The $\pi/12$ model: trimaximal first-column lepton mixing with charged-lepton $\mu$--$\tau$ breaking

Abstract

The first column of the lepton mixing matrix carries two independent magnitude conditions: the trimaximal norm $|U_{e1}|^2=2/3$ and the $\mu$--$\tau$ balance $|U_{\mu1}|=|U_{\tau1}|$. We parameterize their violation by $\varepsilon_1\equiv(3/2)|U_{e1}|^2-1$ and $\varepsilon_2\equiv3(|U_{\tau1}|^2-|U_{\mu1}|^2)$ and show that current data treat the two differently in central value: including the first JUNO measurement, $\varepsilon_1=+0.014\pm0.010$ is consistent with zero, while $\varepsilon_2=+0.29^{+0.07}_{-0.13}$ is large but disfavors zero at only $1.5\sigma$ on the global-fit profile likelihood, the driving phase being still poorly measured. This asymmetry selects the $\pi/12$ model. Its neutrino sector is TM$_1$ with the electron-row condition $|U_{e2}|/|U_{e3}|=2+\sqrt3$ that fixes $\sin^2\theta_{13}=(2-\sqrt3)/12=0.02233$ and $\sin^2\theta_{12}=0.31811$. A charged-lepton 2--3 rotation $R_{23}(\theta_e,\alpha)$ leaves the electron row exactly invariant and generates $\varepsilon_2=\sin2\theta_e\cos\alpha$. In an $S_4\times C_4\times C_3\times C_2$ realization, the rotation arises from a single additional flavon whose alignment forces $\alpha=0$; a natural angle $\theta_e\simeq2^\circ$ then reproduces the departure of $\theta_{23}$ from maximal while pinning the CP phase to $\delta=272^\circ\pm2^\circ$, near-maximal CP violation with $\varepsilon_2=-0.066$. The current $\theta_{23}$--$\delta$ measurements thereby become a sharp test: the predicted $\delta$ sits $1.7\sigma$ from the present central value on the same profile likelihood, the predicted $\varepsilon_2$ has the opposite sign to the measured one, and the Hyper-Kamiokande design precision of $20^\circ$ on $\delta$ resolves the question at $3\sigma$. The mass predictions are inherited unchanged, $\Sigma m_\nu=65.6$ meV and $m_{\beta\beta}=5.4$ meV, at the current DESI bound in $\Lambda$CDM.