Unified linear fluctuation-response theory arbitrarily far from equilibrium

Understanding how systems respond to external perturbations is a fundamental challenge in physics, particularly for nonequilibrium and nonstationary processes. The fluctuation-dissipation theorem provides a complete framework for near-equilibrium systems and various bounds have recently been reported for specific nonequilibrium regimes. Here, we present a compact, trajectory-score-based formulation that synthesizes and generalizes linear fluctuation-response relations for arbitrary Markov processes, decomposing system response into spatial correlations of local dynamical events. This decomposition reveals that response properties are encoded in correlations between transitions and dwelling times across the network, providing a natural generalization of the fluctuation-dissipation theorem and recently developed nonequilibrium linear response relations. Our theory unifies existing response bounds, extends them to time-dependent processes, and reveals fundamental monotonicity properties of the tightness of multiparameter response inequalities. Beyond its theoretical significance, this framework enables efficient numerical evaluation of response properties from sampling unperturbed trajectories, offering significant advantages over traditional finite-difference approaches for estimating response properties of complex networks and biological systems far from equilibrium.