Identifiable Interventions from Nonidentifiable Sparse Codes

Sparse autoencoders may admit multiple nonnegative sparse explanations of one activation, so a feature deletion can depend on the chosen explanation. We study identifying the activation change rather than recovering a unique code. Signed circuits characterize all exactly identifiable diagonal interventions, and a sharp modulus quantifies their noise amplification. An exact quotient theorem shows that certification and minimum-cost selection commute with arbitrary replication of decoder directions. On forest Gram graphs, a balanced-subtree algorithm certifies stability in $O(m{k}^4)$ arithmetic operations with rational counterexamples; positive paths admit optimal binary deletion over all feature subsets. For single-activation codes on arbitrary dictionaries, optimal attenuation has an explicit shortest-path formula. A separate construction proves an unbounded advantage over every stable binary deletion for any sparsity budget and arbitrarily overcomplete, nonidentifiable codes. Conversely, exact certification of one feature is coNP-complete even for well-conditioned, weakly correlated rational dictionaries. Uniform amplitude bounds transfer the structured certificates to densely perturbed dictionaries and control actual collateral change. Reproducible synthetic validation includes exact algorithm checks, 120 whole-input certification instances, and checkable refusal witnesses.