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.

Although value-aligned language models (LMs) appear unbiased in explicit bias evaluations, they often exhibit stereotypes in implicit word association tasks, raising concerns about their fair usage. We investigate the mechanisms behind this discrepancy and find that alignment surprisingly amplifies implicit bias in model outputs. Specifically, we show that aligned LMs, unlike their unaligned counterparts, overlook racial concepts in early internal representations when the context is ambiguous. Not representing race likely fails to activate safety guardrails, leading to unintended biases. Inspired by this insight, we propose a new bias mitigation strategy that works by incentivizing the representation of racial concepts in the early model layers. In contrast to conventional mitigation methods of machine unlearning, our interventions find that steering the model to be more aware of racial concepts effectively mitigates implicit bias. Similar to race blindness in humans, ignoring racial nuances can inadvertently perpetuate subtle biases in LMs.