Gauge Group Structure Theorem, Direct Product Theorem, and the Three-Generation Proposition— The Inevitable Emergence of SU(3)×SU(2)×U(1) from Regime Boundaries and the Topological Constraint on the Number of Heavy Quarks(III)

Building on the empirical findings and mathematical theorems of the previous two papers, this paper completes the theoretical derivation of the gauge group structure of the Standard Model from the Factor Hierarchy Law. The meta-regime factor (color charge) satisfies three conditions: dividing particles into mutually exclusive regimes, having irreducible interaction effects, and modulating all executive factors — these conditions mathematically define exactly the generators of a discrete symmetry group. Given that color charge is a meta-regime factor with three mutually exclusive regime states (the latter being a known fact of the Standard Model), SU(3) is the unique gauge group satisfying the conditions. The orthogonal independence of color charge and flavor quantum numbers (confirmed in Paper 15) implies that the two meta-regime factors act independently on distinct internal spaces, so their gauge groups must form a direct product. The SU(3)×SU(2)×U(1) structure of the Standard Model is not an arbitrarily written assumption but an inevitable algebraic consequence of this rule. The topological constraint of the transition zone width, together with the experimental exclusion of a fourth generation of quarks by the LHC, jointly points to the number of heavy quarks being limited to four. The four regime centers (strange, charm, bottom, top) fill the mass range from approximately 1.5 GeV to approximately 172.6 GeV. The spacing from bottom to top (162.4 GeV) already far exceeds those from strange to charm (0.8 GeV) and from charm to bottom (7.17 GeV); the spacing of a fifth regime center would be even larger, and its transition zone width would cover a vast region above the electroweak unification scale. The two theorems and one proposition proved in this paper — the Gauge Group Structure Theorem (Theorem 4), the Direct Product Theorem (Theorem 5), and the Three-Generation Proposition (Proposition 6) — together constitute the demonstration of the inevitable emergence of the gauge group structure from regime boundaries. Gauge field theory is not the most fundamental theoretical framework — it is the expression of regime boundaries in the language of group theory. The Factor Hierarchy Law is an organizing principle deeper than gauge field theory. As the third paper of the five-paper series, this paper completes this crucial theoretical leap, providing the core theoretical pillar for the conceptual reinterpretations and system legislation of the subsequent two papers. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.