Topological Origin of Neutrino Mass and Mixing: From the SU(3)₃ Modular Tensor Category to the PMNS Matrix

We propose a topological origin for neutrino masses and mixing: neutrinos are boundary topological flows of a string-net condensed phase, and their mass matrix is determined by boundary dynamics that inherits the hierarchical parameters of the bulk SU(3)₃⊗SU(2)₂ modular tensor category. Using the conformal-weight spacing Δh=1/3 and the topological temperature β=π²/14 of this category, we obtain λ=e-14/(3π)≈0.2264 without free parameters, consistent with the experimental Cabibbo angle; the numerical calculations use the standard value λ=0.225. On this basis, we propose a mass matrix whose coefficients are mapped directly to the S-matrix elements of the SU(3)₃ modular tensor category, with diagonalization yielding mixing angles close to the experimental values. Further introducing the CP phase ε=λ² arising from the quadratic effect of topological suppression, and reading the charged-lepton mixing parameters (α, β, γ) from the quantum-dimension ratios and conformal weights of the category, we can simultaneously fit the four observables of NuFIT 6.0. The parameters α=-3λ/5, β=-λ²/2, and γ=2λ/3 are read from the quantum-dimension ratios d₈/(d₈+d₃)=3/5, and d3/d8=2/3, as well as the conformal weight h8=1/2. The CP phase η=π/10 has a proposed reading mechanism η=π/Nsimple, pending confirmation; we have systematically excluded several possible algebraic origins (T-matrix phases, R-matrix phases, topological entropy change) and leave its full derivation as an open problem.The chirality of the neutrino sector arises from anomaly matching: the bulk is parity-symmetric, but its unique fermion (1,1) carries a fermion-parity anomaly, and the boundary must retain exactly one chirality—ψL, selected by the sign of the bulk anomaly—to cancel it. This identifies parity non-conservation as a boundary effect, not a fundamental law, and explains why only left-handed neutrinos participate in the weak interaction.