Algebraic Stability and Cosmological Structure B, The Geometric Origin of Three Generations of Fermions

This paper reveals the deep origin of the three-generation fermion structurewithin the Standard Model within a unified, falsifiable mathematical physics framework. We demonstrate that three independent paths——the geometric-topologicalpath (based on the Atiyah–Singer index theorem on G2 manifolds), the informationalgebraic path (based on Jones subfactors and information phase transitions), andthe self-referential dynamics path (based on spectral gap analysis of recursive stability)——reduce ontologically to the same set of algebraic stability primitives. This primitive system consists of six irreducible mathematical elements: the algebraic stability functional, the Jones discrete spectrum, the minimal information potentialdifference, the spectral gap constant, the total Betti number constraint, and thegauge bundle rank constraint. This system uniquely locks the topological invariants (b2,b3) = (21,77) of a seven-dimensional G2 manifold, and from this naturallyderives the gauge group SU(3) × SU(2) × U(1) and three generations of chiralfermions. All derived parameters originate from mathematical constants or integerratios, with no freely adjustable parameters. This framework yields seven quantitative, cross-scale testable predictions and establishes a clear three-tiered falsificationstructure: the rigid core (Tier I), phenomenological bridges (Tier II), and parametervalues (Tier III). We publicly commit: the framework is falsified as a whole if andonly if the algebraic solver in Appendix D, given standard primitive inputs, outputsa solution other than (21,77). This work elevates the deep structure of the Standard Model from a soft anthropic explanation to a rigid expression of mathematicalnecessity