The Extended Alpha Group and Integrable Foliations in Four-Dimensional Geometry

This work presents the structural framework of the Alpha Group and its associated integrable foliations in four-dimensional geometry. The formulation is based on the angular matrix M(θ) together with the idempotent projective operator μ, allowing the decomposition of the dynamics into symmetric and antisymmetric components. The symmetric sector is shown to generate invariant structures satisfying the Frobenius integrability condition, leading to a projective family of foliated manifolds. The antisymmetric sector, although not preserving classical local invariants, is organized by the projective operator μ, which governs global topological restructuring and coupled topological dynamics. This decomposition establishes a unified geometric framework in which local invariance and global topological reconfiguration coexist within the Alpha Group.