Sphere Numbers and the Number Sphere Field - A Consolidated Algebraic-Geometric Framework

This paper introduces the theory of Sphere Numbers and the associated Number Sphere Field, a unified algebraic–geometric framework based on a three-dimensional real algebra equipped with multiplication, conjugation, and norm structures. Sphere Numbers are represented as points in a three-dimensional space, where elements of equal norm form concentric spheres, giving rise to a geometric field of spherical layers. The framework develops the algebraic foundations of Sphere Numbers, establishes their geometric interpretation, and constructs an intrinsic differential and spectral calculus on the resulting Number Sphere Field. Connections with spherical harmonics, spectral operators, and selected structures from analytic number theory are explored within a unified mathematical setting. This work forms part of a broader research program on Sphere Numbers and their geometric, analytical, and computational properties. The present article consolidates and refines several earlier formulations into a single self-contained framework intended as a foundation for subsequent developments.