A Parameter-Free Geometric Origin for the Fermion Mass Hierarchy, Gauge Symmetries, and Spacetime Curvature

The Standard Model of particle physics and General Relativity rely on over two dozen empirical parameters, including fermion masses, mixing angles, and the cosmological constant, that must be inserted by hand. This monograph presents the Operator-Derived Dimension (ODD) framework, demonstrating that these parameters are not arbitrary axioms, but strict topological invariants of a 5D vacuum geometry.

By modeling the physical vacuum as the stable BPS domain wall of a triple-hypersphere intersection (S3⊂R5S3⊂R5), we formally derive the structure of quantum mechanics and the Standard Model from a single dimensionful input: the 5D Planck Mass (MPl).

Key Geometric Derivations:

• Gauge Structure: The SU(3)×SU(2)×U(1) group emerges as the inescapable isometry of the boundary triality and Hopf fibration.

• The Fundamental Constants: The fine-structure constant (α−1≈137.06) and Strong coupling (gs2=4/3) are evaluated from topological flux integrals over the soliton variance.

• The Mass Spectrum: The framework resolves the 14-order-of-magnitude hierarchy, from the Top Quark (173 GeV) to the Neutrino (<0.1eV), via a universal integer resonance lattice. The historical anomaly of the first-generation mass inversion (Down>>Up) is derived as the kinematic transition from 5D transverse shear to 4D centrifugal pooling.

• General Relativity: The Einstein Field Equations emerge from the Gauss-Codazzi projection of the bulk, with the cosmological constant (Λ4D=0) vanishing due to a dynamic self-tuning of the bulk curvature.

The ODD framework offers a complete, mathematically explicit transition from an unparameterized 5D geometry to the observable 4D universe, replacing parameter fitting with geometric necessity.