Representation Rigidity in Face Embeddings: Orthogonal Identifiability and Backfill-Free Compatibility

Upgrading a deployed face-recognition model usually makes new query embeddings incompatible with the legacy gallery, forcing costly backfilling of stored embeddings. We show that this incompatibility is often much simpler than it appears: independently trained angular-margin face models exhibit representation rigidity, where their embedding spaces are well approximated by a single global orthogonal transformation. We formalize this phenomenon as Procrustes rigidity, derive a finite-calibration recoverability bound, and estimate the orthogonal adapter from a small paired calibration set via closed-form Procrustes alignment. The resulting deployment protocol is backfill-free: transport upgraded queries into the legacy coordinate system, then reuse the existing gallery without re-embedding stored identities. Across 45 models spanning three architectures, two losses, three data scales, and five seeds, we find strong rigidity within architecture families, non-monotonic scaling with data size, and robust cross-model retrieval. On the 1.6M-image IFRT benchmark, Procrustes-aligned queries recover at least $99.8\%$ of mono-model Rank-1 accuracy for same-family upgrades, while lightweight residual correction substantially narrows the cross-architecture gap.