Perron-Weighted Cluster Consensus Optimizer

Integrate this as a communication step between local optimizer updates for parallel replicas, MoE experts, or independently initialized transformer heads. First choose a partition of N replicas into K clusters. Construct a sparse directed nonnegative matrix Omega with the desired edges, then numerically solve for its Perron eigenpair (lambda,v); rescale edge weights or use constrained projected optimization so that entries of v are equal within each cluster. At every iteration, compute local gradients, perform the usual AdamW or SGD update, then stack parameters theta_i and apply theta <- Q_epsilon theta. To prevent cross-cluster collapse, use a block-diagonal Omega in the first experiment, or add only weak edges between cluster leaders. Monitor within-cluster disagreement D_k(t)=mean_{i in k}||theta_i-theta_bar_k||^2 and inter-cluster distance. Estimate lambda and non-Perron eigenvalues with a sparse eigensolver; gradient-noise statistics are empirical. Start with four ResNet-18 replicas on CIFAR-10, divided into two clusters, comparing independent SGD, global averaging, and this method. Sweep epsilon. The falsifiable prediction is a sharp instability boundary when rho_perp(Q_epsilon) crosses one, with D_k(t) approximately proportional to rho_perp(Q_epsilon)^{2t}; the measured stable epsilon interval should match the spectral prediction within 20%. Within-cluster disagreement should decrease while inter-cluster parameter distance remains nonzero.