A Mathematical Theory of Hyper-simplex Fractal Network for Blockchain: Part I

Abstract: Blockchain technology holds promise for Web 3.0, but scalability remains a critical challenge. Here, we present a mathematical theory for a novel blockchain network topology based on fractal N-dimensional simplexes. This Hyper-simplex fractal network folds one-dimensional data blocks into geometric shapes, reflecting both underlying and overlaying network connectivities. Our approach offers near-infinite scalability, accommodating trillions of nodes while maintaining efficiency. We derive the mathematical foundations for generating and describing these network topologies, proving key properties such as node count, connectivity patterns, and fractal dimension. The resulting structure facilitates a hierarchical consensus mechanism and enables deterministic address mapping for rapid routing. This theoretical framework lays the groundwork for next-generation blockchain architectures, potentially revolutionizing large-scale decentralized systems. The Part I work was conducted between March and September 2024.