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 $\mathcal{S}_n(M,G)=M^n/(G\times S_n)$ and a formation matching metric $d_{M,G}$ obtained by optimizing a worst-case assignment error over ambient symmetries $g\in G$ and relabelings $σ\in S_n$. This metric is a structured, physically interpretable relaxation of Gromov--Hausdorff distance: the induced inter-agent metric spaces satisfy $d_{\mathrm{GH}}(X_x,X_y)\le d_{M,G}([x],[y])$. Composing this bound with stability of Vietoris--Rips persistence yields $d_B(Φ_k([x]),Φ_k([y]))\le d_{M,G}([x],[y])$, providing persistence-stable signatures for reconfiguration monitoring. We analyze the metric geometry of $(\mathcal{S}_n(M,G),d_{M,G})$: 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 $H_0$ signature is locally bi-Lipschitz to $d_{M,G}$ up to an explicit factor, yielding two-sided control. Examples on $\mathbb{S}^2$ and $\mathbb{T}^m$ illustrate satellite-constellation and formation settings.