[SUPERSEDED] The PMNS CP Phase from Seesaw Geometry

PRELIMINARY VERSION — SUPERSEDED This paper presents an earlier derivation within the Canvas Model framework. The CP phase prediction \delta_{\text{CP}} = \pi(1+\alpha) \approx 220^\circ remains valid. However, the supporting values have been updated in the definitive work: · \theta_{13} = \arcsin(1/(3\sqrt{5})) \approx 8.57^\circ (replaces 1/6 \approx 9.5^\circ)· M_R = M_P \cdot \alpha_0^2 \approx 1.5 \times 10^{14} GeV (replaces \alpha_0 M_P) See "The Emergence Canvas Model: A Unified Framework for Fundamental Physics - The Machine and the State" (2026) https://doi.org/10.5281/zenodo.20795774 for the complete, self-consistent treatment. ------------------------------------------------- The PMNS matrix describes neutrino flavor mixing. Its three mixing angles have been derived in the canvas model from gauge subspace dimensions: \theta_{12} = 1/\sqrt{3} \approx 33.1^\circ, \theta_{23} = \pi/4 = 45^\circ, \theta_{13} = 1/6 \approx 9.5^\circ. All three match observation. The CP-violating phase \delta_{\text{CP}} has remained an open problem. This paper derives \delta_{\text{CP}} from the seesaw mechanism geometry in the internal coupling space. What this paper provides: · The seesaw mechanism in the internal space. The right-handed neutrino is a singlet under all Standard Model gauge groups—its coupling vector is the fully symmetric direction \vec{c}^{(\nu_R)} = (1,1,1)/\sqrt{3}. The seesaw mass scale is M_R \sim \alpha_0 M_P \sim 10^{17} GeV, giving the correct light neutrino mass scale m_\nu \sim 0.1 eV.· The neutrino Higgs direction. The effective neutrino Higgs direction \vec{v}_H^{(\nu)} is a mixture of the charged lepton direction (from the Dirac coupling) and the symmetric direction (from the Majorana coupling). The seesaw rotation angle is \theta_{\text{seesaw}} \sim m_D/M_R \sim 10^{-12}, negligible. The neutrino direction is therefore essentially the fully symmetric direction with azimuthal angle \phi_\nu = 45^\circ = \pi/4.· The CP phase from azimuthal misalignment. The charged lepton Higgs direction defines the reference axis in the (1,2) plane, \phi_e \approx 0. The azimuthal difference is \Delta\phi = \phi_\nu - \phi_e = \pi/4. The seesaw rotation adds a geometric modulation from the waveform asymmetry parameter \alpha = (\pi-2)/(\pi+2) \approx 0.222, giving: \delta_{\text{CP}} = \pi + \alpha\pi = \pi\left(1 + \frac{\pi-2}{\pi+2}\right) \approx 220^\circ· Relation to the CKM phase. The CKM phase \delta_{\text{CKM}} \approx 68^\circ arises from the same geometric mechanism. The PMNS phase is larger because the neutrino Higgs direction is maximally misaligned—the fully symmetric direction in the internal space—while the quark Higgs directions are within or near the (1,2) plane.· Testable predictions. \delta_{\text{CP}} \approx 220^\circ = 3.84 rad, the Jarlskog invariant for leptons J_{\text{PMNS}} \approx 0.033, and large CP violation in the lepton sector—comparable to the quark sector. These predictions are testable with DUNE and Hyper-Kamiokande within the coming decade. Why this matters: This completes the PMNS matrix derivation. All three mixing angles and the CP phase are now determined from the canvas model primitives. The prediction is specific, falsifiable, and within reach of upcoming neutrino experiments. Keywords: PMNS matrix, CP violation, seesaw mechanism, neutrino oscillations, canvas model, internal coupling space, Higgs direction, waveform asymmetry, DUNE, Hyper-Kamiokande