Hyper-Pascal Lattice: Infinite-Zoom Universal Coordinate System for Telescopes and Microscopes

We present the Hyper-Pascal Lattice (HPL) — a revolutionary coordinate system with theoretically infinite zoom capability, enabling continuous magnification from galactic scales (~10²¹ m) to subatomic scales (~10⁻¹⁵ m) and beyond, without any mathematical bound. Unlike conventional coordinate systems that fix resolution at a given scale, the HPL employs Pascal’s Triangle as a hierarchical combinatorial network in which each zoom level reveals new binary subdivisions ad infinitum. Topologically, the HPL reinterprets Pascal’s Triangle: each position (n, k) represents not a binomial coefficient, but a node in an infinite directed acyclic graph with binary connectivity. Every spatial point receives a potentially infinite binary address (b₀, b₁, b₂, …), where each additional digit doubles the spatial resolution. This defines a truly scale-free addressing mechanism—the same structural principles hold whether observing galaxy clusters through telescopes or molecular structures through electron microscopes. Key Innovation.The zoom operator Z : (b₀ … b_N) ↦ (b₀ … b_N b_{N+1})can be applied indefinitely, with each iteration halving the uncertainty radius. This architecture enables: Astronomical mapping: hierarchical star catalogs with arbitrarily fine positional precision; Microscopy: continuous cellular or molecular tracking at unbounded magnification; Multi-scale unification: seamless coordinate transformations between macroscopic and microscopic regimes; Future-proof addressing: expanding resolution simply by appending binary digits as instrumentation improves. We prove that the resulting address space forms a complete ultrametric space homeomorphic to the Cantor set, while its geometric projection achieves full two-dimensional coverage (Hausdorff dimension = 2.0). This dimensional bridge between one-dimensional encoding and two-dimensional geometry produces a unique form of compression: infinite information capacity within a discrete hierarchical structure. Efficient algorithms support O(N) coordinate transformations and O(log n) spatial queries. Solid-Sierpinski Universal Coordinate System: A Complete Framework for Multi-Scale 3D Spatial Localization Paper Link: https://doi.org/10.5281/zenodo.18050925