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We measure 8 such properties with the same harness around a multitask LoRA operating point, on 9 transformers (82M-7B), with a prospectively registered property list, thresholds, and test split. We find a shared one-direction validity window up to the tested scale $10^{-2}$, but no universal radius for pairwise composition or update ordering. Along individual directions, changes of the probe loss remain first-order predictable throughout the grid: a perturbation","title":"First-Order Predictable but Pairwise Fragile: Local Task Adaptation in Trained Transformers","url":"https://arxiv.org/abs/2607.16821","vendor":"arxiv_cs_ai"},"summary":"arXiv:2607.16821v1 Announce Type: cross \nAbstract: Task arithmetic, sequential fine-tuning, activation steering, and first-order random search all operate through relatively small perturbations around an already trained checkpoint, and they rely on different local approximations: individual perturbations should be first-order predictable, task updates should compose with controlled interference, useful tangent structure should be stable and possible to estimate, and weight edits should have counterparts in representation space. 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