Mod--Three Quantization of Chiral Matter from a $\mathbb Z_3$ Two--Group Extension of the Standard Model Gauge Group

This work studies higher-group (two–group) extensions of the global Standard Model gauge group arising from torsion classes in group cohomology. Using the global topology of ((SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6), the analysis identifies a nontrivial (\mathbb{Z}_3) Postnikov class that is compatible with all admissible gauge bundles and matter representations. Coupling gauge theories to this two–group background leads to anomaly inflow whose coefficients are quantized modulo three. As a consequence, anomaly-free chiral matter sectors must occur in multiples of three within this framework. The result is conditional on the realization of the (\mathbb{Z}_3) two–group structure, but if realized it provides a topological explanation for the observed triplication of chiral matter. The argument relies only on global and cohomological properties of the gauge group and does not assume grand unification, flavor symmetries, or a specific ultraviolet completion.