GRRR: The Geometry of Reshaping, Rotation, and Routing in Decoder LLM Post Training

We study where the useful weight change of post training lives in decoder LLMs. We analyze 12 public training chains spanning supervised fine-tuning (SFT) and reinforcement learning (RL) style updates, and our causal claims use a five benchmark average dominated by math reasoning. We write each update in the pretrained model's SVD frame and split it into three parts: diagonal reshaping, rotation, and routing. Across the 25 transitions where the full model improves over base by at least 1 point, keeping only the rotation and routing terms stays within 4.3 points of the full post trained model in 24 cases on this math-heavy evaluation suite. In these settings, the useful part of the final weight difference lies mainly in rotation and routing, not in changing singular values.