Sphere Numbers and the Number Sphere Field: A Consolidated Algebraic–Geometric Framework

This paper presents a complete and consolidated formulation of the theory of Sphere Numbers and the associated Number Sphere Field. The work introduces a three-dimensional real algebra endowed with a norm-preserving associative product and a natural geometric realization as concentric spheres of constant curvature. This framework unifies, corrects, and rigorously extends a research program previously developed across six Zenodo preprints by the author. Those earlier formulations are superseded here by a single coherent algebraic–geometric structure, a complete differential and spectral calculus, and a mathematically closed framework suitable for further applications in analysis and mathematical physics. The algebraic foundations include a rigorous definition of Sphere Numbers as a three-dimensional real division algebra with explicit structure constants, norm multiplicativity, conjugation, and a closed-form inverse. The Number Sphere Field is constructed as a foliation of concentric Riemannian spheres of constant positive curvature, providing a geometric stratification where algebraic magnitude corresponds to radial position and directional information is encoded on spherical shells. A complete intrinsic calculus on the field is developed, including gradient, divergence, Laplace–Beltrami operator, spherical harmonics, spectral decomposition, and Taylor series expansions in both Cartesian and spherical coordinates. The framework establishes a precise correspondence with analytic number theory: prime numbers are encoded as radial positions via logarithmic scaling, the von Mangoldt function defines spherical wave packets, the Chebyshev function is represented as a radial integral, and a prime-counting operator is introduced. The non-trivial zeros of the Riemann zeta function are given a spectral interpretation, and a geometric reformulation of the Riemann Hypothesis is proposed. Computational implementations in Python are provided, including numerical simulations of the heat equation on spheres, approximation of prime-counting functions using spherical harmonic expansions, and visualization tools. Numerical results demonstrate rapid convergence and stability. The paper consolidates, refines, and supersedes six earlier preprints, providing the first fully unified, self-contained, and journal-ready exposition of the Sphere Number paradigm. This work opens new avenues for research in hyperdimensional algebra, spectral geometry, analytic number theory, and mathematical physics.