Active prime criterion

Statement

At the reduced attractor $\mu^* = 15$, a prime $p$ is called active if and only if its anomalous dimension satisfies

The active prime set is exactly $\mathcal{P}_{\text{act}} = \{3, 5, 7\}$. The exact rational formula is

Why it matters

The criterion $\gamma_p > s$ is the filter that selects the three primes physically relevant to the Standard Model. It simultaneously delivers:

• the number of colours $N_c = |\mathcal{P}_{\text{act}}| = 3$ ($N_c$ theorem, ch. 8),
• the number of generations $N_{\text{gen}} = 3$ (via Fisher–Koide),
• the fixed point $\mu^* = 3 + 5 + 7 = 15$ (T5),
• the product form of the coupling $\alpha_{\text{sieve}} = \prod_{p \in \{3,5,7\}} \sin^2\theta_p$ (BA5).

Without the active prime criterion, there would be no canonical discrimination between primes — and so no reason for the physical observables to organise around exactly three channels.

Proof — outline

(i) Exact rational computation (fractions.Fraction) at $\mu^* = 15$:

The first three values are strictly $> 1/2$, the last two strictly $< 1/2$. The boundary falls between $p = 7$ and $p = 11$.

(ii) Analytic monotonicity argument for $p \geq 7$: write $\gamma_p = F(p)\,G(p)$ where $F$ is exponentially decreasing (factor $q^{p-1}$) dominating the polynomial growth of $G$. Hence $\gamma_p < \gamma_7 < 1/2$ for every $p \geq 11$.

(iii) Robustness: any threshold $\tau \in [0.43, 0.59]$ yields the same active set. The physics is structurally stable under ±17 % variation of the threshold.

Full details: monograph §6.6, file PT.Holonomy.ActivePrimeAnalyticMonotonicity.

See also

• T6 — Holonomy — $\sin^2\theta_p$ formula
• T5 — Attractor $\mu^* = 15$ — $\mu^* = \sum_{\text{active}} p$
• $N_c = 3$ — colour from the sieve
• BA5 — Pontryagin product — product over the active primes
• All theorems