Activation steering often averages activation differences between positive and negative prompts to construct a steering vector. This difference-in-means (DiM) estimator is simple, but a contrast pair can coactivate the intended concept together with topic, style, length, refusal, or truthfulness-related nuisance features. We model each pair as revealing the target vector through a quotient that removes its active nuisance span. The resulting polysemantic information operator determines which target components are identifiable and the number of contrast pairs needed to recover them. We establish identifiability, sample-complexity bounds, a coactivation phase diagram, and sparse-dictionary recovery guarantees. Synthetic and controlled experiments support the predicted quotient-recovery behavior. Public-data and Pythia-410M pilots instead evaluate recovery of the quotient estimand induced by chosen nuisance supports, not an externally validated semantic target. A controlled TruthfulQA counter-confounder illustrates the main NLP lesson: raw behavioral margin can reward a style-entangled DiM vector even when quotienting removes the injected style component.