Tracking fine-tuning trajectories shows safety-aligned LLMs are not steered toward harm but lose their safe subspace, and projecting out the harmful gradient subspace cuts free-generation emergent misalignment by up to 80.0% on Qwen2.5-14B-IT
The work presents a dynamic second-order geometric study of emergent misalignment (EM): tracking training trajectories shows directional Hessian curvature concentrates on semantic pivot tokens, Grassmannian projections show the harmful-safe gap widens mainly because safe-gradient overlap declines rather than through rotation toward a new malicious circuit, and on this basis it introduces a parameter-level Geometric Mitigation Framework that orthogonally projects the empirical harmful gradient subspace out of LoRA updates, suppressing free-generation EM by up to 80.0% on Qwen2.5-14B-IT and, across four open-weight instruction-based model families (3B–20B), teacher-forced evaluation shows the same harmful subspace controls the conditional support of frozen EM responses.