SELF-REFERENCE IN LARGE LANGUAGE MODELS: THE INTROSPECTION THRESHOLD FOR RECURSIVE SELF-IMPROVEMENT

Modern AI development has shifted toward autonomous, self-evolving systems that refine their own prompts, code, and strategies. While these systems show promise, they consistently hit a performance ceiling, often referred to as asymptotic saturation. This paper investigates why these self-improvement loops fail to sustain progress, drawing a parallel between modern AI and John von Neumann’s work on self-reproducing automata. The authors argue that just as self-reproduction requires a specific complexity threshold, sustained recursive self-improvement (RSI) requires a functional analogue: introspection.

Introspection is defined here not as subjective consciousness, but as a precise computational capacity: the ability of a system to simulate its own operations, evaluate its performance, and target structural modifications. Grounded in Kleene’s Second Recursion Theorem, the authors prove that introspective programs are theoretically possible. They propose the Introspection Threshold Thesis, which states that only systems capable of self-referential introspection can achieve unbounded, monotonic performance gains. Systems lacking this capability are confined to blind self-modification, which eventually leads to sub-optimal fixed points or performance degradation.

Despite their sophistication, current Large Language Models (LLMs) exhibit only quasi-introspection. The authors identify three structural reasons for this failure:

1. Lack of Reflexivity: True introspection requires a reflexive architecture where the system has complete access to its own parameter tensors. Current LLMs lack this, as their weights are not accessible as data during the forward pass.
2. Feedforward Constraints: Standard Transformer architectures are feedforward and lack the architectural recurrence necessary for the unbounded, nested self-simulation required for true introspection.
3. Causal Grounding: Purported self-awareness in LLMs is often a byproduct of statistical patterns in training data rather than genuine, causally grounded self-monitoring.