Topological trapping in circular midpoint opinion dynamics

We study a discrete-time asynchronous midpoint dynamics on the circle in which, at each step, a uniformly chosen neighboring pair moves to the midpoint along the shortest arc. Although the update rule is locally contractive, we show that the global relaxation mechanism depends sharply on the boundary topology. Under open boundary conditions the system converges almost surely to consensus through pure contraction. Under periodic boundary conditions the graph contains a single cycle, and the wrapped edge increments define an integer-valued winding number. While consensus remains the unique absorbing state for every fixed system size, we show that topology profoundly reshapes the transient dynamics. We prove that branch–crossings are the only mechanism capable of modifying the winding number and compute explicitly their probability for disordered initial data. Local averaging rapidly suppresses large gradients and drives the system into a no-branch-crossing regime where the winding number freezes. Inside a fixed winding sector we construct an adaptive co-moving frame in which the dynamics becomes an exact Euclidean midpoint process and establish strict contraction in a co–moving frame toward a twisted linear profile determined by the winding number. Our results isolate a minimal mechanism by which a single cycle induces sector locking and escape, even though the final equilibrium remains unchanged.