A Complex and Triplex Framework for Encoding the Riemannian Dual Space-time Topology Equipped with Order Parameter Fields

In this work, we forge a powerful, easy-to-visualize, flexible, consistent, and disciplined abstract vector framework for particle and astro physics that is compliant with the holographic principle. We demonstrate that the structural properties of the complex number and the sphere enable us to introduce and define the triplex number—an influential information structure that is similar to the 3D hyper-complex number by D. White and P. Nylander—which identifies a 3D analogue of (2D) complex space. Consequently, we engage the complex and triplex numbers as abstract vectors to systematically encode the state space of the Riemannian dual 3D and 4D space-time topologies, where space and time are dual and interconnected; we use the triplex numbers (with triplex multiplication) to extend 1D and 2D algebraic systems to 3D and 4D configurations. In doing so, we equip space-time with order parameter fields for topological deformations. Finally, to exemplify our motivation, we provide three example applications for this framework.