Hyperdimensional Extensions of Sphere Numbers: Clifford Algebraic Structures and Transdimensional Mathematics

This work develops a hyperdimensional extension of the Sphere Number framework through the construction of Sphere Number Algebras in arbitrary dimensions. The resulting structures establish a direct connection with Clifford algebras, providing a unified algebraic setting that combines geometric interpretation, norm theory, and higher-dimensional algebraic operations. The framework introduces discrete-continuous duality operators, lattice substructures, complexified extensions, Hermitian formulations, and transdimensional embeddings between spaces of different dimensions. Isometric embeddings into Euclidean spaces are constructed, revealing geometric and spectral properties of the associated algebraic structures. Applications to geometric algebra, quantum information theory, and mathematical models of transdimensional systems are discussed. The proposed