|V_us|

V_us and the Cabibbo angle

$|V_{us}| \approx \sin\theta_C$ where $\theta_C \approx 13.0°$ is the Cabibbo angle. In PT, this angle derives from the cascade of anomalous dimensions on the q_- branch (geometry / quarks).

PT formula

$$ \sin\theta_{12}^{\rm CKM} = \sqrt{\frac{\delta_5(q_-)}{\delta_3(q_-) + \delta_5(q_-)}} \cdot K_{12} $$

with $K_{12} = 1 - O(\alpha_{\rm EM})$ correcting for electroweak feedback. At $\mu^* = 15$, $q_- = e^{-1/15}$:

$$ \delta_3(q_-) = (1 - q_-^3)/3 = 0.06453, \qquad \delta_5(q_-) = 0.05751. $$

Computation

$$ \sin\theta_{12} = \sqrt{\frac{0.05751}{0.06453 + 0.05751}} = \sqrt{0.4713} = 0.2244. $$

Times $\cos\theta_{13} \approx 0.99975$ (angle-13 correction):

$$ |V_{us}| = 0.2244 \cdot 0.99975 = 0.22421. $$

PT value: 0.224 21. PDG: 0.2243 ± 0.0008. Gap: 0.038%.

Why this structure?

The Wolfenstein hierarchy $|V_{us}| \sim \lambda \approx 0.22$ emerges naturally as a gap-fraction ratio between the two adjacent active primes (3, 5) on the geometric branch. No fitted parameter.