The Law of Self-Maintained Invariance

This paper presents an axiomatic formulation of the Law of Self-Maintained Invariance, a proposed general principle governing the persistence of identifiable systems in open environments. The formulation begins from four foundational postulates. First, an identifiable system persists as a distinct entity only if at least one organizational invariant remains stable across interactions with its surroundings. Second, open systems undergo continual interactions with their environments that are capable of altering, dispersing, or eroding those organizational invariants. Third, whenever organizational invariants persist despite continual environmental interaction, persistence necessarily requires internally generated processes that continually counteract environmental alteration. Fourth, at the effective dynamical level, the net evolution of organizational invariants is determined by the additive competition between internally generated restoration and environmentally generated disruption. On this basis, the paper defines λ_self as the effective aggregate rate at which internally generated processes reconstruct or preserve organizational invariants inherited from previous system states, and λ_env as the effective aggregate rate at which environmental interactions alter, disperse, or erase those invariants. These effective rates provide a complete coarse-grained description of the competing processes governing organizational persistence. Their dimensionless ratio, R = λ_self / λ_env, is introduced as the canonical measure of the balance between internally maintained organization and environmental disruption. Because ratios of quantities with identical dimensions remain invariant under changes of scale and units, this ratio provides the natural effective comparison between restoration and disruption. From the postulates, it follows that the unique balance point occurs at R = 1. For R 1, organizational invariants undergo net accumulation because restoration exceeds disruption. Above this threshold, organizational invariants inherited from previous system states are reconstructed more rapidly than they are erased. Consequently, accumulated organizational history remains causally active through successive interactions, allowing earlier organizational states to continue constraining later system states and causing persistence to become increasingly determined by accumulated internal history rather than by immediate environmental influence. Because the formulation depends only upon the existence of internally maintained organizational invariants in an open system and not upon any particular material substrate or implementation, the resulting law is substrate-independent. It therefore provides a general formal framework for analyzing self-maintained organization, persistence, historical constraint, and the accumulation of inherited organizational structure across biological, cognitive, artificial, ecological, and other complex dynamical systems.