Extremal Hypergraph Saturation and Asymptotic Density Thresholds in Self-Replicating Dyson Swarm Architectures

Mathematical Applications of Science Fiction We present a rigorous graph-theoretic framework for optimizing the structural integrity and energythroughput of self-replicating Dyson Swarms modeled as k-uniform hypergraphs. By investigating theextremal saturation number sat(n, F) for forbidden sub-configurations F representing catastrophicresonance cascades, we derive tight asymptotic lower bounds for resource allocation. We introduce theFlux-Weighted Adjacency Tensor to account for stochastic stellar flux anomalies and prove that thestability of the swarm converges to a limit distribution governed by the spectral gap of the Laplacian ofthe underlying hypergraph. Furthermore, we establish the "Pax-Dyson Threshold," a critical densitybeyond which self-replication triggers a phase transition in connectivity, ensuring global coherencedespite local node failures. Our results utilize methods from probabilistic combinatorics, spectralhypergraph theory, and homological algebra.