How Much Must a Ridge Learner Remember to Delete Data?

An offline ridge learner retains a finite-bit summary; a decoder receives deleted records and approximates the retained-data optimum without accessing the retained records. How do regularization, accuracy, and deletion size determine the necessary memory? For bounded grid inputs, $n=\Theta (d)$, and a fixed deletion fraction, the worst-case memory is $d\min \{d,(\lambda {\epsilon }^{1/3}{)}^{-1}\}$ bits within one factor $\log d$, for sufficiently small additive or relative error $\epsilon$. When $\lambda =1/4$ and the deletion budget $\rho$ vanishes, the additive law becomes $d\min \{d,\rho /\sqrt{\epsilon },(\rho /\epsilon {)}^{1/3}\}$ within a precision logarithm. For bounded noiseless teachers, a joint rank law is matching when $\rho /\lambda$ is bounded. A determinantal response envelope extends its upper bound to every budget with an $\exp (O(\rho /\lambda ))$ overhead. A routing construction proves that an $\exp (\Omega (\rho /\lambda ))$ overhead is necessary along explicit families, despite a common teacher, original optimizer, and legal requests. Thus the law persists within polylogarithmic factors for $\rho =O(\lambda \log \log d)$, while extensions beyond that scale can fail. The lower bounds allow pointwise randomized success; deterministic summaries attain the upper bounds. Residual importance sampling gives a polynomial bit encoder for a general-label DPP bound. Controlled and frozen-feature experiments assess accuracy and storage. These guarantees concern optimization utility, not distributional certification of unlearning.