Beyond Fixed Directions: Adaptive Representation Analysis of Reasoning and Memorization in LLMs

Recent work has proposed that reasoning and memorization in language models can be characterized by a single representation direction, including methods that keep this direction fixed during reinforcement learning. We test two assumptions behind this view. First, are reasoning-oriented and factual-recall task groups approximately single-direction separable? Second, does the resulting geometry remain stable after GRPO? Using Qwen3-0.6B and a controlled 400-example dataset, we find that a one-dimensional projection can match a full 1024-dimensional linear probe with AUROC = 1.00 on the studied task groups. However, after GRPO, the corresponding direction is substantially reorganized: mean-direction cosine averages 0.453, probe-direction cosine 0.445, while direct representation drift reaches 0.511 at the final layer. Probe AUROC nevertheless remains 1.00. The evidence therefore supports single-direction decodability for the studied task groups but challenges fixed-direction stability: the information persists while its geometric realization changes.