Exactly Three Colors: N_c = 3 from Primorial Depth
The number of color charges N_c = 3 is derived from the primorial depth d = 3 of the strong sector.
Overview
IV.T70 proves that the number of color charges N_c = 3 follows from primorial depth d = 3 of the strong (η) sector. The strong sector operates at the third primorial level (the C-sector in the ABCD chart), where primorial depth d means the sector occupies the d-th primorial orbit. Exactly three independent color charges arise because the third primorial orbit has exactly three primitive elements.
Detail
The number of color charges N_c = 3 in QCD is an experimental observation with no theoretical derivation in the Standard Model. Various arguments (cancellation of anomalies, statistical mechanics of baryons) make N_c = 3 consistent but do not derive it. Book IV derives N_c from the position of the strong sector in the primorial hierarchy. The primorial sequence is p# = 2, 6, 30, 210, …; the d-th primorial level is the level at which d prime factors appear. The η-sector (strong force) is assigned to the third primorial orbit because it has the third-highest binding energy among the four primitive sectors. At primorial depth d = 3, the relevant lattice structure has exactly three primitive elements modulo the orbit action of ρ. These three elements are the three color charges. The derivation is unique: at d = 2 there would be two colors, at d = 4 there would be four. The empirical value N_c = 3 selects depth d = 3 uniquely.
Result Statement
IV.T70: N_c = 3 from primorial depth d = 3. The strong sector operates at the third primorial level, forcing exactly three color charges.