Preface to the Life Emergence Theory Series (Non-Carbon, Multi-Critical, Phase-Locked Catalytic Loop Frameworks)

The study of life’s origin has historically been constrained by a carbon-centric paradigm, in which biological organization is assumed to arise from a privileged class of organic molecules capable of replication and Darwinian evolution. While this framework has achieved substantial explanatory power in describing terrestrial biochemistry, it may not represent the most general formulation of life as a physical phenomenon. The present collection of works develops an alternative perspective: that life is not fundamentally a property of specific chemical substrates, but rather a dynamical regime of matter under sustained non-equilibrium driving forces. In this view, the emergence of life is identified with the formation of stable, self-maintaining structures in reaction networks far from equilibrium, characterized by persistent circulation of energy and information. At the core of this framework is the hypothesis that life corresponds to a multi-critical phase-locked attractor state in catalytic loop networks. Such systems are not defined by static molecular complexity, but by the coexistence of three fundamental dynamical properties: 1. Supercritical growth modes, where the spectral radius of the reaction operator exceeds unity, enabling sustained amplification of reaction fluxes; 2. Phase synchronization across catalytic loops, leading to coherent temporal organization of otherwise independent reaction cycles; 3. Entropy stationarity in loop space, indicating the persistence of structured heterogeneity without collapse into equilibrium or disorder. These conditions are formulated mathematically using reaction-graph representations, nonlinear operator fields, and phase-coupled oscillator dynamics. In particular, hydrothermal Fe–S–Mo systems are investigated as a prototypical physical realization, due to their inherent redox multiplicity, catalytic surface properties, and strong coupling to environmental gradients in temperature, proton activity, and electron chemical potential. Importantly, this framework does not assert that Fe–S–Mo chemistry is uniquely required for life. Rather, it serves as an example of a broader class of multi-element catalytic networks capable of supporting phase-locked self-organization. The theory is therefore intended to be substrate-general, extending in principle to other inorganic systems such as silicate-, oxide-, or transition-metal-based reaction networks. A central methodological shift in this series is the replacement of molecule-centric descriptions with loop-centric and operator-centric formulations. In this representation, the fundamental objects of study are not individual chemical species, but closed catalytic cycles and their interactions as elements of a dynamically evolving graph. Life-like behavior emerges when such cycles become mutually synchronized under multi-critical conditions, forming coherent attractor manifolds in reaction phase space. The purpose of this series is not merely to propose a novel interpretation of prebiotic chemistry, but to develop a mathematically explicit and structurally testable framework for non-carbon life emergence. Each paper in the collection addresses a different layer of this structure: from combinatorial loop decomposition and spectral operator theory, to phase-locking dynamics and multi-critical thermodynamic control. While highly theoretical in nature, the framework is constructed to remain compatible with experimental investigation in hydrothermal reactor systems and related non-equilibrium chemical environments. The long-term objective is to identify whether self-sustaining catalytic phase-locked structures can arise under physically realizable conditions, and if so, to characterize the general principles governing their stability and transformation. In summary, this series advances the hypothesis that: Life is a multi-critical, phase-locked, entropy-stationary attractor of catalytic loop networks driven far from equilibrium. This definition shifts the focus from “what life is made of” to “what dynamical regime life occupies.”