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. 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's effect on the loss is essentially its projection onto the gradient, which is also what makes local random search work. Pairwise structure, however, proves to be far more fragile: on over a third of the measured (model, task pair) combinations, two-update order sensitivity sets in strictly inside that window; task-gradient subspaces rotate within tens of steps; additivity under our fixed activation probe fails at full task-vector scale on several models, including both held-out 7B models; and no model median passes the registered global mean-vector weight-to-steering correspondence bar. For two sequential task-gradient steps, the leading order-dependent term is the Lie bracket $H_B\textbf{g}_A-H_A\textbf{g}_B$; its normalized prediction $c(η)=ηκ+O(η^2)$ tracks the measured defect at median ratio 1.002, while the onset scale $η^\dagger\approx0.10/κ$ spans three orders of magnitude across models and task pairs.
Language models are often adapted in stages: a public skill phase, a private memory phase, and a later safety phase that learns to refuse outputs tied to the remembered entities. Revoking the memory after the safety phase is not the same problem as subtracting the memory update: the later safety optimizer has transported the memory direction. We introduce process sidecars, a two-coefficient edit family $\hatθ(λ,γ)=θ_{\mathrm{AMS}}-λΔ_{\mathrm{M}}-γ\hat{R}_{\mathrm{S}\leftarrow\mathrm{M}}$, with $\hat{R}_{\mathrm{S}\leftarrow\mathrm{M}}=\hat{J}_{\mathrm{S},\varepsilon}(Δ_{\mathrm{M}})-Δ_{\mathrm{M}}$, where $\hat{J}_{\mathrm{S},\varepsilon}$ is a centered secant through the realized future AdamW safety-training process. The implementation uses $\varepsilon=1$ at the natural memory-edit scale; it reuses $θ_{\mathrm{AMS}}$ as the positive endpoint and computes one additional safety trace at $θ_{\mathrm{A}}-Δ_{\mathrm{M}}$. We prove two things. First, the exact sidecar, using the true transported direction $R_{\mathrm{S}\leftarrow\mathrm{M}}$ rather than the secant estimate, at $(λ,γ)=(1,1)$ recovers the counterfactual safety-only oracle $θ_{\mathrm{AS}}$ up to second order; the proof treats AdamW as an augmented-state map over parameters, first moments, and second moments. Second, this process information is necessary: whenever future safety training bends the memory direction, every scalar task-arithmetic edit leaves first-order counterfactual error, while the process-sidecar edit is second-order accurate. Across three models, the validation-selected 2D edit improves held-out refusal closure over naive task arithmetic in all trials, and over the $γ=λ$ process-JVP subfamily, the diagonal slice of the cached 2D grid, in all paired trials.