The Complete Fermion Mass Spectrum, Spin-Charge Quantization, and Flavor Mixing in Covariant Bitemporal Geometrodynamics: Topological Derivation of N_gen = 3, Koide's Q = 2/3 Invariant, Quark-Lepton Masses, and the CKM/PMNS Matrices from M_{3,2}

Fifth and culminating foundational monograph (Paper V) completing the Covariant Bitemporal Geometrodynamics (CBG) Pentalogy:- Paper I (Galaxies & Field Laws): DOI: 10.5281/zenodo.23443512- Paper II (Black Hole Mechanics): DOI: 10.5281/zenodo.23451974- Paper III (Cyclic Bounce Cosmology): DOI: 10.5281/zenodo.23458462- Paper IV (Quantum-Gauge Unification): DOI: 10.5281/zenodo.23464846- Paper V (Complete Fermion Spectrum & Flavor Mixing): This Work ABSTRACT:In the first four foundational monographs of this series (Papers I–IV), we formulated Covariant Bitemporal Geometrodynamics (CBG) on the 5-dimensional manifold M_{3,2} ≃ (R_{t_1} ×_Φ S^1_{t_2}) × Σ_3 with exterior bitemporal signature (-,-,+,+,+) and integer dimensions (d_t, d_s, D) = (2, 3, 5) (N_Cartan = d_t d_s D = 30, D_config = 17), proving that a single topological invariant Φ_∞ = (α_EM / 30)(1 - (5/3) α_EM) = 2.403 × 10^{-4} (Ω = c Φ_∞ = 71.95 km/s) and its instanton ladder E_n^{(bare)} = Φ_∞^(2n) E_P (Ẽ_n^{(CBG)} = χ_loop Φ_∞^(2n) E_P with χ_loop = 1.0000 × 10^{-3} for n ≥ 2) resolve galactic rotation curves, nonsingular Σ_4 black holes, cyclic bounce cosmology, ghost-free BRST unitarity, the fine-structure constant (α_EM^{-1} = 137.036), and the electroweak gauge/Higgs hierarchy (Ẽ_2 = 40.507 GeV ⟹ m_W = 81.01 GeV, m_Z = 92.37 GeV, m_H = 127.26 GeV, v_EW = 248.23 GeV). However, the deepest structural mystery of particle physics—the Standard Model Flavor and Spin-Charge Puzzle, comprising the origin of N_gen = 3 fermion generations, fractional electric charges (Q = 0, -1, +2/3, -1/3), the 9 charged fermion masses spanning six orders of magnitude from the electron (0.511 MeV) to the top quark (172.69 GeV), the 3 individual neutrino masses, and the CKM and PMNS mixing angles—remains unexplained in (3+1) quantum field theory, requiring 19 arbitrary free parameters. In this fifth and culminating treatise (Paper V), we derive the complete spin, charge, generation multiplicity, mass spectrum, and flavor mixing matrices of all fundamental particles from the pure geometry of M_{3,2} with zero free parameters. Specifically, we prove that: (1) Spin Classification & N_gen = 3 Fermion Generations: Under the bitemporal spin group Spin(3,2) ≃ Sp(4, R), the Atiyah–Singer Dirac index theorem on Σ_3 × S^1_{t_2} locks the number of chiral fermion families identically to the spatial dimension N_gen = dim(Σ_3) = d_s = 3, while classifying all physical fields into spin s = 0 (Φ holonomy scalar), s = 1/2 (Sp(4, R) symplectic spinors), s = 1 (S^1_{t_2} and Σ_3 fiber connections), and s = 2 (n = 0 luminal graviton). (2) Fractional Electric Charge & SU(3)_C Color Confinement: Partitioning the U(1) Kaluza–Klein Noether winding number on S^1_{t_2} across the d_s = 3 orthogonal spatial axes of Σ_3 forces color confinement (SU(d_s) = SU(3)_C), cancels all chiral gauge anomalies (Tr(Y) = Tr(Y^3) = 0), and locks fractional quark charges to Q_d = -1/d_s = -1/3 and Q_u = +d_t/d_s = +2/3. (3) Charged Lepton Masses & Topological Proof of Koide's Q = 2/3 Invariant: Anchoring the light fermions to the third instanton rung E_3^{(bare)} = Φ_∞^6 E_P = 2.3332 MeV derives the electron mass m_e^{(CBG)} = (2/9) E_3^{(bare)}(1 - (3/2) α_EM) = 0.5128 MeV, the muon mass m_μ^{(CBG)} = ((3/2) α_EM^{-1} + 1) m_e = 105.55 MeV, and proves that Koide's empirical lepton ratio is the exact dimensional invariant Q_Koide^{(CBG)} ≡ d_t / d_s = 2/3, locking the tau mass at m_τ^{(CBG)} = 1,776.97 MeV (PDG: 1,776.86 MeV; 0.006% accuracy). (4) The Six Quark Masses (u, d, s, c, b, t) & Nucleon Splitting: Projecting E_3^{(bare)} = 2.3332 MeV and Ẽ_2 = 40.507 GeV across the Cartan, Clifford (2^D = 32), and Lorentz (dim Spin(3,2) = 10) multiplicities of M_{3,2} yields m_u = 2.22 MeV, m_d = 2 E_3^{(bare)} = 4.67 MeV, m_s = 40 E_3^{(bare)} = 93.33 MeV, Δm_np = 1.293 MeV, m_c = Ẽ_2 / 32 = 1.266 GeV, m_b = (1/10) Ẽ_2(1 + (2/3) Ω_{b,0}) = 4.184 GeV, and m_t = (v_EW / √2)(1 - Ω_{b,0}/3) = 172.65 GeV. (5) CKM/PMNS Mixing Matrices, Individual Neutrino Masses & the LHC Energy Desert: Overlap integrals of the S^1_{t_2} harmonic wavefunctions derive the Cabibbo angle |V_us| = √(m_d / m_s) = 1/√20 = 0.22361 (PDG: 0.2243), |V_cb| = 0.0410, |V_ub| = 0.00367, δ_CP = arctan(2) = 63.43°, J_CP = 3.01 × 10^{-5}, the PMNS angles (sin^2(θ_13) = 0.0220, sin^2(θ_12) = 0.311, sin^2(θ_23) = 0.548), and normal-ordered neutrino masses (Δm_21^2 = 7.53 × 10^{-5} eV^2, Δm_31^2 = 2.515 × 10^{-3} eV^2; screened cosmological sum Σ m_ν = 0.0346 eV), while explaining exact proton stability (τ_p = ∞) and the LHC energy desert below E_2^{(bare)} = 40.51 TeV. Complete synthesis tables replacing all 26 Standard Model + Neutrino free parameters are provided.