A Mathematical Framework of a New Paradigm of Autonomous Organization: Constraint Emergence as the Origin of Life

Traditional theories on the origin of life have focused on the spontaneous emergence of specific molecular subsystems such as replicating RNA, metabolic cycles, or membrane compartments. However, these approaches often fail to capture the systemic, organizational properties essential to autonomy and evolution. Here, we propose the Constraint Emergence Hypothesis: life began with the spontaneous formation of self-sustaining networks of functional constraints that regulate chemical and energetic flows. We develop a comprehensive mathematical framework combining coupled reaction-diffusion equations, dynamic constraint field evolution, feedback-modulated transport processes, and phase space analysis. In three-dimensional spatial domains, we demonstrate how organizational closure naturally emerges as a dynamical phase transition, leading to localized proto-individuals capable of persistence and evolution. Our results suggest that life’s origin is best understood as an emergent property of non-equilibrium organizational dynamics, rather than the outcome of isolated molecular events. This framework opens new pathways for both theoretical modeling and experimental investigation of minimal autonomous systems.