Collective phase leakage from renewing reciprocal noise partners

Equal and opposite random torques conserve the sum of the unwrapped rotor angles, yet the collective direction can still fluctuate. We study a population of N globally coupled rotors whose noise partners are exchanged at random. Each rotor receives Brownian forcing, and the full two-point covariance of that forcing is independent of the exchange rate. In our pair model with a fixed ordered mean field, opposite kicks cancel in the phase projection when the two partners have equal phase sensitivities. The mean squared mismatch decreases as the pair ages, and exchange brings together rotors with uncorrelated sensitivities. For rare exchanges the projected phase variance rate is proportional to the exchange rate, with a coefficient that scales as 4πe−2κ log κ/N at large ordering strength κ. Near the ordering threshold, an explicit crossover relates fast renewal to the initial decay in the limiting sector dynamics and slow renewal to the integrated mismatch. In simulations of 512 interacting rotors, slower exchange increases orientational coherence at both reported lags. At the primary lag, the 95% confidence interval for the change in this coherence difference on halving the time step lies within the prespecified tolerance.