Recursive Emergence Across Scales: Quantifying Intelligence and Consciousness via Ψ and Emergent Recursive Expression

This paper presents a unified framework for quantifying recursive complexity and consciousness across physical, biological, and cognitive systems. Building on the Reaction to Reflection (R2R) model of recursive evolution, we introduce two metrics—Emergent Recursive Expression (ERE) and the Recursive Conscious Phase Index (Ψ)—to capture the structural constraints and dynamic activation of recursive intelligence. ERE defines a system’s capacity to sustain recursive depth given its energy expenditure and modular complexity, while Ψ quantifies phase transitions in recursive activity over time. We show that consciousness is not a static property but a thermodynamic phase state gated by recursive depth, energy availability, and modular architecture. Applying Ψ to empirical sleep EEG data reveals discrete phase transitions aligned with sleep stages, including rare excursions into high-Ψ states consistent with reflexive, conscious processing. ERE, meanwhile, maps system-level viability across a wide range of recursive agents, from molecules to artificial neural networks. Together, Ψ and ERE provide a cross-scale framework for analyzing intelligence, adaptation, and consciousness as emergent properties of recursive thermodynamic organization.