The Hyper-Pascal Lattice: A Combinatorial Framework for Hierarchical Discrete Space

We introduce the Hyper-Pascal Lattice (HPL), a novel mathematical structurethat reinterprets Pascal’s Triangle as an infinite hierarchical network with binaryaddressing. Unlike the classical arithmetic interpretation of binomial coefficients,the HPL treats each position (n, k) as a node in a directed acyclic graph connectedby topological rather than numerical relations. We establish that this structureforms a complete ultrametric space with Hausdorff dimension 1.0 in its addressspace, while achieving full 2-dimensional coverage in its geometric projection. Thisdimensional bridge between discrete encoding and continuous space provides a newframework for hierarchical data structures, fractal geometry, and discrete modelsof spacetime. We prove fundamental theorems on completeness, self-similarity, andscale invariance, and demonstrate that the HPL serves as the combinatorial analogof the Solid-Sierpi´nski system. Applications span computer science, informationtheory, and theoretical physics.