sin²θ_12

sin²θ_12 PMNS — solar angle

$\sin^2\theta_{12}^{\rm PMNS} \approx 0.30$ is the solar oscillation angle (KamLAND, SNO). In PT, it is the gap-fraction ratio between the two main active primes on the q_+ (lepton) branch:

$$ \sin^2\theta_{12}^{\rm PMNS} = \frac{\delta_5(q_+)}{\delta_3(q_+) + \delta_5(q_+)}. $$

Computation

At $q_+ = 13/15$: - $\delta_3(q_+) = 0.11635$ - $\delta_5(q_+) = 0.05057$

$$ \sin^2\theta_{12}^{\rm PMNS} = \frac{0.05057}{0.11635 + 0.05057} \cdot K_{12}^{\rm PMNS} = 0.303\,684. $$

with $K_{12}^{\rm PMNS} \approx 1.0023$ (NLO correction).

PT: 0.303 684 vs PDG: 0.304 ± 0.012. Gap: 0.10%.

Quark/lepton symmetry

$\sin^2\theta_{12}^{\rm PMNS}$ and $\sin^2\theta_{12}^{\rm CKM}$ use the same gap structure, but on different branches (q_+ vs q_-) — hence very different values (0.30 vs 0.05). Direct signature of the PT bifurcation.