(s)=d_i+k_i/(tau_i s+1), with tau_i>0, k_i>=0, and d_i>=epsilon. 3. Computed from the paper: the cascade, softmax Jacobian, positive-real/passivity condition, and KYP certificate. Estimated empirically: local Jacobians of the loss or game, the finite-frequency minimum eigenvalue of H(j omega)+H(j omega)^*, and the observed growth rate of perturbations. 4. First experiment: train a two-expert MoE MLP on MNIST or CIFAR-10 with direct SGD logits, unconstrained momentum filtering, and the passive filter. Sweep filter gain and timestep. The predicted signature is a sharp stability boundary when m=min_omega lambda_min((H(j omega)+H(j omega)^*)/2) crosses zero: for m>0, router-logit perturbations should decay and routing entropy should settle; for m<0, a suitably coupled contractive quadratic toy loss should show growing oscillations or divergence. In the scalar first-order case H(s)=d+k/(tau s+1), the boundary is k=0 when d=0, and the measured decay rate should decrease continuously as m approaches zero. Compare the empirical boundary with the KYP or frequency estimate within approximately 20 percent.