Manifold Alignment, Not Norm, Determines Sparse Autoencoder Fidelity Under Steering

Interpreting a steered activation with a sparse autoencoder (SAE) is routine. Mayne et al. (2024) showed such decompositions are misleading because steering vectors are out-of-distribution, diagnosing the cause as their L2 norm. We show norm is the wrong quantity: on a real GPT-2 SAE, the off-active-subspace fraction c_v dominates Mahalanobis displacement as a predictor of directional fidelity. Alignment ρ_k orders fidelity in both a synthetic tier and a real SAE (three seeds, six concepts), but the method labels swap—a logistic probe and contrastive activation addition are worse-aligned than an isotropic random direction at nearly every layer tested. The remedy this implies, projecting the vector back on-manifold, is costly: measured behaviourally across 36 held-out prompts it retains only ~25-30% of the concept-relevant steering effect and flips its sign in 20-26% of tested conditions. We propose Certified Delta Attribution (CDA), which leaves the intervention untouched and augments the dictionary with the one direction the theory identifies as missing, reporting the unattributable share as a certificate. CDA retains 100% of the effect by construction and cuts spurious feature activation by 52-100%. The inversion replicates on a second, architecturally unrelated TopK SAE. We read this as limitative: a dictionary learned on a distribution cannot represent interventions that leave it. All experiments are CPU-only.