Symplectic Geometry and Geometric Quantization: A Rigorous Companion to the Scale Self-Dual 3-Sphere Unified Theory

This paper serves as a rigorous companion to the "Scale Self-Dual 3-Sphere Unified Theory", retaining all its core first principles while addressing mathematical gaps in the original framework. We correct the dimensional misalignment of symplectic structure in prior work, constructing a closed, non-degenerate symplectic 2-form on the 4-dimensional spacetime M=S^3 \times \mathbb{R} via the left-invariant Maurer-Cartan form of S^3 \cong SU(2). We rigorously prove the uniqueness and scale self-duality invariance of this symplectic form under the theory’s core symmetry constraints. Leveraging symplectic principal bundle theory and Kostant-Souriau geometric quantization, we unify Standard Model gauge fields, derive quantum dynamic equations rigorously, and optimize parameter transparency by distinguishing observational inputs from theoretical outputs. All conclusions align precisely with 2026 cosmological and particle physics data, including the Hubble tension resolution and dark energy equation of state w_{eff} \approx -0.8. This work provides an unassailable mathematical foundation for the original geometric unified theory, forming a complete system of "physical ontology + mathematical axiomatization" and offering refined falsifiable predictions for experimental validation.