Effective-resistance natural-gradient routing

Integrate this into the router-logit update, not into the transformer hidden-state path. For each token or token group, let the router produce logits theta in R^d, choose a fixed budget m, and sample S from the external-field law. A practical exact sampler uses dynamic programming for elementary symmetric polynomials: compute E[k][r] for the first k experts with E[0][0]=1 and E[k][r]=E[k-1][r]+exp(theta_k)E[k-1][r-1], then sample the subset backward conditional on the remaining cardinality. For small d, enumerate all m-subsets; for larger d, use the DP conditional probabilities. At each optimizer step, compute inclusion means mu_i=exp(theta_i)e_{m-1}(w_{-i})/e_m(w), variances v_i=mu_i(1-mu_i), and pairwise probabilities P_ij=exp(theta_i+theta_j)e_{m-2}(w_{-i,-j})/e_m(w), giving Sigma_ij=P_ij-mu_i mu_j for i not equal to j. Project the router gradient g with P, solve Sigma u=P g using conjugate gradient restricted to the sum-zero subspace, and use u as the natural-gradient direction. Add the resistance-based trust-region rescaling in the final formula; rho is a tunable maximum pairwise logit movement. Estimate Sigma from exact DP marginals when d is modest, or from a moving-average minibatch covariance for large or hierarchical routers. First test on a 12-expert, top-2 or top-4 CIFAR-100 or WikiText MoE, comparing ordinary softmax/top-m routing with Adam at equal router FLOPs. Measure validation loss, expert-load coefficient of variation, dead-expert fraction, gradient-norm spikes, and loss variance. The expected signal is fewer routing-collapse events and faster loss reduction at equal communication budget, especially when logits become highly imbalanced.