When Wholes Resist Decomposition: A Spectral Measure of Epistemic Emergence.

Multi-agent and distributed dynamical systems can exhibit coordinated behavior that is difficult to summarize in terms of independent parts. Integrated Information Theory (IIT) provides one influential notion of system-level irreducibility, but exact computation of causal Φ remains intractable except in very small systems. In this work, we introduce Φspectral, a scalable observer-relative statistic defined on pairwise mutual information networks extracted from multivariate time-series data. A normalized graph Laplacian and its Fiedler vector identify a bipartition of the mutual information graph, and Φspectral reports the normalized weight of informational coupling crossing that cut. The measure is inspired by IIT's concern with irreducibility but is not equivalent to intrinsic causal Φ: it is pairwise, undirected, and functional/statistical rather than intervention-based. We evaluate it on four exploratory simulation regimes: random oscillators, a transitional Kuramoto-like synchronization regime, a perfectly synchronized regime, and a combinatorial threshold-linear network (CTLN). Across these cases, Φspectral is most useful as a measure of observer-relative integration under second-order dependencies, separating redundancy-dominated from transiently differentiated regimes. The current results should be read as a proof-of-concept rather than as a formal validation against exact IIT. We discuss relations to weak IIT, Integrated World Modeling Theory (IWMT), and the perturbational complexity index (PCI), and we outline the stationary benchmarking and small-system validation needed for stronger causal claims.