When Self-Reference Fails to Close: Matrix-Level Dynamics in Large Language Models

TL;DR

This paper explores how self-referential prompts affect the internal dynamics of large language models, revealing that paradoxical self-reference induces instability and disrupts attention patterns across multiple models and analysis passes.

Contribution

It provides a detailed empirical analysis of matrix-level dynamics in LLMs under self-reference, identifying specific instability patterns and proposing a conjecture linking NCTR prompts to classical matrix problems.

Findings

Self-reference alone is not destabilizing; paradoxical self-reference causes instability.

NCTR prompts lead to elevated attention effective rank and global dispersion.

A classifier can distinguish NCTR from stable self-reference with high accuracy.

Abstract

We investigate how self-referential inputs alter the internal matrix dynamics of large language models. Measuring 106 scalar metrics across up to 7 analysis passes on four models from three architecture families -- Qwen3-VL-8B, Llama-3.2-11B, Llama-3.3-70B, and Gemma-2-9B -- over 300 prompts in a 14-level hierarchy at three temperatures ($T \in \{0.0, 0.3, 0.7\}$), we find that self-reference alone is not destabilizing: grounded self-referential statements and meta-cognitive prompts are markedly more stable than paradoxical self-reference on key collapse-related metrics, and on several such metrics can be as stable as factual controls. Instability concentrates in prompts inducing non-closing truth recursion (NCTR) -- truth-value computations with no finite-depth resolution. NCTR prompts produce anomalously elevated attention effective rank -- indicating attention reorganization with global dispersion rather than simple concentration collapse -- and key metrics reach Cohen's $d = 3.14$ (attention effective rank) to $3.52$ (variance kurtosis) vs. stable self-reference in the 70B model; 281/397 metric-model combinations differentiate NCTR from stable self-reference after FDR correction ($q < 0.05$), 198 with $|d| > 0.8$. Per-layer SVD confirms disruption at every sampled layer ($d > +1.0$ in all three models analyzed), ruling out aggregation artifacts. A classifier achieves AUC $0.81$-$0.90$; 30 minimal pairs yield 42/387 significant combinations; 43/106 metrics replicate across all four models. We connect these observations to three classical matrix-semigroup problems and propose, as a conjecture, that NCTR forces finite-depth transformers toward dynamical regimes where these problems concentrate. NCTR prompts also produce elevated contradictory output ($+34$-$56$ percentage points vs. controls), suggesting practical relevance for understanding self-referential failure modes.