Learning Dynamics of Chain-of-Thought State Tracking in a Solvable Transformer Model

Chain-of-thought generation can turn a multi-step computation into a sequence of locally checkable state updates, but the training dynamics by which transformers acquire such updates remain poorly understood. We study this question in a solvable setting: a simplified one-block transformer trained by supervised next-token prediction on state sequences generated by composing permutations. The architecture separates fixed-lag action retrieval, learned by RoPE attention, from a specialized MLP logic module that applies the retrieved permutation to the current state. Using a statistical-physics mean-field description, we derive dynamics for three order parameters measuring attention retrieval, teacher-matrix alignment, and off-target logic overlap. These equations quantitatively match simulations for the order parameters and, combined with a logit-distribution approximation, qualitatively predict the sharp transition in final rollout accuracy. The analysis reveals staged learning: the logic module first learns a mixed heuristic; attention then locks onto the relevant action, enabling efficient MLP alignment. Together, these results provide a controlled mechanistic account of how attention-based retrieval and MLP-based logic co-develop during chain-of-thought state tracking.