Algebraic Stability and Cosmological Structure D: From Information Phase Transition to Three Generations of Fermions

This paper, starting solely from the conservation of information distinguishability and critical phase transition conditions within the discrete information hierarchyframework characterized by subfactor theory, and completely independent of anyphenomenological input from particle physics, rigorously derives the unique possiblevalue for the number of chiral fermion generations in the four-dimensional effectivetheory as Ngen = 3. The entire derivation relies only on: (i) the algebraic rigidityof Jones’ discrete spectrum; (ii) the extremal principle of topological degrees offreedom in the information-geometric duality; and (iii) the integrality constraintsimplied by the generalized Atiyah–Singer index theorem on G2 manifolds. We provethat after the cumulative information twistor crosses the critical threshold determined by the tensor product of the Fibonacci category and the SU(2)3 category,the system necessarily gives rise to a seven-dimensional internal space with G2holonomy; the second and third Betti numbers (b2,b3) of this space, together withthe fermion generation number Ngen, are uniquely locked by a system of Diophantine equations, whose solution is (b2,b3,Ngen) = (21,77,3). This result reduces thethree-generation fermion structure from an empirical observation to an arithmeticnecessity of information-dynamical self-consistency, providing the first rigid case ofparameter-free falsifiable predictions in quantum gravity theory.