When Is Rank-1 Steering Cheap? Geometry, Granularity, and Budgeted Search

Activation steering offers a lightweight way to control large language models without retraining, but its effectiveness varies sharply across concepts. Prior work often interprets this variability as evidence that many concepts are not well captured by a single steering direction. We argue instead that much of this variability 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 problem over intervention layer and coefficient. Across the concepts and model families, prompt-boundary directional alignment predicts where effective interventions are likely to 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 both slower convergence and lower best-found steering performance (Pearson $r = 0.44$ with trials-to-95\%, $p < 0.001$, and $r=-0.46$ with best-found utility, $p < 0.001$). These observations suggest a practical workflow rather than a single universal vector-construction rule. We therefore present GRACE, a Granularity- and Representation-Aware Concept Engineering framework that uses activation geometry to diagnose the dominant source of steering difficulty, choose the appropriate remedy, and allocate optimization effort more efficiently. Our results shift the frame of activation steering from "when does rank-1 fail?" to "when is rank-1 cheap and stable?", and turn activation geometry from a descriptive tool into an actionable prior for LLM control.