Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations

Can temporal information recover geometry lost by a persistence diagram? For $n\ge3$ points on the unit sphere evolving under known homogeneous complete consensus, we prove generic local recovery modulo rotations and permutations from two ordinary zero-dimensional Vietoris--Rips persistence snapshots. The structural result applies to every labeled tree: its edge inner products and their first temporal derivatives have differential rank $2n-3$, exactly the dimension of the regular formation quotient. An explicit pole--equator witness proves this simultaneously for all tree types after taking a finite intersection of open dense sets. Although one snapshot has generic local fibers of dimension $n-2$, two sufficiently close snapshots are injective on a common neighborhood. The optimal inverse Lipschitz constant grows as $Θ(1/|h|)$ when the sampling interval tends to zero. A verified left inverse, interval derivative bounds, and explicit persistence-matching margins certify neighborhoods and time intervals at rational reference formations. The results are local and model-specific; reflection ambiguity and centered rank obstructions remain.