The Continuity Vortex Principle: An Axiomatic Framework for Functional Persistence in Dissipative Systems A Mathematical Theory of Functional Identity, Critical Flow, and Information Continuity

Systems subjected to distinct perturbations can follow markedly different transient trajectories while nevertheless converging to a shared late functional state. This pattern recurs across physical, biological, and computational systems, yet existing descriptions of stability, resilience, relaxation, and attractor dynamics do not isolate it as an explicit, quantifiable object of study. This work proposes a mathematical structure for this problem: the Continuity Vortex Principle, an axiomatic framework describing persistence as the dynamic stability of an organizational pattern sustained by irreversible flows of energy, matter, or information, in which functional identity is associated with the state reached under restoration rather than with the specific path by which restoration proceeds. Four axioms are proposed, establishing functional identity, dissipative persistence, transmissibility, and the dependence of restoration capacity on sustaining flow as necessary conditions for observable continuity. Based on these axioms, a functional state space is constructed together with a continuity functional that measures the preservation of identity through time, and a minimal stochastic dynamical model — an Ornstein–Uhlenbeck process describing perturbations around a reference functional state — yields closed-form predictions, including a critical sustaining flow and an inverse scaling law relating functional deviation to restoration flow. A pre-registered experimental program in artificial neural networks, comprising three independent confirmatory protocols across two perturbation mechanisms and multiple severity levels, tested these predictions directly. The transient recovery dynamics were found to be intervention-dependent and did not exhibit a common monotonic structure; the late functional state, in contrast, consistently remained within pre-specified operational equivalence margins across every protocol in which it was tested. Within the broader CLEO–LOP–COUT framework, this pattern is interpreted as observable continuity being a property of the persistent organized state rather than of the specific trajectory leading to it — the claim best supported by the evidence collected to date, and the central organizing result of this article.