Geometry, Not Scale Alone, Predicts Sparse Recovery of Causal Subspaces

Whether high-dimensional causal representations become easier to recover in sparse bases is not determined by model size alone. We study sparse decomposability: the extent to which a dense DAS-localized causal subspace can be expressed by a small set of pretrained SAE latents at a particular model--site--dictionary tuple, measured by causal sparse distillation (CSD), which matches a dense causal teacher with an SAE-constrained student. The central finding is geometric: two compact pre-CSD decoder statistics, the top-$K$ decoder-subspace cosine with the teacher and the decoder stable rank, predict CSD/dense recovery across dense-valid Gemma/Qwen tuples with leave-one-out $R^2=0.89$, leave-one-model-out $R^2=0.80$, and bootstrap 95\% CI $[0.79,0.95]$, while model size alone gives $R^2=-2.00$. This explains why some larger-model sites recover cleanly, others are random-$K$ degenerate, and a 27B dense-valid site remains sparse-limited. A ground-truth synthetic battery and cross-family checks on Gemma, Llama, and Qwen-Scope calibrate the measurement, and matched random-$K$ controls show that high CSD/dense recovery is not by itself evidence of meaningful feature selection. The result is a high-dimensional learning story about how dictionary geometry, scale, and causal subspace orientation jointly control sparse recovery.