A geometrical approach to neutrino oscillation parameters

Abstract

We propose a geometric hypothesis for neutrino mixing: twice the sum of the three mixing angles equals $180^\circ$, forming a Euclidean triangle. This condition leads to a predictive relation among the mixing angles and, through trigonometric constraints, enables reconstruction of the mass-squared splittings. The hypothesis offers a phenomenological resolution to the $\theta_{23}$ octant ambiguity, reproduces the known mass hierarchy patterns, and suggests a normalized geometric structure underlying the PMNS mixing. We show that while an order-of-magnitude scale mismatch remains (the absolute splittings are underestimated by $\sim 10\times$), the triangle reproduces mixing ratios with notable accuracy, hinting at deeper structural or symmetry-based origins. We emphasize that the triangle relation is advanced as an empirical, phenomenological organizing principle rather than a result derived from a specific underlying symmetry or dynamics. It is testable and falsifiable: current global-fit values already lie close to satisfying the condition, and improved precision will confirm or refute it. We also outline and implement a simple $\chi^2$ consistency check against global-fit inputs to quantify agreement within present uncertainties.