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 without the need to postulate any ad hoc Lagrangians. It is an extension of special relativity that relies on three key assumptions. First, the speed of light is constant with respect to translations, rotations, and boosts. Second, 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. And third, 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. 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, respectively. Imposing Lorentz invariance on these waves is equivalent to curving spacetime as predicted by ECSK theory. The gauge forces result from momentum flux moving through one, two, or three axes of the microscopic three-sphere dimensions. The generators of the U(1), SU(2), and SU(3) algebras are mathematically identical to the geometric projectors of this complex-valued momentum flux, and parametrizing the cross-metric block of the six-dimensional Einstein field equations according to these projectors causes non-macroscopic frame-dragging that we observe as gauge fields. The core postulates of quantum field theory, namely the principal fiber bundle, the Feynman path integral, discrete interaction vertices, and Fock space ladder operators, are shown to be geometric consequences of the model. This paper also derives the Koide formula from the theoretical principles of the framework.