Δm²_31

Δm²_31 — large mass splitting

$\Delta m^2_{31} = m_3^2 - m_1^2 \approx 2.5 \times 10^{-3}$ eV² measured by atmospheric oscillations (Super-K, T2K).

In PT, with normal hierarchy ($m_1 \ll m_3$):

$$ \Delta m^2_{31} \approx m_{\nu_3}^2 = (\sqrt{\Delta m^2_{31}})^2. $$

Computation

From the PT cascade (cf. m_ν3, ID 31):

$$ m_{\nu_3} = 0.050\,475\ \text{eV} \Rightarrow m_{\nu_3}^2 = 2.548 \times 10^{-3}\ \text{eV}^2. $$

With correction $m_1^2 \approx 0.034 \times 10^{-3}$ eV² (from Δm²_21):

$$ \Delta m^2_{31} = 2.548 \times 10^{-3} - 0.034 \times 10^{-3} = 2.514 \times 10^{-3}\ \text{eV}^2. $$

PT: 2.514 × 10⁻³ eV² vs PDG: 2.51 × 10⁻³ eV². Gap: 0.17%.

Predicted normal hierarchy

P2 in the list of falsifiable predictions: PT imposes normal hierarchy ($m_1 < m_2 < m_3$). If JUNO (~2027) measures inverted hierarchy at > 5σ, PT falls.