We study how post-training changes the weights of Large Language Models (LLMs) relative to their pretrained weights.
Across 12 post-training chains with supervised fine-tuning (SFT) and reinforcement learning (RL), we express each weight update in the pretrained matrix's singular value decomposition (SVD) frame.
This decomposition separates the changes of three geometrically distinct components:
- Diagonal values, which reshapes singular values;
- Off-diagonal values, which rotates the coupling between pretrained input and output directions;
- Null-space values, which routes outside the matrix's original nonzero SVD core.
On a math evaluation suite, we find that removing the diagonal component usually preserves most of the gains from post-training.
These results suggest that post-training gains are carried primarily by reconfiguring and extending pretrained pathways rather than by substantially changing singular values of pre-trained models.