Steered LLM Activations are Non-Surjective
For interpretability and safety researchers, this shows that activation steering does not imply prompt-based vulnerabilities or interpretability, requiring decoupled evaluation protocols.
The paper proves that activation steering in LLMs produces internal states that are not reachable by any textual prompt, establishing a formal separation between white-box steerability and black-box prompting. Empirical results across three LLMs confirm this non-surjectivity.
Activation steering is a popular white-box control technique that modifies model activations to elicit an abstract change in output behavior. It has also become a standard tool in interpretability (e.g., probing truthfulness, or translating activations into human-readable explanations and safety research (e.g., studying jailbreakability). However, it is unclear whether steered activation states are realizable by any textual prompt. In this work, we cast this question as a surjectivity problem: for a fixed model, does every steered activation admit a pre-image under the model's natural forward pass? Under practical assumptions, we prove that activation steering pushes the residual stream off the manifold of states reachable from discrete prompts. Almost surely, no prompt can reproduce the same internal behavior induced by steering. We also illustrate this finding empirically across three widely used LLMs. Our results establish a formal separation between white-box steerability and black-box prompting. We therefore caution against interpreting the ease and success of activation steering as evidence of prompt-based interpretability or vulnerability, and argue for evaluation protocols that explicitly decouple white-box and black-box interventions.