This paper presents a complete and consolidated formulation of the theory of SphereNumbers and the associated Number Sphere Field, introducing a three-dimensionalreal algebra endowed with a norm-preserving associative product and a natural geometric realization as concentric spheres of constant curvature. The present workunifies, corrects, and rigorously extends a research program previously developedacross six Zenodo preprints by the author. Those earlier formulations are superseded here by a single coherent algebraic–geometric structure, a complete differentialand spectral calculus, and a mathematically closed framework suitable for furtherapplications in analysis and mathematical physics.Keywords: Sphere Numbers, Number Sphere Field, Geometric Algebra, NormedDivision Algebras, Spherical Geometry, Spectral Theory, Prime Number Theory,Riemann Zeta Function, Spherical Harmonics
Mohammad Reza Alaei Jordehi (Sat,) studied this question.