Most AI research concerns itself with what a system can do at a single moment: a benchmark score, a task completion rate, a held-out test accuracy. Far less formal attention has been paid to what happens when a system operates across many cycles of interaction, update, and adaptation over an extended horizon. This paper, available on arXiv, takes that neglected question seriously and attempts to answer it with genuine mathematical rigour. The central claim is both precise and counterintuitive: indefinite cyclic operation does not necessarily imply unbounded structural aging. The paper provides a formal framework under which this can be proved rather than merely asserted.
What Problem Is Being Solved?
The motivating concern is straightforward. As AI systems are deployed in long-running, iterative contexts rather than one-shot settings, a natural question arises: does repeated operation cause a system to accumulate structural burden without limit? The intuitive answer might be yes, by analogy with physical wear. The paper argues this intuition is not formally necessary, and builds a mathematical framework to demonstrate why.
The foundation is the Artificial Age Score (AAS), described here as originating in prior work by Kayadibi (2026). In its static form, AAS is an evaluative snapshot. The contribution of this paper is to extend it into a cycle-level functional, generating an age sequence across repeated operation. Each cycle's structural age is computed via a weighted, redundancy-adjusted logarithmic penalty over component consistency levels. The redundancy correction, expressed through a factor of (1 - R_{n,i}) for each component, is designed to prevent double-counting deterioration when multiple weaknesses share a common underlying cause. This is not a superficial adjustment; it reflects a substantive claim about how organised systems fail.
Key Contributions
The paper establishes several results of increasing strength, structured as a hierarchy of asymptotic regimes:
- Uniform boundedness: The cycle-level age functional is shown to be well-defined, nonnegative, and uniformly bounded across all cycles. Explosive pointwise aging is excluded by construction, not by assumption about the system's behaviour.
- Asymptotic regime hierarchy: Four regimes are formally distinguished: burdened persistence (cycle-level age converges to a positive limit), zero-burden persistence (burden converges to zero), oscillatory persistence (bounded but non-convergent), and cumulative terminal burden (divergent cumulative sum).
- Finite total variation implies convergence: If the age sequence has finite total variation, it converges. This connects the framework to classical real analysis results and provides a tractable sufficient condition for stabilisation.
- Geometric damping: Componentwise perturbations that decay geometrically between cycles produce geometrically stabilised age sequences. This gives a concrete structural condition under which strong persistence holds.
- Zero-burden characterisation: Under nondegenerate redundancy conditions, zero-burden persistence is equivalent to asymptotically perfect consistency across all active components. The strongest regime therefore has a precise componentwise interpretation.
The paper also establishes comparative ordering and sensitivity bounds, which allow the framework to be used for ranking systems and bounding the effect of perturbations in component consistency or weighting.
Methodology and Formal Structure
The approach is axiomatic and analytic rather than empirical. No experiments are reported; the paper is a work of mathematical theory. This is appropriate given its aims, though it does mean the connection to actual deployed systems remains indirect. The formal structure draws on standard tools from real analysis, including convergence theorems, variation arguments, and asymptotic comparison, applied to a novel object: the age functional over an indexed sequence of operational cycles.
The redundancy-adjusted logarithmic penalty is the technical core. Logarithmic penalties are well-motivated in information-theoretic contexts, where they arise naturally in entropy and coding arguments. The redundancy correction is less standard and constitutes one of the paper's more original formal choices. The authors draw an explicit parallel to Shannon's insight that structure matters more than raw quantity in interpreting uncertainty, and to von Neumann's reliability work on systems with redundant components. These are reasonable intellectual ancestors, though the formal connection to either body of work is analogical rather than derived.
The paper situates itself in a broader intellectual tradition running from Turing and McCulloch-Pitts through Ashby's cybernetics and von Neumann's reliability theory. This contextualisation is thoughtful and not merely decorative. The framing of repeated operation as a cybernetic problem, concerned with the evolution of structural burden rather than isolated success events, is genuinely useful for understanding what the theory is trying to do.
Limitations and Open Questions
The paper is careful about its scope and this care deserves acknowledgement. It explicitly does not claim to provide a theory of intelligence, cognition, or semantics. The framework applies to structural burden under formal assumptions on consistency, redundancy, and weighting; what these quantities correspond to in a real system is left largely unspecified.
This is also the framework's most significant limitation. The cycle-level age functional is mathematically clean, but the mapping from its components to observable properties of an actual AI system is not addressed. What counts as a component? How is consistency measured? How is redundancy estimated between components? These are not trivial questions, and without answers the framework cannot be applied, only interpreted. A reader looking for guidance on how to operationalise AAS for a real system will not find it here.
There is also a question about the AAS itself. The framework is built on top of a prior construct (Kayadibi, 2026) that is not fully derived within this paper. Readers unfamiliar with that prior work will need to take the foundational score partly on trust. The logarithmic penalty form is plausible but not uniquely motivated; it would be worth understanding how sensitive the asymptotic results are to the choice of penalty function.
The regime hierarchy is formally clean but the boundaries between regimes may be difficult to diagnose empirically. Distinguishing burdened persistence from oscillatory persistence, or determining whether a system is approaching zero-burden persistence, requires knowledge of the asymptotic behaviour of the age sequence, which is not observable in finite time. The paper does not address how one would infer regime membership from finite operational data.
These gaps point toward natural extensions. Connecting the formal framework to measurable system properties, developing estimation methods for AAS components, and studying robustness of regime classification under model misspecification would all substantially increase the practical reach of the theory.
Despite these open questions, the paper makes a genuine contribution. It demonstrates formally that indefinite cyclic operation and bounded structural aging are mathematically compatible, replaces a binary working-versus-failing picture with a graded hierarchy of asymptotic burden profiles, and provides a set of sufficient conditions for the strongest persistence regime. For researchers working on the theoretical foundations of long-run AI operation, this framework offers a precise vocabulary and a set of proved results to build on. The full paper is available at https://arxiv.org/abs/2608.04012.