A Unified Theory Based on Three-Dimensional Complex Geometric Structure

This paper proposes a dynamical symmetry-breaking unified field theory model based on a three-dimensional complex manifold $M_{C}^3$ and its symmetry group $U(3)$. Distinct from mainstream approaches in traditional Grand Unified Theories (GUTs) that expand group dimensions (e.g., $SU(5)$, $SO(10)$) or introduce extra spacetime dimensions in string theory, this framework deeply explores the intrinsic geometric structure of $U(3)$ and signature-breaking dynamics, achieving a unified structural explanation for the four fundamental interactions, the fermion spectrum, and dark energy. The core innovation lies in rigorously distinguishing three hierarchical levels of symmetry: (1) \textbf{Ontological Level} ($U(3)$): The irreducible representation $\mathbf{3}\oplus\mathbf{1}$ naturally defines the classification of quarks and leptons, unifying the strong force with the primordial hypercharge interaction; (2) \textbf{Dynamical Level} ($U(2,1)\supset U(2)$): Gravitational geometry emerges through signature breaking, with its maximal compact subgroup $U(2)$ unifying the weak force and hypercharge interaction; (3) \textbf{Dual Level} ($U(1,2)$): The two-time branch, suppressed by causality, manifests as dark energy in the form of geometric tension. While maintaining mathematical minimality, this theory naturally circumvents the proton decay problem of traditional GUTs and the vacuum landscape dilemma of string theory, providing a self-consistent geometric unified framework for core puzzles in particle physics and cosmology.