Representation Splitting Theorem and the Geometric Origin of Three Fermion Generations: A First-Principles Derivation from Scale Geometry

The origin of three fermion generations in the Standard Model (SM) has remained a foundational unsolved puzzle for over half a century, with conventional approaches treating the generation structure as an ad hoc input. This work is the fourth paper in a 7-part series built on the Scale Geometry unified framework, rooted in the foundational S³ Quantum Deformation Geometry Operator (Paper 1), geometric locking of the fine-structure constant (Paper 2), and elimination of ultraviolet divergences (Paper 3). As a component of the parent monograph The Scale Geometry Framework: Topology, Projection, and Static Universe (https: //doi. org/10. 5281/zenodo. 19368089), we provide a rigorous first-principles solution to the generation puzzle, in homage to Albert Einstein's vision of geometric unification of all fundamental physical structures. We formally state and prove the Representation Splitting Theorem: when the fundamental representation of the quantum group SUq (3) is restricted to the right coideal subalgebra of the SUq (3) /T quantum flag manifold, it decomposes into three inequivalent 1-dimensional irreducible representations—identified with the three SM fermion generations. We establish a one-to-one mapping between these subrepresentations and generation degrees of freedom, explaining identical gauge quantum numbers across generations as a natural geometric consequence. We derive the Casimir eigenvalues of the subrepresentations, providing a geometric explanation for the extreme intergenerational mass hierarchy, and predict right-handed neutrinos as a geometric necessity (supporting the Type-I Seesaw mechanism). We further propose falsifiable experimental predictions, including neutrino normal ordering and tiny flavor-changing neutral current corrections. This work solves the SM generation puzzle, laying a self-consistent geometric foundation for flavor physics within Scale Geometry. V2;After adopting the RP³ = S³/Z₂ manifold as the underlying spacetime topology, the intrinsic Z₂ quotient breaks global chiral symmetry and filters the representation spectrum at the geometric level; as a result, right-handed neutrinos can no longer exist as physical propagating degrees of freedom. This implies that nature is fundamentally left-handed, and such intrinsic chiral asymmetry directly originates the arrow of time. Both right-handed neutrinos and the corresponding seesaw mechanism are theoretically consistent within the original S³ external manifold framework; however, under the RP³ topology, physical right-handed degrees of freedom are geometrically excluded from the observable particlespectrum. (https: //doi. org/10. 5281/zenodo. 19504572) Author: Xinyu Zheng (郑心宇) ORCID: 0009-0000-3175-1681Correspondence: wxsq1638@outlook. comDOI: To be Assigned by Zenodo