Three-Dimensional Number Algebra System — A Non-Associative Algebra Unifying Complex Plane and Hyperbolic Phase Space with Applications in Differential Geometry and Number Theory

Abstract This paper constructs a complete axiomatic system of three-dimensional numbers, defined as ordered triples . The basis identities hold simultaneously without algebraic contradictions, and the algebra fully inherits all operational properties of standard complex numbers. We systematically derive closed formulas for addition, subtraction, scalar multiplication, multiplication, division, three types of conjugates, algebraic norm and Euclidean norm, and classify eight octants, subalgebras and zero-divisor singular regions. This 3D number space naturally couples the horizontal complex plane and vertical hyperbolic phase surface, providing independent vertical phase freedom missing in 2D complex analysis. We further prove that the algebra is compatible with Riemannian metric tensors and Laplace–Beltrami operators on 3D manifolds, and can be applied to construct the full 3D phase manifold of the Riemann zeta function. Extensive interdisciplinary applications are proposed for analytic number theory (Riemann Hypothesis CLC potential well), mathematical physics, and economic dynamical systems. This self-consistent algebraic framework retains the full analytical power of complex numbers while adding native geometric descriptions of phase deformation, offering a fundamental new tool for phase space geometry, arithmetic gravity and spectral geometry.Keywords: three-dimensional numbers; non-associative algebra; hyperbolic phase; complex plane; Riemannian metric; zeta-gravity metric; phase manifold; critical line convexity