Observable choice and temporal resolution determine when spectral gaps predict relaxation

This work develops a general theory of when spectral gaps successfully predict observable relaxation in coarse-grained stochastic dynamics. The framework shows that observable relaxation depends not only on the operator spectrum, but also on the geometry of the observable and on temporal observability within the fitting window. For reversible transfer operators, the observable autocorrelation decomposes as a weighted spectral mixture. Gap-based relaxation inference succeeds only when two conditions are simultaneously satisfied: (1) geometric alignment between the observable and the leading relaxation mode, quantified by modal leakage, and (2) temporal observability of the leading timescale within the empirical fitting window, quantified by temporal mismatch. The work introduces the observable-relative effective dimension, interpreted as the number of effective relaxation modes induced by the observable on the operator spectrum. The theory is developed using reversible transfer operators, spectral mixture analysis, total positivity, Gantmacher-Krein oscillation theory, and Koopman spectral methods. Empirical and numerical 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 suggest that observable relaxation is fundamentally a problem of measurement geometry: spectral gaps are predictive only when the observable can both align with and resolve the dominant relaxation mode.