$N_c = 3$ — Colour from the sieve

Statement

The colour number $N_c$ and the spatial dimension $N_{\text{spatial}}$ are the unique positive integer simultaneously satisfying the spatial dimension equation:

The unique integer solution is $(N_c, N_{\text{spatial}}) = (3, 3)$.

Why it matters

This is one of the most striking PT results: the colour number $N_c = 3$ of QCD and the spatial dimension $N_{\text{spatial}} = 3$ of our universe are jointly forced by a single elementary arithmetic equation, with no adjusted parameter.

The theorem is purely arithmetic. Its physical interpretation ($N_c = 3$ colours, $N_{\text{spatial}} = 3$ dimensions) is a BRIDGE identification — but the combinatorial uniqueness itself is a THM.

Combined with the active prime criterion ($\mathcal{P}_{\text{act}} = \{3,5,7\}$, hence $|\mathcal{P}_{\text{act}}| = 3$) and the generation count $N_{\text{gen}} = 3$ (theorem N_gen_from_active_spectrum), PT delivers the triad

from the single fixed point $\mu^* = 15$.

Proof — outline

Direct enumeration:

For $N_c \geq 5$, $(N_c+1)!/(N_c+3)$ is neither an integer nor a power of 2 — verified exhaustively by enumeration up to $N_c = 20$, then closed off by factorial growth dominating the polynomial decay of the denominator.

The unique pair $(N_c, N_{\text{spatial}}) = (3, 3)$ survives.

See also

• Active prime criterion — where “3” primes come from
• T5 — Attractor $\mu^* = 15$ — $15 = 3 + 5 + 7$
• T6 — Holonomy — phases $\sin^2\theta_p$ over the 3 active primes
• All theorems