Activation-based steering provides control of LLM behavior at inference time, but the dominant paradigm reduces each concept to a single direction whose geometry is left largely unexamined. Rather than selecting a single steering direction, we use conceptors: soft projection matrices estimated from activations pooled across both poles of a bipolar concept, which preserve the concept's full multidimensional subspace. A geometric analysis shows the bipolar subspace strictly subsumes the single-vector baseline. We further show that the conceptor quota provides a parameter-free layer-selection diagnostic, predicting concept separability with Pearson correlations up to r=0.96 across three instruction-tuned models and three semantic dimensions. Beyond selection, conceptors admit a closed-form Boolean algebra (AND, OR, NOT): we evaluate conceptor compositionality on thematically related sub-concepts. Across a systematic five-axis design-space evaluation, conceptors match or outperform additive baselines at layers where concept subspaces are multi-dimensional while producing substantially fewer degenerate outputs. Conceptor steering is a geometrically principled, compositional, and practically safer alternative to single-direction steering from a limited number of contrastive pairs.
Activation steering offers a lightweight way to control LLMs without retraining, but its effectiveness varies sharply across concepts. Prior work often reads this variability as evidence that many concepts are not captured by a single steering direction. We argue instead that much of it reflects search difficulty: a useful rank-1 intervention often exists, but finding it can be expensive. We formalize rank-1 steering as a budget-constrained optimization over intervention layer and coefficient. Across concepts and model families, prompt-boundary directional alignment predicts where effective interventions occur, enabling geometry-guided search that reaches high utility with substantially fewer evaluations, reducing the trials needed to recover 95% of best-found utility by 39.8% on average across three model families. To explain why some concepts remain expensive even under better search, we introduce concept granularity, a measure of directional heterogeneity across contrastive contexts. Granularity distinguishes concepts whose difference vectors share a stable global direction from those where prompts agree locally within each input but the utility-maximizing direction rotates systematically across inputs. Higher granularity is associated with slower convergence and lower best-found performance (Pearson r=0.44 with trials-to-95%, r=−0.46 with best-found utility, both p<0.001). We present GRACE, a Granularity- and Representation-Aware Concept Engineering framework that uses activation geometry to diagnose the dominant source of steering difficulty, select the appropriate remedy, and allocate optimization effort efficiently. Our results shift the frame from "when does rank-1 fail?" to "when is rank-1 cheap and stable?", turning activation geometry from a descriptive tool into an actionable prior for LLM control.
Applying steering vectors to large language models (LLMs) is an efficient and effective model alignment technique, but we lack an interpretable explanation for how it works--specifically, what internal mechanisms steering vectors affect and how this results in different model outputs. To investigate the causal mechanisms underlying the effectiveness of steering vectors, we conduct a comprehensive case study on refusal. We propose a multi-token activation patching framework and discover that different steering methodologies leverage functionally interchangeable circuits when applied at the same layer. These circuits reveal that steering vectors primarily interact with the attention mechanism through the OV circuit while largely ignoring the QK circuit. Freezing all attention scores during steering drops performance by only 8.83% across three model families. A mathematical decomposition of the steered OV circuit further reveals semantically interpretable concepts, even in cases where the steering vector itself does not. Leveraging the activation patching results, we show that steering vectors can be sparsified by up to 85-96% while retaining most performance, and that different steering methodologies agree on a subset of important dimensions.
Linear activation steering has gained popularity as a simple and empirically effective way to control language model behavior. More recently, spherical steering paradigms have been proposed to address limitations of additive interventions, often motivated by the assumption that hidden-state norm does not carry concept-relevant information. In this work, we revisit this assumption through a controlled empirical study designed to disentangle the roles of angular and radial components. We show that steering methods differ mainly in how they couple two geometric effects: changing a token's angular alignment with a concept direction and changing its hidden-state norm. Across seven language models, we find that concepts are represented primarily in angular structure, supporting the motivation for spherical methods, but that norm remains important for the stability and downstream effects of steering. Our results explain why interventions with similar concept-level effects can behave differently, and suggest that activation steering should be parameterized by interpretable angular and radial components of the intervention, rather than by a single additive coefficient that entangles these two effects.