[SUPERSEDED] Majorana Phases in the PMNS Matrix from Seesaw Geometry

This paper has been superseded by "The PMNS Matrix from Information Theory: Why Lepton Mixing Is Large" (2026), which contains the complete derivation of all PMNS parameters including the Majorana phases. The PMNS paper corrects the neutrino mass spectrum and provides a unified treatment of all mixing angles and CP phases. This paper is retained for archival purposes. Please cite the PMNS paper for the current results. ------------------------------------------------- If neutrinos are Majorana particles — identical to their own antiparticles — the PMNS matrix contains two additional CP-violating phases beyond the Dirac phase \delta_{\text{CP}}. These Majorana phases \alpha_{21} and \alpha_{31} are not measurable in neutrino oscillation experiments but affect neutrinoless double beta decay and leptogenesis. They have no explanation in the Standard Model. This paper derives both Majorana phases from the seesaw mechanism geometry in the internal coupling space of the canvas model. What this paper provides: · A geometric origin for the Majorana phases. The phases are determined by the relative orientation of the neutrino Higgs modulation direction \vec{v}_H^{(\nu)} and the charged lepton Higgs direction \vec{v}_H^{(e)}. The neutrino direction is the fully symmetric direction (1,1,1)/\sqrt{3} in the internal 3D space — the only direction that treats all three spatial axes equally.· The seesaw mass matrix in the generation basis. Using the three generation eigenvectors \vec{c}^{(1)} = (1,1,0)/\sqrt{2}, \vec{c}^{(2)} = (1,-1,0)/\sqrt{2}, and \vec{c}^{(3)} = (0,0,1), the Yukawa couplings to the symmetric direction are computed. The second generation has zero overlap — a striking result: the second generation neutrino has no Dirac coupling to the right-handed neutrino at leading order.· The mass matrix diagonalization. The tree-level seesaw produces eigenvalues m_1 = v^2 y_0^2/(2M_R), m_2 = 0, m_3 = v^2 y_0^2/(6M_R). The vanishing of m_2 predicts that the second generation neutrino mass must arise from sub-leading effects (mixing, radiative corrections, or Planck-scale effects).· Radiative generation of m_2. The tau Yukawa loop generates a small imaginary contribution to the (2,2) element of the mass matrix. This radiative correction makes the second eigenvalue complex, producing the Majorana phase \alpha_{21}.· The Majorana phase values: \alpha_{21} = \pi/2 (maximal) and \alpha_{31} = 0. The first Majorana phase is maximal — a consequence of the radiative origin of m_2. The second vanishes at this order.· Testable predictions for neutrinoless double beta decay. The effective Majorana mass \langle m_{\beta\beta} \rangle is computed with these phases. For normal mass ordering, \langle m_{\beta\beta} \rangle \approx 0.022 eV — within the sensitivity of next-generation experiments (LEGEND-1000, nEXO, CUPID).· Implications for leptogenesis. The maximal phase \alpha_{21} = \pi/2 maximizes the interference between first and second generation contributions to the CP asymmetry in right-handed neutrino decays, potentially enhancing the efficiency of leptogenesis. Why this matters: This paper completes the PMNS matrix derivation. All three mixing angles, the Dirac CP phase (\delta_{\text{CP}} \approx 220^\circ), and both Majorana phases (\alpha_{21} = \pi/2, \alpha_{31} = 0) are now determined from the canvas model primitives — with no free parameters. The predictions are testable through neutrinoless double beta decay experiments. Keywords: Majorana phases, PMNS matrix, seesaw mechanism, neutrinoless double beta decay, leptogenesis, canvas model, internal coupling space, generation eigenvectors, radiative corrections, \alpha_{21} = \pi/2