Self-Referential Introspection in Large Language Models: The Critical Threshold for Recursive Self-Improvement.

The pursuit of self-evolving AI raises a critical question: when is autonomous self-improvement sustainable rather than degenerative? Drawing an analogy to von Neumann's complexity threshold for self-reproducing automata, we argue that sustainable recursive self-improvement in large language models (LLMs) requires a functional analogue: introspection-the system's capacity to simulate its own operations and target modifications. Grounded in Kleene's Second Recursion Theorem, we construct such introspective self-improvement programs and prove their key properties: completeness of self-modification, necessity of the reflective architecture, undecidability of improvement in general, and equivalence with Schmidhuber's Gödel machine under a rewrite-equivalence notion, which transfers the global optimality guarantee. An empirical review, organized around these functional criteria, suggests that current LLMs exhibit only quasi-introspection.The available evidence does not establish complete introspection in the formal sense developed here, while pointing to several candidate structural bottlenecks, including incomplete self-access, feedforward processing, and limited computational depth. We outline architectural paths toward the threshold and discuss the safety implications of crossing it.