Many papers try to explain fundamental constants in the Standard Model in terms of deeper relationships. This attempt to gain insight into the neutrino oscillation parameters is more thought provoking than most.
We propose a geometric hypothesis for neutrino mixing: twice the sum of the three mixing angles equals 180∘, 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 θ23 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 ∼10×), 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 χ2 consistency check against global-fit inputs to quantify agreement within present uncertainties.
Mohammad Ful Hossain Seikh, "A geometrical approach to neutrino oscillation parameters" arXiv:2510.06526 (October 7, 2025).
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