The $π/12$ model: trimaximal first-column lepton mixing with charged-lepton $μ$--$τ$ 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 $μ$--$τ$ balance $|U_{\mu1}|=|U_{\tau1}|$. Parameterizing their violation by $\varepsilon_1\equiv\tfrac32|U_{e1}|^2-1$ and $\varepsilon_2\equiv3(|U_{\tau1}|^2-|U_{\mu1}|^2)$, we show that current data treat the two differently in central value: with the first JUNO measurement included, $\varepsilon_1=+0.014\pm0.010$ is consistent with zero, while $\varepsilon_2=+0.29^{+0.07}_{-0.13}$ is large but disfavors zero at only $1.5σ$ on the profile likelihood of the global fit, since the phase that drives it remains poorly measured. This asymmetry selects a specific framework, the $π/12$ model. Its neutrino sector is $\mathrm{TM}_1$, supplemented by the electron-row condition $|U_{e2}|/|U_{e3}|=2+\sqrt3$ that fixes $\sin^2θ_{13}=(2-\sqrt3)/12=0.02233$ and $\sin^2θ_{12}=0.31811$. A charged-lepton 2--3 rotation $R_{23}(θ_e,α)$ leaves the electron row exactly invariant and generates $\varepsilon_2=\sin2θ_e\cosα$. In a concrete $S_4\times C_4\times C_3\times C_2$ realization, a single additional flavon alignment forces $α=0$; a natural angle $θ_e\simeq2^\circ$ then reproduces the observed departure of $θ_{23}$ from maximal while pinning $δ=272^\circ\pm2^\circ$, near-maximal CP violation with $\varepsilon_2=-0.066$. The result is a sharp test: the predicted $δ$ sits $1.7σ$ 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 $δ$ resolves the question at the $3σ$ level. The mass predictions are inherited unchanged, $Σm_ν=65.6$ meV and $m_{ββ}=5.4$ meV, placing the model at the current DESI bound in $Λ$CDM.