The PMNS Matrix from Information Theory: Why Lepton Mixing Is Large

This paper derives the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix—the lepton mixing matrix—from the information capacities of the gauge subspaces in the canvas model. No experimental input is required. The fundamental difference between quarks and leptons is that neutrinos do not couple to SU(3). Quark mixing is suppressed by both SU(2) and SU(3): \lambda_{\text{CKM}} = 1/(2+3) = 1/5. Lepton mixing involves only SU(2) and U(1): \lambda_{\text{PMNS}} = 1/(1+2) = 1/3. This larger base parameter explains why lepton mixing angles are systematically larger than quark mixing angles. The three PMNS angles are derived in closed form: \theta_{12} \approx \frac{1}{\sqrt{3}} \approx 33.1^\circ \quad \text{(obs: } 33.4^\circ \pm 0.7^\circ\text{)} \theta_{23} \approx \frac{\pi}{4} = 45^\circ \quad \text{(obs: } \sim 45^\circ\text{)} \theta_{13} = \arcsin\!\left(\frac{1}{3\sqrt{5}}\right) \approx 8.57^\circ \quad \text{(obs: } 8.57^\circ \pm 0.13^\circ\text{)} The (1,3) mixing angle \theta_{13} uses an arcsine form because it describes an out-of-plane rotation. The argument 1/(3\sqrt{5}) combines the spatial dimensionality n = 3 and the CKM mixing scale \sqrt{5} = \sqrt{2+3}. This corrects the earlier 1/6 approximation, which gave 9.5^\circ and a 7\sigma tension with observation. The new prediction matches the observed central value exactly. The Dirac CP phase is \delta_{\text{CP}} = \pi(1+\alpha) \approx 220^\circ, where \alpha = (\pi-2)/(\pi+2) is the waveform asymmetry parameter. The Majorana phases are \alpha_{21} = \pi/2 (from the radiative origin of the second-generation neutrino mass) and \alpha_{31} = 0 (from the tree-level reality of the first and third generation masses). The effective Majorana mass for neutrinoless double beta decay is predicted to be \langle m_{\beta\beta} \rangle \approx 0.048 eV, within reach of next-generation experiments (LEGEND-1000, nEXO, CUPID). Why this matters: The PMNS matrix is not an arbitrary unitary matrix. It is a geometric consequence of the fact that neutrinos feel only the SU(2) and U(1) subspaces of the internal 3D space. The integers \{1,2,3\}—the dimensions of the gauge subspaces—determine all mixing angles. No free parameters. The difference between quark and lepton mixing follows from a single fact: neutrinos do not feel the strong force. Keywords: PMNS matrix, lepton mixing, neutrino oscillations, canvas model, information theory, gauge subspaces, CP violation, Majorana phases, neutrinoless double beta decay, theta13