The Linear Representation Hypothesis associates high-level concepts with directions in language models, but it remains unclear how these concept-related linear structures are organized within the model. We propose the Answer-Basin Representation Hypothesis: the probability measure induced over answers by the model's continuation distribution organizes these linear structures, with its statistics represented along linear directions shared across questions. All continuations yielding the same answer form an answer basin, whose mass is their total probability. These basin masses define the pushforward probability measure over answers. We posit that concept-related linear structure emerges from differences in the answer measure rather than being determined by changes in concept labels. Experiments across models and tasks link concept-consistent effects and their reversals in probing and steering to the alignment between concept labels and the answer measure.
Steering is a widely used technique for controlling large language models, yet its effects are often unstable and hard to predict. Existing theoretical accounts are largely based on the Linear Representation Hypothesis (LRH). While LRH assumes that concepts can be orthogonalized for lossless control, this idealized mapping fails in real representations and cannot account for the observed unpredictability of steering. By relaxing LRH's orthogonality assumption while preserving linear representations, we show that overlapping concept contributions naturally yield a sample-specific axis-orthogonal structure. We formalize this as the Cylindrical Representation Hypothesis (CRH). In CRH, a central axis captures the main difference between concept absence and presence and drives concept generation. A surrounding normal plane controls steering sensitivity by determining how easily the axis can activate the target concept. Within this plane, only specific sensitive sectors strongly facilitate concept activation, while other sectors can suppress or delay it. While the surrounding normal plane can be reliably identified from difference vectors, the sensitive sector cannot, introducing intrinsic uncertainty at the sector level. This uncertainty provides a principled explanation for why steering outcomes often fluctuate even when using well-aligned directions. Our experiments verify the existence of the cylindrical structure and demonstrate that CRH provides a valid and practical way to interpret model steering behavior in real settings: https://github.com/mbzuai-nlp/CRH.
Fine-grained control of language model behaviors (e.g., steering) is among the more actionable outcomes of interpretability research. For binary concepts such as refusal, a single direction in activation space often suffices for steering. However, many concepts are not binary: Animals and Countries contain many subcategories, each with multiple instances. For these concepts, the search space over possible representation geometries is far larger than for binary concepts; it is thus not clear what geometries are most appropriate, nor what methods are most effective at recovering them. In this work, we introduce MAxBench, a geometry-agnostic evaluation framework for multinomial concept representations based on sampling from the recovered concept representation. We use MAxBench to compare 10 localization methods (covering 5 geometry types) across 6 concepts and 4 models. Using this framework, we find that (i) affine subspaces steer more reliably and have greater recall than rank-one or linear subspaces; (ii) much of this advantage is due to better non-zero offsets rather than the choice of bases; (iii) manifold steering is competitive with the best methods when applicable; and (iv) no method consistently outperforms prompting, in alignment with prior findings on binary concepts. These findings underscore the importance of expanding the scope of interpretability research and meta-evaluation to concepts with more varied structure.
Large language models often appear to reason reliably, yet on many questions repeated sampling yields both correct and incorrect answers, revealing an underlying fragility in how final decisions are formed. We study whether this fragility can be exploited through implicit reasoning steering: using natural-language text to bias a model toward a designated answer without explicit instructions, triggers, or direct answer cues. Our approach, Concept Chaining, generates a short connection paragraph that links question entities to a target option through one or two intermediate concepts. We then continue pretraining a victim model on these connection paragraphs and evaluate whether its answer preference shifts on the original multiple-choice questions. Our results show that indirect, natural-looking text can systematically steer model predictions while remaining substantially less inferable than direct paraphrases, which shows that reasoning brittleness is not merely an evaluation artifact: it creates a practical channel through which latent biases can be amplified by ordinary-looking text to covertly redirect model decisions.