The PMNS Matrix and Majorana Masses from the Fibonacci Structure of the Lepton Coset (IGPS IX)

This paper presents the ninth installment of the Information-Geometric Physics System (IGPS) series, focusing on a geometric and topological approach to the lepton sector. It explores the derivation of the lepton mixing matrix (PMNS), neutrino masses, and CP phases from first principles, utilizing the lepton sector vertex algebra Vₗₑₚₜₒₙ = SU(2)₁⊗ 3/SU(2)₃ with a central charge of $c = 6/5$. While the Standard Model typically relies on phenomenological fitting for neutrino parameters, this work investigates whether these structures can emerge analytically from the spectral geometry of a BPS lepton kink. Key Explorations & Theoretical Results: Lepton Mixing Angles: The study suggests that Tribimaximal Mixing (TBM) emerges at leading order due to the S₃ permutation symmetry of the parent algebra. At next-to-leading order, the framework calculates θ₁₃ ≈ 9.15^∘ and θ₂₃ ≈ 47.30^∘ through Quark-Lepton Complementarity (QLC) via W-boson exchange effective field theory. Fibonacci Structure and the Solar Angle: The framework models the primary spectrum of Vₗₑₚₜₒₙ as a Fibonacci category associated with the quantum group SU(2)q at q = eiπ/5 (pentagon symmetry). This topological feature proposes a correction to the solar angle, yielding θ₁₂ ≈ 33.77^∘, which closely aligns with current global oscillation fits. Topological CP Violation: The paper derives a topological Dirac CP phase of δPMNS = -90^∘ based on T-matrix holonomy, falling within the current 3σ experimental range. Majorana Neutrinos without Seesaw: The research proposes that neutrinos are Majorana fermions as a topological consequence of a "doubled vacuum" ($h=0$) structure and crosscap boundary conditions. Furthermore, it shows that Majorana phases vanish (α₁ = α₂ = 0) due to S₃ degeneracy. Absolute Neutrino Mass Scale: By connecting non-orientable topology to bulk volume dilution, the mass of the heaviest neutrino is estimated at m₃ = mₑ h_ν² α³ ≈ 49.64 meV, consistent with cosmological bounds. By analyzing the distinct Fibonacci topological characteristics of the lepton coset compared to the Abelian nature of the quark sector, this work aims to provide a parameter-sparse, structural explanation for the large mixing angles and mass hierarchies observed in neutrino physics. The paper clearly details its mathematical derivations while explicitly acknowledging remaining open problems for future exploration.