The Three-Generation Problem in the Metric Bundle Framework

Paper 5 of 5 in the Metric Bundle Programme. We investigate the three-generation problem within the metric bundle framework, where the Pati-Salam gauge group \SU(4) × \SU(2)_L × \SU(2)_R and one generation of fermions emerge from Y^{14} = \Met(X^4). The algebraic construction yields exactly one \mathbf{16}-plet of \Spin(10); the observed three generations require a topological or geometric mechanism. We examine five candidate explanations: (i) the index theorem on Y^{14}, constrained by the contractibility of the fibre; (ii) a quaternionic structure on the positive-norm sector, providing a natural multiplicity of three from the three independent complex structures of \HH; (iii) topological sectors of the base…