Observable choice and temporal resolution determine when spectral gaps predict relaxation

This work establishes a general framework for understanding when spectral gaps successfully predict observable relaxation in coarse-grained stochastic dynamics. For any centered observable and leading non-trivial eigenfunction of a reversible transfer operator, the observable autocorrelation decomposes as a weighted mixture of spectral modes. The work identifies two complementary conditions required for successful gap-based relaxation inference. The first condition is geometric alignment between the observable and the leading eigenmode. This is quantified by modal leakage, which measures how much of the observable lies outside the dominant relaxation direction. The second condition is temporal observability of the leading timescale within the fitting window. This is quantified by temporal mismatch, which measures whether the empirical decay rate estimated from the observable is compatible with the leading operator timescale. The theory proves that modal leakage bounds the relative error of gap-based relaxation prediction, while temporal mismatch determines whether the dominant relaxation mode can actually be resolved from finite observations. The framework is developed using reversible transfer operators, total positivity, Gantmacher-Krein oscillation theory, and spectral mixture analysis. Numerical and empirical validation is provided across multiple physically unrelated systems, including Gaussian AR(1) and Ornstein-Uhlenbeck processes, GW150914 gravitational-wave ringdown data, neural surrogate dynamics, resting-state EEG recordings, and financial market time series. The results show that relaxation inference is fundamentally an observability problem: the predictive value of a spectral gap depends not only on the operator itself, but also on what the observable can resolve.