Sparse autoencoders (SAEs) decompose LLM activations into sparse dictionary atoms so that each concept obtains a distinct feature. A recurring issue is feature absorption, where a parent concept and its children (e.g., fruit and {apple, banana, pear}) collapse into a shared direction, violating the one‑concept‑one‑feature premise. Prior work reported this phenomenon only empirically and lacked a closed‑form condition for when the shared direction becomes cost‑optimal. We consider a hierarchical Bernoulli generator with $k$ active children and residual scale $\alpha$. For the $L_0$‑penalized reconstruction objective we derive the phase boundary $$\lambda_c(k,\alpha)=\frac{\alpha^2 k}{k-1}$$; when the regularization weight $\lambda$ exceeds $\lambda_c$, pure parent absorption is strictly cheaper than encoding children separately. Building on this result, we introduce the HiPACE evaluation protocol. It measures parent‑child decoder structure in real SAE dictionaries over WordNet families, freezes the discovery‑selected statistic before testing on unseen families, and contrasts genuine families with randomized sibling nulls. The boundary is sharp in its native regime, predicting the synthetic transition within $\pm15\%$ across all 30 tested cells. In Pythia‑160m SAEs, the parent‑child decoder gap recovers the predicted ordering with partial correlations up to $-0.93$, which persist on the locked holdout and reject sibling nulls ($p=0.002$). Controlled activation composition links the theoretical active‑child count to recovered family directions, and residual‑stream interventions show that signed family directions increase parent‑category logits, reverse under sign flip, and vanish under random controls—establishing causal sufficiency at the family‑subspace level.
Review