The neutrino mass spectrum is derived from first principles within the Quantum Geometric Unification (QGU) framework based on M-theory compactification on the Joyce orbifold T⁷/(Z₃ ⋉ I*) with G₂ holonomy. Right-handed neutrino Majorana masses are generated by M2-brane instantons wrapping associative three-cycles at the three conical singularities, with the intermediate scale fixed by the SUGRA hierarchy relation as M_{R,2} = √α_{EM} × M_comp = 1.71 × 10¹⁵ GeV. The instanton action gap δS = 1.746 is derived from the N=1 SUGRA scalar potential with flux superpotential W₀ = -n_f dim(G₂)/(2π) K₀^{-3/7} = -0.486, determined by the G₂ identity ∫*φ∧φ = 7 Vol(X₇) and flux quantisation with N = 3 quanta. The complete mass spectrum m₁ = 1.56, m₂ = 8.87, m₃ = 50.6 meV reproduces Δm²₂₁ and Δm²₃₂ to 1.3% and 1.2% respectively. PMNS angles emerge from Z₃ symmetry: sin²θ₁₂ = 0.303 (0.3σ) and sin²θ₂₃ = 0.542 (0.2σ). Compatibility with baryogenesis via leptogenesis is demonstrated (η_B^max/η_B^obs ≈ 9), with CP violation protected at leading order by the even rank of E₈ (spectral flow = 8 ≡ 0 mod 2). Five testable predictions include Σm_ν = 61 meV (within DESI sensitivity). All mass ratios employ zero adjustable parameters beyond three geometric constants established independently in eight companion publications.