Self-Reference in Large Language Models: The Introspection Threshold for Recursive Self-Improvement
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.
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BibTeX
@misc{zhang2026selfreference,
title = {{Self-Reference in Large Language Models: The Introspection Threshold for Recursive Self-Improvement}},
author = {Jiang Zhang and Bing Yuan and Qian Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04277},
eprint = {2607.04277},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04277},
}
RIS
TY - UNPB
TI - Self-Reference in Large Language Models: The Introspection Threshold for Recursive Self-Improvement
AU - Jiang Zhang
AU - Bing Yuan
AU - Qian Zhang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.04277
UR - https://arxiv.org/abs/2607.04277
AB - 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.
ER -