Exact Phenomenology of the Neutrino Cuboid: Normal Mass Ordering, the Tribimaximal Limit, and Cosmological Constraints

Abstract

We investigate the exact phenomenology of a geometric neutrino cuboid defined by $m_1=m_0\sin\xi$, $m_2=m_0\cos\xi\sin\zeta$, and $m_3=m_0\cos\xi\cos\zeta$, whose cubic point corresponds to mass degeneracy and the tribimaximal values of the solar and atmospheric mixing angles. Imposing the mass-mixing alignment $\xi=\theta_{12}$ and $\zeta=\theta_{23}$, we derive closed-form consistency relations without relying on an expansion about the cubic limit. These relations show that the observed condition $\sin^2\theta_{12}<1/3$ selects normal mass ordering and, in the resulting normal-ordering regime, the physical condition $0<\Delta m_{21}^2/\Delta m_{31}^2<1$ requires $\theta_{23}$ to lie in the lower octant. Representative JUNO-based inputs give $\sin^2\theta_{23}\simeq0.440$, $(m_1,m_2,m_3)\simeq(0.0938,0.0942,0.1063)\mathrm{eV}$, and $\sum_i m_i\simeq0.294\mathrm{eV}$. We demonstrate that the first-order expansion around the tribimaximal point is numerically unstable for the solar-to-atmospheric mass-splitting ratio because of a leading-order cancellation. Present atmospheric-angle data yield only mild tension with exact alignment, whereas stringent neutrino-mass bounds in baseline $\Lambda$CDM cosmology strongly disfavor it; its viability under extended cosmological models remains model dependent. We finally formulate geometric alignment residuals as general null-test observables, providing a systematic framework for testing the neutrino cuboid with future oscillation and absolute-mass measurements.