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.

翻译：区块链技术为Web 3.0带来希望，但可扩展性仍是关键挑战。本文提出一种基于分形N维单纯形的新型区块链网络拓扑数学理论。该超单纯形分形网络将一维数据块折叠为几何形状，同时反映底层与覆盖层的网络连接性。我们的方法提供了近乎无限的可扩展性，能够容纳数万亿节点同时保持效率。我们推导了生成和描述这些网络拓扑的数学基础，证明了节点数量、连接模式和分形维数等关键性质。所得结构促进了分层共识机制，并实现了确定性地址映射以实现快速路由。该理论框架为下一代区块链架构奠定了基础，可能彻底改变大规模去中心化系统。第一部分工作于2024年3月至9月期间完成。