Ambient Dimension Governs Whether Frozen Single-Cell Embeddings Can Be Edited and Interpreted

Frozen single-cell foundation models are increasingly used as off-the-shelf embeddings, and linear-probe benchmarks report that perturbation phenotypes are well “contained” in them. Yet the same embeddings rarely help unsupervised analysis, where perturbed and control cells fail to separate. We resolve this tension by distinguishing two axes the field conflates: recoverability, whether a supervised probe can read a label, and surfacing, whether the label occupies embedding variance without supervision. Across seven representations and six datasets, recoverability is high everywhere and, perturbation by perturbation, scales with the true transcriptional effect, whereas surfacing is low in every one of them, and whether the two axes come apart is governed by one geometric quantity, the ambient dimension of the representation. Ambient dimension thus leaves recoverability unchanged but gates whether a frozen embedding can be edited and interpreted, and this single quantity predicts three otherwise-separate phenomena. In high-ambient embeddings surfacing rather than recoverability predicts unsupervised usability (Spearman 0.77 vs 0.27), while at low ambient dimension the two axes coincide. Surfaced nuisances are removable at zero parameters everywhere, whereas aggressive edits and selective steering are safe only when ambient room exceeds the number of entangled directions: a compressed latent fails this criterion, a high-ambient one satisfies it, and an intermediate one sits on the boundary. And sparse-autoencoder features become perturbation-specific and recover the perturbed regulatory program only as ambient dimension grows (win rate 34% to 92%; regulon recovery 5% to 22%). One axis thus tells a practitioner in advance which operations a frozen embedding will support.