Dynamic Closure Theory: A Minimal Variational Law for Coherence-Based Self-Consistency

Critical transitions—abrupt, system-wide reorganizations triggered by smooth parameter changes—pervade nature from molecular folding to cognitive systems. We introduce Dynamic Closure Theory (DCT), a minimal framework grounded in three physical postulates: (1) systems evolve via coherence fields, (2) transitions occur when accumulated energy satisfies a functional closure condition, and (3) the closure dynamic follows a logistic evolution. From these, we derive a universal scaling law predicting an inverse coupling between effective structural density and critical field strength: Phic is proportional to rhoₑff^ (-1). Quantitative validation using a benchmark of proteins covering all major SCOP fold types, and explicitly accounting for literature-reported variance, yields an experimental exponent of sₑxp = -0. 96 (6) (R² = 0. 975). This result is statistically indistinguishable from the theoretical prediction of -1. 0. At the systems level, anesthesia-induced consciousness loss exhibits a metabolic threshold of 42 +/- 3%, remarkably consistent with DCT's half-closure criterion C* approx 0. 5. These results span ten orders of magnitude in spatial scale and fundamentally different substrates, suggesting that DCT captures a fundamental topological coherence constraint governing conformational stability across scales. Significance Statement This work introduces Dynamic Closure Theory (DCT), a novel theoretical framework that provides a unified thermodynamic explanation for critical transitions across disparate biological and physical scales. While traditional models of phase transitions often rely on system-specific parameters, DCT derives a universal inverse scaling law between effective structural density (rhoₑff) and critical field strength (Phic). The originality of this contribution lies in the derivation of the Dynamic Closure Equation (DCE) from a minimal set of physical postulates, bridging the gap between molecular thermodynamics and systems-level emergence. We provide rigorous quantitative validation across ten orders of magnitude, demonstrating that the same topological coherence constraints govern the stability of protein manifolds (where we find an experimental scaling exponent of s approx -0. 96 +/- 0. 06) and the metabolic thresholds of human consciousness during anesthesia. By identifying a universal half-closure criterion (C* approx 0. 5) as the tipping point for system-wide reorganization, this research advances the field of complex systems by offering a predictive, substrate-independent tool for auditing structural stability. This framework has immediate implications for protein engineering, AI interpretability, and the study of collective dynamics in neural and social networks. Updated with refined mapping parameters and extended mathematical appendix for enhanced theoretical consistency Aspects of this theoretical framework are subject to pending patent applications by Yunaverse, Inc.