The Cylindrical Representation Hypothesis for Language Model Steering

Steering is widely used for controlling large language models, yet its effects are often unstable and difficult to predict. Existing theoretical accounts are largely based on the Linear Representation Hypothesis (LRH), which assumes that concepts can be orthogonalized for lossless control. However, this assumption rarely holds in practice and cannot explain the variability of steering outcomes. We propose the Cylindrical Representation Hypothesis (CRH), a geometric extension of LRH that relaxes the orthogonality assumption while preserving linear concept representations. We show that overlapping concept contributions naturally induce a sample-specific cylindrical structure consisting of a central axis, a normal plane, and sensitive sectors. The central axis captures the primary semantic transition associated with a target concept, while the normal plane governs steering sensitivity. Within this plane, some sectors facilitate concept activation, while others suppress or delay it. CRH reveals an asymmetry in steering predictability: the normal plane can be inferred from difference vectors, but the sensitive sectors cannot, introducing an intrinsic source of uncertainty. This explains why steering outcomes vary across samples even when intervention directions are well aligned. Experiments spanning 100 concepts, multiple models, and diverse steering methods provide consistent evidence for the predicted cylindrical structure, suggesting that steering variability arises from representation geometry rather than imperfect steering vectors. Our code is available at: https://github.com/mbzuai-nlp/CRH.