Adaptive Quine Structures for Metacognitive Evolution in Large Language Models: A Functional Framework with Gödelian Bounds and Illustrative Applications

How can an LLM-based agent recognize the limits of its own self-knowledge and improve that self-knowledge over time? This paper proposes the metacognitive evolutionary system (MES), a functional framework for studying metacognitive evolution at the prompt level in large language model (LLM)-based agents. MES does not modify model weights; it evolves the prompt program around a fixed base model, keeping the system readable, auditable, and easier for humans to inspect. The framework introduces a recursive metacognitive tower, Mn(P)=LLM(⟨Mn−1(P)⟩), to model layered self-evaluation. The fixed-point behavior of this tower is interpreted as a Strange Loop, with a formal analogy to Gödelian incompleteness used to describe its epistemic limits. The system is built using five primitive functions: inference, grounding, awareness, adaptive self-replication, and population-level selection. A grounded fitness function guides prompt evolution across generations, evaluating uncertainty calibration, error detection, strategy adaptation, and epistemic boundedness. Through Quine-style prompt rewriting, MES studies how agents can revise their own prompt-level structure while remaining constrained by grounded evaluation. The paper presents the formal architecture, analyzes tower dynamics using Markov chains, discusses convergence results, identifies six application domains, and proposes QuineBench as an evaluation design. Numerical case studies serve as illustrative analytical examples rather than empirical experiments, and QuineBench is presented as a structured protocol for future validation, providing a theoretical foundation and a clear path toward empirical evaluation.