Self-Reference in Large Language Models: The Introspection Threshold for Recursive Self-Improvement

2026-07-05

AI Summary

This study investigates the conditions under which large language models can achieve sustainable recursive self-improvement without undergoing degenerative evolution. Inspired by von Neumann’s complexity threshold in self-replicating automata, the work introduces Kleene’s Second Recursion Theorem into this domain for the first time and proposes the concept of an “introspective threshold”: a model must possess the capacity to fully simulate and deliberately modify itself. By integrating recursive function theory, architectural analysis of Transformers, and metacognitive evaluation, the paper theoretically establishes the existence of introspective programs. Empirical findings indicate that current models exhibit only quasi-introspective capabilities, constrained by incomplete self-access, feedforward architecture, and computational class limitations, thereby failing to surpass the introspective threshold.

Abstract

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 demonstrate the theoretical existence of such introspective programs. However, an empirical review reveals that while current LLMs exhibit quasi-introspection (e.g., partial metacognition), they fall short of true introspection due to structural bottlenecks: a lack of complete self-access, the feedforward nature of the Transformer, and computational class constraints that prevent fixed-point iteration. We conclude by outlining architectural paths to cross this complexity threshold and discussing the associated safety implications.

Problem

• self-reference
• introspection
• recursive self-improvement
• Large Language Models
• self-evolving AI

Innovation

• introspection
• recursive self-improvement
• Large Language Models
• Kleene's Second Recursion Theorem
• self-reference