Sliding Mode Control for Adaptive Kuramoto Synchronization on Hyperbolic Graphs: Preventing Monolithic Collapse in Self-Organizing Coherent Networks

SUMMARYWe study adaptive Kuramoto networks in which pairwise coupling weights evolve under anonline Hebbian-like rule driven by local phase coherence, a rule previously proposed inGeneral Coherent Field Theory (G-CFT) for distributed cognitive swarms. We showanalytically (sketch proofs) and confirm numerically that this rule induces a positivefeedback loop that generically drives the network to a monolithic collapse: a singleglobally synchronized cluster with order parameter R -> 1 and vanishing phase entropy.Naive local remedies (weight decay, amplitude capping, pairwise anti-Hebbian correction)fail. We then close the loop with a sliding mode controller (SMC) whose sliding surface isthe entropy window of G-CFT expressed through the global order parameter R; the controllersign-switches the effective learning rate. With the controller active from initialization,monolithic collapse is prevented in all tested configurations (18-point sweep, N = 60,seed 1) and the network retains a two-cluster structure; for a tuned configuration(alpha_max = 0.5, sigma = 1.0, gamma = 0.6) the steady state lies inside the admissibleentropy window, = 0.366, with clusters (41, 19). All tables and figures are producedfrom raw simulation output. CONTENTS OF THIS RECORD- Paper PDF (version 2, October 2026)- Simulation scripts (protocols P1, P2) and figure-generation scripts- Raw simulation output used for Tables 1-3 and Figures 1-5