SO(16,3) Symmetry: Unification of Gravity and Gauge Forces

This paper proposes and systematizes a four-force unification model based on the non-compact orthogonal Lie group $\mathrm{SO}(16,3)$. We demonstrate the mathematical consistency of the group decomposition chain\[\mathrm{SO}(16,3) \supset \mathrm{SO}(6,3) \times \mathrm{SO}(10) \supset \mathrm{SO}(3,1) \times \mathrm{SO}(2,1) \times \mathrm{U}(1)_Y \times \mathrm{SU}(5) \supset \text{Standard Model},\]and construct a purely geometric MacDowell–Mansouri-type action\[S_{\text{fund}} = \frac{1}{2\kappa^2} \int d^4x\,\epsilon^{\mu\nu\rho\sigma} \,\epsilon_{I_1 I_2 I_3 I_4 J_1 \cdots J_{15}} \,\mathcal{R}_{\mu\nu}{}^{I_1 I_2} \,\mathbb{E}_{\rho}{}^{I_3} \,\mathbb{E}_{\sigma}{}^{I_4} \,\Phi^{J_1} \cdots \Phi^{J_{15}}\], which involves only the $\mathrm{SO}(16,3)$ curvature and the frame field, and \textbf{introduces no fundamental Yang–Mills terms}. All low-energy physics—including general relativity, vector/scalar gravity, and the gauge dynamics of the Standard Model—emerge naturally as \textbf{induced effects} following spontaneous symmetry breaking of $\mathrm{SO}(16,3)$. In particular, the unified gauge coupling $g_{\text{uni}}$ and the gravitational constant $\kappa$ are related through a single geometric scale $v$ via\[g_{\text{uni}} = \frac{\kappa}{v},\]realizing a \textbf{genuine dynamical unification}. The central innovation of this work lies in interpreting the $\mathrm{SO}(6,3)$ subgroup as a generalized theory of gravity encompassing three fundamental gravitational degrees of freedom—tensor gravity (from $\mathrm{SO}(3,1)$), vector gravity (from $\mathrm{SO}(2,1)$), and scalar gravity (from $\mathrm{SO}(1,1)$)—and in viewing the $\mathrm{SO}(10)$ gauge forces as effective projections of gravitational geometry onto the internal space, thereby establishing a profound unification between gravity and gauge interactions.