Deriving the Pontecorvo–Maki–Nakagawa–Sakata Matrix from Koide’s Mass Formula and Brannen’s Neutrino Mass Hypothesis: Resolving the Mystery of the θ₁₃ Rotation

Koide's mass formula, originally proposed for charged leptons, has been hypothesized by Carl A. Brannen to also apply to neutrinos. Assuming this hypothesis' validity, two three-dimensional mass models were constructed based on the proposed neutrino masses. This paper demonstrates that the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix can be derived by introducing an intermediate set of hypothetical states, referred to as mass negative eigenstates, which mediate the transformation between mass and flavor eigenstates. This framework naturally reproduces the tribimaximal mixing structure and yields a PMNS matrix with elements close to those obtained using global fits. Neutrino oscillation probability predictions were further compared with results from the Tokai-to-Kamioka (T2K) and Daya Bay collaborations. While the proposed model captures key structural lepton mixing features, a deviation of approximately −3σ in sin²(2θ₁₃) highlights its limitations in terms of reproducing current data. This discrepancy may indicate the involvement of additional mechanisms or physics beyond the current framework. Future theoretical refinements and more precise experimental tests are crucial to assess whether the Koide--Brannen framework can serve as a meaningful step toward a deeper understanding of neutrino phenomenology.