A Hyperbolicity Atlas of Large Language Model Hidden States

Abstract:LLM hidden states are ordinary vectors, but the distances among those vectors may still show hierarchical structure. To our knowledge, this paper is the first systematic study of whether prompt-token hidden states in contemporary LLMs exhibit Gromov Hyperbolicity (GH), a distance-based measure of tree-likeness. Using 818,904 sample-layer measurements from ten open-weight models across MATH500, HumanEval, WinoGrande, and TruthfulQA, we build a GH map over four axes: parameter scale, layer depth, model family, and input domain. The clearest pattern is depth, not scale: middle layers usually form a high-relative-hyperbolicity plateau, while final layers often become substantially more tree-like. Scale effects are weak and non-monotonic, matched 7/8B model families differ strongly, and domains interact with model specialization. These findings make GH useful as a practical diagnostic: it shows where hierarchical distance structure appears, how specialization changes it, and which model-layer-domain comparisons deserve closer analysis.

Computation and Language (cs.CL)

arXiv:2609.07053 [cs.CL]

(or arXiv:2609.07053v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2609.07053 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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