The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason
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| Format: | Preprint |
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2026
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| _version_ | 1866911599264006144 |
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| author | Liu, Yi |
| author_facet | Liu, Yi |
| contents | We discover that large language models exhibit \emph{spectral phase transitions} in their hidden activation spaces when engaging in reasoning versus factual recall. Through systematic spectral analysis across \textbf{11 models} spanning \textbf{5 architecture families} (Qwen, Pythia, Phi, Llama, DeepSeek-R1), we identify \textbf{seven} core phenomena: (1)~\textbf{Reasoning Spectral Compression} -- 9/11 models show significantly lower $α$ for reasoning ($p < 0.05$), with larger effects in stronger models; (2)~\textbf{Instruction Tuning Spectral Reversal} -- base models show reasoning $α< $ factual $α$, while instruction-tuned models reverse this relationship; (3)~\textbf{Architecture-Dependent Generation Taxonomy} -- prompt-to-response shifts partition into expansion, compression, and equilibrium regimes; (4)~\textbf{Spectral Scaling Law} -- $α_\text{reasoning} \propto -0.074 \ln N$ across 4 Qwen base models ($R^2 = 0.46$); (5)~\textbf{Token-Level Spectral Cascade} -- per-token alpha tracking reveals local synchronization that decays exponentially with layer distance, and is weaker for reasoning than factual tasks; (6)~\textbf{Reasoning Step Spectral Punctuation} -- phase-transition signatures align with reasoning step boundaries; and (7)~\textbf{Spectral Correctness Prediction} -- spectral $α$ alone achieves AUC $= 1.000$ (Qwen2.5-7B, late layers) and mean AUC $= 0.893$ across 6 models in predicting correctness \emph{before} the final answer is generated. Together, these findings establish a comprehensive \emph{spectral theory of reasoning} in transformers, revealing that the geometry of thought is universal in direction, architecture-specific in dynamics, and predictive of outcome. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15350 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Liu, Yi Machine Learning We discover that large language models exhibit \emph{spectral phase transitions} in their hidden activation spaces when engaging in reasoning versus factual recall. Through systematic spectral analysis across \textbf{11 models} spanning \textbf{5 architecture families} (Qwen, Pythia, Phi, Llama, DeepSeek-R1), we identify \textbf{seven} core phenomena: (1)~\textbf{Reasoning Spectral Compression} -- 9/11 models show significantly lower $α$ for reasoning ($p < 0.05$), with larger effects in stronger models; (2)~\textbf{Instruction Tuning Spectral Reversal} -- base models show reasoning $α< $ factual $α$, while instruction-tuned models reverse this relationship; (3)~\textbf{Architecture-Dependent Generation Taxonomy} -- prompt-to-response shifts partition into expansion, compression, and equilibrium regimes; (4)~\textbf{Spectral Scaling Law} -- $α_\text{reasoning} \propto -0.074 \ln N$ across 4 Qwen base models ($R^2 = 0.46$); (5)~\textbf{Token-Level Spectral Cascade} -- per-token alpha tracking reveals local synchronization that decays exponentially with layer distance, and is weaker for reasoning than factual tasks; (6)~\textbf{Reasoning Step Spectral Punctuation} -- phase-transition signatures align with reasoning step boundaries; and (7)~\textbf{Spectral Correctness Prediction} -- spectral $α$ alone achieves AUC $= 1.000$ (Qwen2.5-7B, late layers) and mean AUC $= 0.893$ across 6 models in predicting correctness \emph{before} the final answer is generated. Together, these findings establish a comprehensive \emph{spectral theory of reasoning} in transformers, revealing that the geometry of thought is universal in direction, architecture-specific in dynamics, and predictive of outcome. |
| title | The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2604.15350 |