Abstract
This paper studies the local geometry of semantic meaning in large language models. Current representation analyses ask whether meanings are close in embedding space, or whether linear probes separate semantic classes, but they do not characterize the within-concept local geometry: what is invariant across realizations of a single semantic factor, what varies, and what changes when meaning is refined. This paper characterizes that geometry through two controlled probes evaluated on four architecturally diverse decoder-only LLMs (Gemma 4, Llama 3.2, Qwen 2.5, Pythia). A cross-lingual probe uses language as a perturbation to separate exact meaning matches, same-factor realizations, nearby concepts, and out-of-domain contrasts. A refinement probe separates specificity-bearing semantic narrowing from high-probability attributive elaboration and adversarial controls; both probes are validated by blind human judgments. Four findings recur across all models: under mean pooling, once dominant variance directions are removed, sentence pairs order by semantic relatedness (same factor, then same domain, then different domain) without overlap over a band of layers; semantic factors organize around an invariant core surrounded by structured realization-specific variation; refinement produces directed displacement into tighter neighborhoods, including against controls built on the same base noun; and sentence-level and token-local readouts expose complementary geometry (volume contraction at the sentence level, sharper anisotropic endpoints at token-local readout). We interpret these results as evidence that LLM semantics are organized as local anisotropic neighborhoods in which meaning contracts under refinement.
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URL
https://arxiv.org/abs/2610.30715
