Solid-Sierpinski Universal Coordinate System: A Complete Framework for Multi-Scale 3D Spatial Localization

We present a revolutionary universal coordinate system for multi-scale spatial localization using a modified Sierpinski Triangle as a 2D projection surface for 3D space. Unlike classical Sierpinski constructions that remove central triangles (creating incomplete coverage), we introduce the Solid-Sierpinski variant where central regions are filled with inverse-oriented recursive subdivisions, achieving total planar coverage with Hausdorff dimension 2.0. The Master Sierpinski Triangle (MST), anchored at a spatial origin P₀, defines a reference plane. Any 3D point P receives a hybrid address: (1) quaternary planar coordinates (base-4: digits 0,1,2,3) within the 2D hierarchy, and (2) perpendicular depth (logarithmic layers) orthogonal to the plane. This framework spans 38 orders of magnitude—from galactic structures (10²³ m) to subatomic particles (10⁻¹⁵ m)—using identical mathematical principles with standardized reference scales to ensure universal consistency. We establish rigorous proofs of completeness, uniqueness, continuity, and scale invariance. Computational algorithms enable practical implementation. The hierarchical structure facilitates efficient spatial indexing with O(log n) query complexity. Applications include astronomical mapping, biological anatomy indexing, molecular structure encoding, and quantum state localization. Hyper-Pascal Lattice: Infinite-Zoom Universal Coordinate System for Telescopes and Microscopes Paper link : https://doi.org/10.5281/zenodo.18135213