Human memory consolidation involves the gradual stabilization and reorganization of memory traces over time. Despite numerous empirical and computational accounts emphasizing different aspects of this process, an integrated framework for evaluating consolidation theories against brain data remains limited. We propose a biologically informed, data-driven framework based on Koopman operator analysis to examine latent dynamical structure in fMRI signals associated with memory consolidation. The Koopman framework lifts nonlinear brain dynamics into a linear function space, enabling spectral characterization of persistence and stability. In practice, we employ Dynamic Mode Decomposition (DMD) together with an observability-aware extension tailored to consolidation-related neural dynamics. We organize existing theories into three functional clusters: standard consolidation, episodic replay during rest, and distributed long-term storage, and then align open-access fMRI datasets with each cluster to assess their dynamical plausibility. Across datasets, delayed or repeated retrieval conditions generally tend to show greater spectral persistence than early encoding-related conditions. Among the three analyses, Cluster 3 yielded the clearest statistically reliable subject-level contrast, with semantically abstracted autobiographical content exhibiting higher mean eigenvalue magnitude and more near-unit modes than event-specific episodic content. This finding is compatible with transformation-oriented and distributed-storage accounts but does not constitute a direct temporal test of consolidation. Replay-related conditions show strong spectral differentiation across task states, although part of this separation likely reflects task structure in addition to consolidation-related dynamics. For the standard consolidation cluster, effects are directionally consistent with theory but remain small and not statistically significant at the subject level. Overall, the proposed framework provides an interpretable operator-theoretic approach for linking memory consolidation theory to latent brain dynamics and for comparing competing accounts in a common spectral language.

Nonlinear systems can be lifted to higher-dimensional spaces where their dynamics evolve linearly, enabling linear analysis and control. However, most existing approaches prioritize the linearity of the lifted dynamics and accurate state reconstruction, while overlooking whether the features of interest are actually observable from the measured outputs. To address this limitation, we propose an observability-aware Koopman lifting framework that incorporates a short-horizon observability check alongside standard invariance and reconstruction objectives. The formulation separates roles by employing a linear output map for identifiability and a nonlinear reconstruction map for high-fidelity recovery. Rather than verifying observability post hoc, the proposed method embeds it directly into the training objective to actively shape the learned model. Experiments on benchmark nonlinear systems, namely the Van der Pol and Duffing oscillators, demonstrate that the proposed approach yields better-conditioned lifted-to-output mappings, stronger identifiability, and a more reliable spectral profile than invariance-only and fixed-dictionary baselines, while revealing a trade-off in long-horizon rollout accuracy.