Fine-tuning updates' orthogonality to safety-critical directions is structurally unstable — loss-landscape curvature systematically steers later training into a low-rank, sharply-curved alignment-sensitive subspace regardless of initial direction

safety-alignmentfine-tuningjailbreakintrinsic-dimensiondimensionalitylarge-language-models

Springer, Lee, Metevier, Castleman, Turbal, Jung, Shen & Korolova (Princeton), "The Geometry of Alignment Collapse: When Fine-Tuning Breaks Safety" (arXiv:2602.15799, 2026-02-17), analyze why benign fine-tuning — math tutoring, creative writing, code generation, with no harmful training data — nonetheless degrades safety. They reject the reassuring account: "The prevailing explanation, that fine-tuning updates should be orthogonal to safety-critical directions in high-dimensional parameter space, offers false reassurance: we show that this orthogonality is structurally unstable and collapses under the very dynamics of gradient descent." Their geometric result is that "alignment concentrates in low-dimensional subspaces with sharp curvature, creating a brittle structure that first-order methods cannot detect or defend." On the mechanism: "While initial fine-tuning updates may indeed avoid these subspaces, the curvature of the fine-tuning loss generates second-order acceleration that systematically steers trajectories into alignment-sensitive regions."

This complicates the naive within-manifold reading of Qi et al.'s jailbreak. It is not that ordinary fine-tuning already sits inside the alignment-sensitive low-dimensional subspace from the first step (a clean "within-manifold, therefore easy" story); rather, that subspace is a distinct geometric structure early updates avoid, and training dynamics — not starting position — pull trajectories into it. Whether Qi et al.'s much shorter, more targeted ten-example attack reaches the subspace by this same curvature-driven route or a more direct one is not addressed. The finding stands in tension with claim-ponkshe-2025-safety-subspaces-not-linearly-distinct-entangled-with-general-learning (safety entangled from the start, not separable-but-unstable) and complements the repair-side geometry of claim-zhang-2026-safety-alignment-low-rank-subspace-regardless-of-model-size. The three-part condition the paper introduces to explain the drift is stubbed at entity-alignment-instability-condition. That no source draws the Sadtler within-/outside-manifold comparison is recorded at observation-no-primary-source-links-qi-2023-jailbreak-to-sadtler-2014-within-manifold-asymmetry; bears on question-low-dimensional-subspace-one-object-or-analogy.