Geometric Calculation of Neutrino Masses from a Six-Dimensional Compact Manifold (S³×S³)/ℤ₃

This preprint presents a unified geometric approach to neutrino mass generation based on a fundamental scalar field PPP defined on a compact six-dimensional manifold Σ=(S3×S3)/Z3\Sigma = (S^3 \times S^3)/\mathbb{Z}_3Σ=(S3×S3)/Z3.The framework naturally explains the three-generation structure of fermions through topological cycles and provides parameter-free predictions for neutrino masses, mixing angles, and the CP-violating phase.The mass formula mν=ℏΓΣ/c2m_\nu = \hbar \Gamma_\Sigma / c^2mν=ℏΓΣ/c2 yields Σmν=0.059±0.001\Sigma m_\nu = 0.059 \pm 0.001Σmν=0.059±0.001 eV and mβ≈0.009m_\beta \approx 0.009mβ≈0.009 eV, both consistent with cosmological and experimental constraints.The model establishes a link between geometry, topology, and entropy, suggesting that the reduction from SU(2)×SU(2) to ℤ₃ symmetry corresponds to an entropic symmetry breaking leading to the observed three-family structure. This work serves as the reference preprint for the version submitted to Entropy (MDPI) in November 2025.