LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals

TL;DR

This paper models large language model reasoning as trajectories in representation space, revealing step-specific structures and enabling prediction and correction of reasoning correctness.

Contribution

It introduces the concept of reasoning trajectories, analyzes their structure in base and trained models, and proposes trajectory-based steering for reasoning correction.

Findings

Reasoning traverses step-specific subspaces that become more separable with depth.

Correct and incorrect solutions diverge late in the reasoning process.

Trajectory-based steering enables inference-time correction and length control.

Abstract

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.