Random-cluster anti-correlated routing

Integrate this at the branch-selection tensor of a small tree-MoE or hierarchical MLP. Given an input x, compute one feature vector per branch expert, but multiply expert outputs by a sampled connectivity gate B_j. Build a finite rooted d-ary tree of depth L; identify all leaves as a wired boundary, assign each tree edge an open/closed variable, and sample an edge configuration from the random-cluster law. For an MVP, use depth L=3 and enumerate all edge masks for very small trees, computing each mask weight from the displayed \(p^{|\omega|}(1-p)^{|E|-|\omega|}q^{\kappa(\omega)}\); for larger trees, use Gibbs sweeps with the exact local update: an edge is open with probability p if its endpoints are already connected after deleting that edge, and with probability \(\widehat p=p/[p+q(1-p)]\) otherwise. Normalize the selected expert sum by \(\max(1,\sum_j B_j)\), and use several Monte Carlo masks per minibatch or a straight-through Bernoulli relaxation. Initialize p near but below \(p_{\mathsf c}\) for sparse routing, then optionally learn a sigmoid parameter p while constraining q\in[0.1,0.9]. Compute from the mathematics the threshold \(p_{\mathsf c}\), the fixed-point prediction r* by one-dimensional bisection on \(p=\psi(r)\), and the expected activity from sampled masks; estimate covariance empirically over minibatches. The first experiment should compare this router against independent Bernoulli dropout and standard noisy top-k routing on CIFAR-100 with a 3-level 8-branch hierarchical MLP at matched expected active experts. Measure validation accuracy, expert-output cosine redundancy, pairwise covariance of branch activations, and seed-to-seed variance. Success means covariance is negative as intended and the model achieves equal or better accuracy with lower redundancy and no increase in wall-clock cost; a stronger result is improved accuracy at the same expected number of active experts.