When can an interpretability claim be trusted? Stability and certification of feature dimensionality, ridge leverage and the feature recovery rate

Interpretability tools increasingly support safety claims, such as that a sparse dictionary has recovered most of a model's features. Such claims rest on measurements with arbitrary thresholds, and small changes to the model weights, a different training seed or a different threshold can reverse them. We study two widely used measurements in toy models and synthetic dictionaries with known ground truth: the feature dimensionality of Elhage et al. and the feature recovery rate (FRR) with a cosine threshold. We characterize exactly where feature dimensionality is discontinuous and replace it with a regularized (ridge) version with sharp stability bounds: the ridge leverage score changes by at most $3\sqrt{3}/(8\sqrtλ)$ times the operator-norm change of the weights, and this constant cannot be improved; for the regularized dimensionality the stability constant grows like $ρ/4$ in the scale $ρ=R/\sqrtλ$, with explicit upper and lower bounds. We also prove certificates stating when a claim "feature $i$ is recovered" must survive a given change of the dictionary. The certificates held in every test and are informative for small perturbations, but certify no feature non-trivially across independently trained dictionaries; at the standard threshold, whether a feature counts as recovered changed with the training seed for 55 percent of features. Experiments use toy models only. The proofs, code and text were drafted by AI agents directed by the authors and checked by an independent blind AI reviewer and by numerical tests; the authors take full responsibility for the content.

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70 pages. Also archived at https://doi.org/10.5281/zenodo.23842568. The AI use statement is in the paper