N_c

The number of colours

N_c = 3 is the QCD colour count (red, green, blue). In PT, it is a theorem, not a postulate:

$$ N_c = |\{p\,:\,\gamma_p(\mu^*) > 1/2\}| = |\{3, 5, 7\}| = 3. $$

Why exactly 3?

At $\mu^* = 15$, the active primes (those with anomalous dimension exceeding $s = 1/2$) are precisely {3, 5, 7}. No others. T5 proves this by exhaustive rational scan.

The consistency between N_c = 3 (quark colours) and N_gen = 3 (fermion generations) is not a coincidence: both count the same set {3, 5, 7}. This is what makes the Standard Model gauge-anomaly-free — required for quantum consistency.

Epistemic status

Exact. Not a measurement, not a probabilistic prediction. An unconditional theorem: $|\{3, 5, 7\}| = 3$ by enumeration.

See essays [Why three dimensions?](/en/essays/why-3-dimensions) and [Why three generations?](/en/essays/why-3-generations) for plain reading.