Last Base Mathematics: A Constructible Radial Framework for Number Representation and Compression

This paper introduces Last Base Mathematics (LxB), a novel numerical and geometrical framework based on an alternating base system rooted in base-12 with a secondary recursive base, commonly base-5. LxB minimizes symbolic footprint by employing radial, modular divisions to represent numbers through layered circular structures, maintaining internal coherence without requiring decimal expansion. We formalize the arithmetic operations intrinsic to LxB, including native addition, subtraction, multiplication, and division rules respecting alternating bases. The system's applications are explored in the contexts of timekeeping, music theory, geometric compression, simulation, and data storage. Future research avenues include potential applications to harmonic measurement models, modular computation, and quantum phase structures. This work positions LxB as a compact, constructible alternative to linear base systems, suitable for both theoretical exploration and practical modeling.