Gravitational Wave in an $A_4\otimes Z_4$ Neutrino Model with Generalised CP

We extend the Standard Model with an $A_4\times Z_4$ flavor symmetry and generalized CP (GCP) symmetry, realized through two flavon triplets that spontaneously break $A_4$ and generate the observed lepton mixing pattern. A new $A_4$-singlet flavon couples to one of these triplets through a complex quartic interaction that breaks GCP and fixes the reactor mixing angle; this same coupling biases the domain walls formed when that triplet breaks $A_4$. A global fit to oscillation data reproduces all mixing angles and mass splittings, predicting a maximal Dirac CP phase of ${270}^{\circ }$ and an effective Majorana mass near $15\text{ meV}$, testable by upcoming neutrinoless double-beta-decay ($0\nu \beta \beta$) experiments. The same coupling generates a stochastic gravitational-wave background peaking in the pulsar-timing-array and LISA bands, with peak frequency and amplitude predicted to scale as $\sin {\theta }_{13}$ and ${\sin }^{-4}{\theta }_{13}$, a falsifiable link between neutrino oscillation and gravitational-wave (GW) observations.