Spectral Energy of Mori–Zwanzig Commutation Defects in Non-Markovian Model Reduction

This preprint develops a quantitative framework for non‑Markovian model‑reduction error based on the commutation defect in the Mori–Zwanzig formalism. We show that the defect coincides with the initial orthogonal forcing and fully generates the memory term through the orthogonal dynamics. Using the spectral decomposition of the fast fiber generator, we define two coordinate‑independent functionals: the spectral defect energy, which measures the magnitude of the instantaneous non‑Markovian forcing, and the spectral entropy, which quantifies its modal dispersion. We prove that the short‑time growth of the memory kernel is controlled by the spectral energy of the defect, yielding a stable and functional‑analytic characterization of reduction error in multiscale systems. This approach avoids discrete closure hierarchies and provides a geometric description of the structural complexity of memory generation.