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Revocable Learned State via Process Sidecars

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.

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