Observable modal leakage in coarse-grained stochastic dynamics: a geometric identity and its physical consequences

This work establishes an exact geometric identity governing observable relaxation in reversible stochastic dynamics. For any centered observable f and leading non-trivial eigenfunction psi_1 of a self-adjoint Markov operator, the normalized leading modal weight equals the squared cosine of the angle between f and psi_1. The complementary modal leakage equals the squared sine of the same angle. The manuscript shows that effective one-dimensional relaxation is controlled not by the spectral envelope of the transfer operator, but by the geometric alignment between observables and leading relaxation modes. The theory is developed for Gaussian AR(1) and Ornstein-Uhlenbeck processes using total positivity, Gantmacher-Krein oscillation theory, and Hermite spectral representations. Numerical verification is provided across synthetic systems and coarse-grained physical pipelines, including GW150914 ringdown data and neural-population surrogate dynamics. The manuscript further shows that large fitted spectral envelopes may arise as coarse-graining artifacts and do not by themselves characterize observable relaxation dynamics.