A Long-Run Persistence Theory for AI Systems under the Redundancy-Adjusted Artificial Age Score (AAS)

Artificial intelligence systems are increasingly expected to operate over repeated cycles of interaction, adaptation, and update rather than through isolated one-shot outputs. This raises a fundamental theoretical question: can an AI system persist indefinitely without incurring unbounded structural aging? This paper develops a long-run persistence framework for AI systems based on the redundancy-adjusted Artificial Age Score (AAS). The model extends AAS from a static evaluative measure into a cycle-level functional that generates an age sequence across repeated operation. At each cycle, structural age is defined through a weighted, redundancy-aware logarithmic penalty over component consistency levels. Within this framework, cycle-level age is shown to be well defined and uniformly bounded, thereby excluding explosive pointwise aging. On this basis, the paper defines a hierarchy of asymptotic regimes, including burdened persistence, zero-burden persistence, oscillatory persistence, and cumulative terminal burden. It also establishes comparative ordering, sensitivity bounds, convergence under componentwise stabilization, persistence under finite total variation, geometric stabilization under damped inter-cycle perturbations, and a zero-burden characterization under nondegenerate redundancy conditions. The main result is that indefinite cyclic continuation does not require unbounded structural aging: an AI system may pass through infinitely many cycles while its structural age remains bounded, while under stronger regularity conditions its marginal aging vanishes and, in the strongest regime, its cycle-level burden converges to zero. The framework thus provides a formal basis for analyzing long-run artificial persistence as a problem of bounded structural burden rather than inevitable cumulative deterioration.