An Irreducible Dynamical Grammar for Minimal Living Systems: Synthesizing Non-Equilibrium Thermodynamics, Autopoietic Closure, Biosemiotics, and Grounded Heredity

Abstract

An Irreducible Dynamical Grammar for Minimal Living Systems: Synthesizing Non-Equilibrium Thermodynamics, Autopoietic Closure, Biosemiotics, and Grounded Heredity Overview:This theoretical preprint establishes a unified, irreducible, and linear-algebraic dynamical model of minimal life. Historically, theoretical biology has been divided between physiological/autopoietic paradigms (metabolic closure, homeostasis, far-from-equilibrium dissipation) and informational/evolutionary paradigms (Darwinian evolution, tape copying). This work unifies both traditions into a single, computationally executable mathematical grammar. Core Mathematical Architecture:An organism is modeled as an n-dimensional physical state vector V ∈ S bound within a compact Viability Kernel (V). The minimal living state is governed by five tightly coupled operators: Open Dissipative Exchange (e ο V): Thermodynamic coupling supplying free-energy influx and negative entropy export. Autopoietic Self-Repair (∇V): Active metabolic gradient counteracting spontaneous thermodynamic structural degradation. Biosemiotic Allostatic Control (z + V* → δ): Triadic sign evaluation against an internal allostatic reference core, generating homeostatic boundary-avoidance actions without infinite regulatory regress. Grounded Hereditary Tape (T): Physically instantiated molecular memory satisfying semantic closure (translation into catalysts and replication into daughter tapes). Mutable Replication (M → V + (V ⊕ ε)): Irreversible division yielding offspring with stochastic variation for Darwinian natural selection. Physical and Thermodynamic Grounding:The algebraic grammar is strictly constrained by three foundational physical axioms: 1st Law Continuity: Mass and energy conservation with the environmental reservoir. 2nd Law Dissipation: Strictly positive internal entropy production (σ > 0) and non-equilibrium entropy export. Information-Thermodynamic Limits: Finite free-energy dissipation bounds for decision-making and proofread tape replication (Landauer's principle). Key Theoretical Findings: Proof of Irreducibility (Knockout Analysis): A rigorous component-knockout matrix proves that removing any single operator collapses the entity into a recognized non-living state (thermal equilibrium, passive wear-and-tear, passive dissipative waves, sterile transient chemistry, or rigid crystal formation). Linear-Algebraic Accessibility: By formulating the theory using standard vector spaces, the ontology of life is made directly accessible to numerical computing, GPU vectorization, and agent-based simulation. Scientific Implications: Synthetic Biology: Provides the minimal operational closure criteria for engineering autonomous bottom-up protocells. Astrobiology: Establishes a substrate-neutral, agnostic biosignature metric based on non-equilibrium semiotic dynamics rather than Earth-specific chemistry. Artificial Intelligence: Solves the sensorimotor and symbol grounding problems by embedding cognitive agents inside existential allostatic viability constraints. Theoretical Oncology: Formulates neoplastic transformation (cancer) as an allostatic decoupling event between cellular replication and host viability. Note: This manuscript presents the foundational mathematical and physical theory. Numerical and experimental agent-based simulations validating the framework are included in accompanying releases.