Operational Emergence of a Global Phase under Time-Dependent Coupling in Oscillator Networks

Abstract

Time-dependent coupling is often interpreted as introducing competition between a protocol rate and an intrinsic synchronization rate. We show that this interpretation is unavailable in the identical, overdamped Kuramoto model when time dependence enters only through a scalar multiplier of a fixed coupling field. The accumulated coupling $S(t)=\int_0^tK(u)\mathrm{d} u$ is then an exact dynamical clock: equal-exposure protocols traverse the same autonomous orbit, and integrable decays produce finite-exposure arrest rather than deterministic loss of adiabatic tracking. This clock suggests a classification under time reparametrization. Deterministic orbit observables and winding sectors are invariant; an additive perturbation $\varepsilon G$ has a leading response that is a linear functional of the reciprocal schedule $w(s)=1/K(t(s))$ against a kernel fixed by the autonomous orbit; and inertia or noncommuting graph generators lie outside this reciprocal-weighted class. Laboratory-time white noise has the corresponding $w$-weighted covariance. We derive exact mode covariances, prove a universal terminal bang--bang optimum under bounded coupling and fixed exposure, and validate both results in nonlinear networks. We also separate information in the full phase sample from information retained by the order parameter. Conditional observation error of $\arg Z$ follows a $1/(NR^2)$ law with a configuration-dependent prefactor, whereas a known-template experiment has Fisher information $N/\sigma^2$ independently of $R$; the efficiency of $\arg Z$ is $R^2/q_2$. Finally, periodic rings exhibit stable winding sectors with strong local but vanishing global order.