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This paper analyzes chain-of-thought reasoning in LLMs as trajectories in representation space, revealing step-specific subspaces that become more separable with network depth. Reasoning training primarily accelerates convergence towards termination subspaces rather than creating new representational structures. Leveraging late-stage trajectory divergence between correct and incorrect solutions, the authors achieve up to 0.87 ROC-AUC in predicting final-answer correctness mid-reasoning and introduce trajectory-based steering for inference-time correction and length control.
LLMs don't learn fundamentally new reasoning representations during training; they just get faster at converging to the right answer.
This work characterizes large language models'chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become increasingly separable with layer depth. This structure already exists in base models, while reasoning training primarily accelerates convergence toward termination-related subspaces rather than introducing new representational organization. While early reasoning steps follow similar trajectories, correct and incorrect solutions diverge systematically at late stages. This late-stage divergence enables mid-reasoning prediction of final-answer correctness with ROC-AUC up to 0.87. Furthermore, we introduce trajectory-based steering, an inference-time intervention framework that enables reasoning correction and length control based on derived ideal trajectories. Together, these results establish reasoning trajectories as a geometric lens for interpreting, predicting, and controlling LLM reasoning behavior.
