This preprint is intended for publication in \textit{IPI Letters} and presents a mathematical model of spontaneous and explicit symmetry breaking in six-dimensional oscillatory geometry. The framework treats spacetime as a compact toroidal manifold $\mathcal{M}_6 = T^2_1 \times T^2_2 \times T^2_3$, whose internal degrees of freedom support stable oscillatory modes. By introducing a symmetry-breaking quartic potential and position-angle coupling terms, we demonstrate how SO(3) rotational symmetry reduces to SO(2), and U(1) number symmetry is explicitly violated. The resulting anisotropic dispersion relations and geometric phase shifts provide a natural explanation for direction-dependent mass in semi-Dirac fermions and align with the observed muon $g\!-\!2$ anomaly. This work lays the theoretical foundation for understanding how mass, forces, and quantum corrections may emerge from the internal geometry of compact space.