Prefix-tuning is a parameter-efficient fine-tuning method for large language models (LLMs) that steers model outputs by prepending task-specific prefix representations to the input sequence. Despite its practical success, the theoretical understanding of the expressiveness of prefixes remains incomplete. In particular, the following question remains open: Given a pretrained autoregressive Transformer, for any input-output sequence pair $(\mathtt{x},\mathtt{y})$, does there always exist a prefix $\mathcal{S}$ such that prepending $\mathcal{S}$ to $\mathtt{x}$ induces the model to generate the target output $\mathtt{y}$? In this work, we prove that under the assumption that $\mathtt{y}$ is in the range of the pretrained Transformer, such a prefix $\mathcal{S}$ exists and can be explicitly computed by solving multiple linear systems. We further show that this assumption is necessary, in the sense that such a prefix may not exist when the assumption fails, thereby providing a complete answer to the above question. Our analysis is constructive and the obtained theoretical results hold under fairly general assumptions on model architecture. Extensive experiments on GPT-2 and Qwen3-4B demonstrate that our algorithm produces prefixes with strong control over model outputs.

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