When Wholes Resist Decomposition: A Spectral Measure of Epistemic Emergence

Multi-agent systems often exhibit emergent behavior that appears coordinated, intelligent, and irreducible to the behavior of individual components. Yet quantifying the degree to which such systems form integrated wholes remains a major challenge. While Integrated Information Theory (IIT) was originally developed to explain consciousness, its core concept - measuring how much a system resists decomposition - has broader relevance for understanding informational integration in complex systems. However, the exact computation of IIT’s central quantity, Φ, is intractable for all but the smallest networks. In this paper, we propose a scalable spectral approximation, Φspectral, which estimates system-level irreducibility by applying spectral graph theory to pairwise mutual information networks derived from time-series data. The Fiedler vector of the normalized Laplacian defines a minimal bipartition, across which we sum mutual information to estimate informational disintegration. We interpret this measure as a proxy for epistemic emergence (specifically the diachronic case)—capturing practical, observer-relative irreducibility without invoking ontological claims. We also explore the implications of this metric for measuring emergence and consciousness. We evaluate Φspectral across four classes of simulated systems: random, transitional, and synchronized oscillator networks, as well as a biologically inspired combinatorial threshold-linear network (CTLN) governed by graph-theoretic dynamics. Our results show that Φspectral is sensitive not merely to coordination, but to the emergence of structurally differentiated and functionally integrated behavior. In particular, it detects transient integration in transitional systems and reveals the collapse of complexity in fixed-point–convergent CTLNs. These findings suggest that Φspectral offers a principled and computationally efficient tool for analyzing irreducibility and emergence in distributed systems, with potential applications in AI safety, neural computation, and collective intelligence.