Steering vectors cannot reach most of the activation manifold

Mishra et al. show that adding a fixed direction to residual-stream activations only reaches a thin, structured slice of the activations that occur naturally across contexts [mishra-etal-2026-steered-llm-activations-are-non-surjective] Across Llama-3.2-1B-Instruct, Qwen2.5-0.5B-Instruct, and Gemma-3-1B-it (plus three validation models), the reachable set is measurably non-surjective onto the empirical activation manifold [mishra-etal-2026-steered-llm-activations-are-non-surjective] Many naturally-occurring target-behavior activations lie off the steering line entirely, and increasing the coefficient pushes activations off-manifold rather than toward the unreached regions [mishra-etal-2026-steered-llm-activations-are-non-surjective] This bounds what single-direction steering can achieve and implies off-line behaviors need multi-directional or nonlinear interventions [mishra-etal-2026-steered-llm-activations-are-non-surjective]