Quantifying Observability Limits in Decentralized Swarms: An Information-Geometric Approach

Decentralized multi-agent systems must maintain collective safety using local, incomplete observations rather than global state knowledge. This paper develops an information-geometric framework for quantifying when a swarm agent's local observations cease to distinguish safety-relevant changes in its neighborhood. Each agent constructs a Neighborhood Fisher Information Matrix (N-FIM) from relative positions, bearings, and velocities of nearby agents. Because these measurements are invariant to common translations and rotations, the framework parameterizes the local state space using identifiable relative quantities. A Safety-Relevant Observability Boundary is defined as the point at which local Fisher information no longer supports estimation of a safety-relevant relative quantity within a prescribed tolerance. The framework distinguishes observability degradation from safety failure by treating local statistical identifiability and physical safety as separate constructs. Safe, dangerous, and hazardous state spaces are independently defined and used as ground-truth labels during validation. The proposed methodology employs local Fisher information geometry, Cramér-Rao variance bounds, graph-theoretic connectivity measures, and Kullback-Leibler divergence-based observational margins. Validation is proposed through simulation studies of obstacle avoidance, communication-induced fragmentation, and noise-driven order-disorder transitions in Vicsek-type swarms. The central hypothesis is not that Fisher-information decay causes swarm failure, but that reductions in local statistical identifiability may provide early warning of safety-critical collective transitions without requiring global state information. The paper establishes a testable framework for decentralized observability monitoring in autonomous multi-agent systems and provides a foundation for future empirical investigation.