Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations

Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring multi-agent configuration data. We introduce a quotient formation space Sn(M,G)=Mn/(G× Sn) and a formation matching metric dM,G obtained by optimizing a worst-case assignment error over ambient symmetries g∈ G and relabelings σ∈ Sn. This metric is a structured, physically interpretable relaxation of Gromov--Hausdorff distance: the induced inter-agent metric spaces satisfy dGH(Xx,Xy)≤ dM,G([x],[y]). Composing this bound with stability of Vietoris--Rips persistence yields dB(Phik([x]),Phik([y]))≤ dM,G([x],[y]), providing persistence-stable signatures for reconfiguration monitoring. We analyze the metric geometry of (Sn(M,G),dM,G): under compactness/completeness assumptions on M and compact G it is compact/complete and the metric induces the quotient topology; if M is geodesic then the quotient is geodesic and exhibits stratified singularities along collision and symmetry strata, relating it to classical configuration spaces. We study expressivity of the signatures, identifying symmetry-mismatch and persistence-compression mechanisms for non-injectivity. Finally, in a phase-circle model we prove a conditional inverse theorem: under semicircle support and a gap-labeling margin, the H0 signature is locally bi-Lipschitz to dM,G up to an explicit factor, yielding two-sided control. Examples on mathbbS2 and mathbbTm illustrate satellite-constellation and formation settings.