Mathematical Foundations of the Tridias Couplet Model

The Tridias Couplet model unites general relativity and the standard model of particle physics into a single framework. It relies on three key assumptions. First, it is possible to establish a frame of reference in which a photon is motionless, and from this frame of reference, we would observe a three-dimensional space locally equivalent to the three-dimensional space that we observe. Second, while our own space is a macroscopic three-sphere (or possibly a three-dimensional Euclidean space), this other space is a microscopic three-sphere with a Planck-length-scale radius. And third, the basic postulates of general relativity apply, but extended to six dimensions. As a result of these assumptions, a transformation from a frame of reference in one space to a frame of reference in the other space is not a one-to-one mapping, which causes frames of reference in one space to move as probability waves in the other. For matter, these waves are described by the six-dimensional Dirac equation, forming standing waves in the microscopic three-sphere. The fermion particle families—leptons, neutrinos, and quarks—are one-axis, two-axis, and three-axis solutions to the Dirac equation in Hopf coordinates. The gauge forces result from momentum flux moving through one, two, or three axes of the microscopic three-sphere dimensions. The foundational tangent vectors of a compact sphere—historically categorized as the U(1), SU(2), and SU(3) algebras—are mathematically identical to the geometric projectors of this momentum flux, and parametrizing the cross-metric block of the six-dimensional Einstein field equations according to these projectors causes structured non-macroscopic frame-dragging that we observe as gauge fields. This paper also derives the Koide formula from the theoretical principles of the framework.