Minimum-KL Activation Steering Through a Normalized Linear Readout

Activation steering interventions can achieve the same increase in a desired output score while changing other next-token probabilities differently. At a fixed context, we use the distribution minimizing steered-to-baseline KL under the score constraint as a method-independent benchmark. This minimum-KL target is the standard exponential tilt of the baseline distribution. We ask whether a bounded activation intervention can realize it. For interventions immediately before RMSNorm and a linear readout, we give necessary and sufficient conditions for exact attainment: the target logit direction must lie in the centered-readout range, normalization must admit a realizing state, and the minimum required intervention norm must not exceed the budget. When the score increase is feasible but the target is unattainable, we prove a positive lower bound on excess KL. For compatible directions, a nontrivial centered-readout nullspace gives a single attainment threshold, whereas an injective readout can lose and later recover attainability as the target increase grows. Experiments on three small pretrained models and four lexical event scores find large range mismatches in all twelve model-score pairs, ruling out every positive minimum-KL tilt at these final readouts. Constructed compatible targets are exactly reachable only over a narrow range of score increases at the tested budget. These results distinguish achieving a target score from achieving it with the smallest distributional change under the specified KL criterion.