TARP: Trace-Anchored Regularization Prior for Retaining Stepwise Reasoning

Reinforcement learning (RL) is increasingly used in post-training to adapt large models to new reasoning tasks. Recent forgetting analyses, however, are mostly answer-centric: they report reward or accuracy on previously learned tasks and implicitly treat stable answers as evidence of stable reasoning. We show that this view is incomplete. In our sequential reasoning settings, externalized reasoning traces can change substantially in structure, strategy, and sample-wise behavior before answer-level metrics visibly collapse, and harder full-parameter staged runs can later degrade into catastrophic forgetting on earlier tasks. We refer to the trace-first regime as \emph{reasoning drift}. To address it, we propose \emph{Trace-Anchored Regularization Prior} (TARP), a trace-centric regularizer that mines reliable reasoning spans from successful rollouts and uses them to build a compact trace-derived quadratic prior for subsequent RL updates. Across multiple reasoning task sequences and model backbones, TARP reduces selected measured drift signals and improves earlier-task preservation in several evaluated settings, while also revealing a retention-verbosity trade-off that answer-only evaluation hides.