Axiomatic Theory of Replicating Agents: A Substrate-Independent Mathematical Framework

Axiomatic Theory of Replicating Agents: A Substrate-Independent Framework We introduce a rigorous, substrate-independent mathematical framework that defines the fundamental architecture required for any system—whether biological or artificial—to replicate and adapt in a complex environment. Rather than relying on empirical observation, we construct an axiomatic system based on first principles of dynamical systems and control theory. By defining the core properties of replication, environmental volatility, and viability (survival bounds), we mathematically derive the structural features that any adaptive agent must possess. Our theorems yield the following core findings: Replicating agents cannot be Hamiltonian: Standard conservative physics (Hamiltonian mechanics) dictates that phase-space volume must be preserved over time (Liouville’s Theorem). However, mathematical replication inherently requires phase-volume expansion. Therefore, we prove that any system capable of sustained replication must be governed by dissipative, non-Hamiltonian dynamics. Viability is the physical basis of decision-making: In complex environments, a system cannot survive by remaining in a single stable state. It must possess multiple stable operating regimes (attractors). What we intuitively call "decision-making" is formally proven to be the control-mediated navigation between these attractor basins to prevent the system from crossing fatal viability boundaries. Degeneracy (Redundancy) is a mathematical necessity, not an evolutionary accident: The mapping from a system's internal parameters (its "genotype" or weights) to its observable behavior (its "phenotype") must be many-to-one. If the mapping were perfectly efficient (one-to-one), any mutation or noise would immediately push the system out of the viable parameter space, guaranteeing extinction. Degeneracy creates "neutral networks" that allow the agent to safely mutate and evolve. Adaptation is governed by absolute scaling laws: We translate these architectural necessities into fundamental closed-form equations, introducing the dimensionless Ashby Number. Akin to the Reynolds number in fluid dynamics, the Ashby number formalizes the exact threshold where an agent's internal complexity matches environmental volatility. If an agent's Ashby number drops below 1, the system undergoes a sharp mathematical phase transition resulting in total viability collapse. Ultimately, this theory strips away biological and chemical specifics to reveal the minimal mathematical laws of persistence. It transitions the study of adaptive behavior from a descriptive science to a prescriptive one, providing a formal, reproducible blueprint for both understanding evolutionary dynamics and engineering resilient artificial intelligence.