Z₃-Locked Majorana Textures: Phases, Moduli, and the Limits of Predictivity

We study Majorana neutrino textures in which the complex phases are locked to the cubic root of unity \(\omega=e^{2\pi i/3}\) by a \(Z_3\) CP seed, while the real moduli are free parameters fixed by data. For the reference point \(|d_1|/a_{12}=1/2\), \(d_2/d_3=9/11\), \(a_{23}/a_{12}=45/22\), \(a_{13}/a_{12}=1/3\), \(a_{12}/d_3=6/19\), six oscillation observables are described with \(\chi^2_{\rm block}=1.44\) against the official NuFIT 5.2 release tables (normal ordering, with Super-K atmospheric data). We show that this agreement does not constitute a prediction of the moduli: an exhaustive scan over the same \(Z_3\)-locked family with rational moduli of denominator \(q\le22\) finds 447 alternatives inside \(\pm3\%\) of the reference values, 15 of which reach \(\chi^2_{\rm block}<2\), one of them fitting better than the reference point. We further prove that the moduli cannot be derived from a minimal discrete flavour symmetry. A denominator theorem shows that CG-generated rational ratios have \(|G|\)-smooth denominators, so the observed \(6/19\) and \(45/22\) would require \(11\cdot19\mid|G|\) and are excluded for \(\Delta(27)\), \(A_4\), \(S_4\) and \(\Delta(54)\); a second theorem shows that a single triplet flavon of any \(\Delta(3n^2)\) group yields a diagonal symmetric mass matrix, while two flavons leave at least four continuous inputs. What remains predictive is the phase pattern and its correlated consequences: the texture predicts \(\delta_{CP}\simeq276.6^{\circ}\) (\(-83.4^{\circ}\) in the NuFIT convention), \(J_{\rm PMNS}\simeq-0.033\), normal ordering, \( \sum m_{\nu}\simeq0.070\ eV\ \), and \( m_{\beta\beta}\simeq0.005\ eV\ \), with the best fit in the lower \(\theta_{23}\) octant; \(1\sigma\) ranges on the absolute scale are \( \sum m_\nu\in[0.069,0.070]\ eV\ \) and \( m_{\beta\beta}\simeq0.005\ eV\ \). The leading renormalization group running of the Weinberg operator preserves the \(Z_3\) phase structure. We discuss future tests by DUNE, Hyper-Kamiokande, cosmological mass bounds, and neutrinoless double-beta decay searches.