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

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 $\mathrm{\mid }d\mathrm{\mid }>0.8$. 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.