This paper derives the neutrino mixing angles and mass-squared splitting ratio from the Modular Entropic Gravity (MEG) framework with zero free parameters beyond the overall Majorana mass scale. The derivation rests on four results: (1) A structural theorem: if both the Dirac mass matrix m_D and the Majorana mass matrix M_R are Z₃-symmetric (circulants), the light neutrino spectrum has at most two distinct masses. Three distinct masses require Z₃ breaking in at least one matrix. (2) A diagnostic: the hypothesis m_D ∝ m_ℓ (Dirac neutrino masses proportional to charged lepton masses) fails decisively — the seesaw squares the charged lepton hierarchy, producing a mass-squared splitting ratio 3700× too large. The Dirac neutrino masses must be nearly generation-independent, motivated by the SU(2)_L singlet nature of ν_R. (3) A structural derivation: with democratic Dirac masses (m_D = κ𝟏), the Z₃-broken Majorana matrix is completely determined by four SO(8)-derived quantities already established in the MEG programme: the golden ratio connection r = φ/2, the base entanglement depth α₁ = 1/10, the instanton ratio ε/A = 1/20, and the Z₃ wavenumber k = 3. The diagonal breaking arises from the tunnelling self-energy (r/2) and instanton correction (ε/A); the off-diagonal tunnelling amplitudes become α₁ and 2α₁. (4) A Z₃ Fourier representation: the complete Majorana matrix decomposes into six Z₃ representation generators with algebraically determined coefficients, every one built from the four SO(8) quantities above. The cleanest coefficient is α₁/√3 for the off-diagonal sine mode — the base entanglement depth divided by the geometric factor from the Z₃ Fourier transform. Including the charged lepton rotation from the corrected Hermitian Fritzsch texture, the zero-parameter prediction gives: θ₁₂ = 34.1° (obs 33.4°, +3.4%) θ₂₃ = 48.4° (obs 49.2°, −2.4%) θ₁₃ = 8.5° (obs 8.6°, −4.5%) Δm²₂₁/Δm²₃₁ = 0.030 (obs 0.029, +1.4%) with Σm_ν ≈ 73 meV. All four observables agree to within 4.5%. This paper is part of the MEG programme (60+ papers). It builds on the CKM/PMNS paper (P55, doi:10.5281/zenodo.19980020), which fitted two Dirac neutrino mass ratios. Those fitted parameters are now eliminated: the neutrino sector joins the charged fermion sector as a zero-parameter prediction from the vacuum distinguishability kernel.