Uncovering the Gradient Geometry of Long CoT: A Spectral-guided Approach to Reasoning Distillation

Large reasoning models (LRMs) achieve remarkable reasoning performance by generating long chains-of-thought (CoT). However, standard supervised fine-tuning (SFT) treats all tokens uniformly, indiscriminately minimizing loss across both essential reasoning steps and those that are noisy, redundant, or instance-specific. This often leads student models to memorize superficial patterns rather than acquire generalizable reasoning capabilities. To better understand this limitation, we introduce \textit{Loss Subspace Attribution}, a gradient decomposition analysis approach that uncovers a striking geometric structure: Gradients corresponding to effective reasoning predominantly lie within a low-rank consensus subspace, while conflicting or unstructured signals dominate the residual subspace. Guided by this insight, we propose \textbf{\textit{Spectral-guided Learning}}, a step-level distillation strategy that uses spectral strength to identify reasoning steps aligned with the consensus subspace and prioritizes their contribution to parameter updates, while suppressing gradients from the residual subspace. Experiments across various LRMs and diverse complex reasoning tasks consistently demonstrate that focusing optimization on the consensus subspace yields more robust and generalizable student models.