Transversal Matroids of Resolvable Group Divisible Designs Modulating the Catastrophic Failure Thresholds in Non-Rigid Dyson Swarm Hypergraphs

Mathematical Applications of Science Fiction We present a rigorous combinatorial and topological analysis of the structural stability of non-rigid DysonSwarm architectures modeled as hypergraphs H. By embedding the communication and power-transfertopology of the swarm into the incidence structure of a Resolvable Group Divisible Design (RGDD), weconstruct a canonical transversal matroid MT (D). We investigate the rank function rkM under stochasticedge percolation, representing component failure. We derive the exact catastrophic failure threshold pc,defined as the critical probability at which the rank of the matroid collapses, leading to a loss of globalrigidity. We prove that the resolvability of the design—specifically the partition of blocks into parallelclasses—provides a stabilizing homological invariant that mitigates the percolation phase transition. Weculminate in the derivation of the Valerius Stability Criterion, a necessary and sufficient condition for swarmpersistence in the limit of infinite component density.