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Derivation of Standard Model Mixing Angles from a 44-Vector Discrete Vacuum Lattice

by Yuxuan Zhang, Weitong Hu, and Wei Zhang

Posted: June 18, 2026
Server: Preprints.org
DOI: 10.20944/preprints202606.1401.v1

Abstract

We derive the mixing angles and CP-violating phases of the PMNS and CKM matrices from a 44-vector discrete vacuum lattice—the 3D flavor-space projection of a 19-dimensional $\mathbb{Z}_3$ -graded Lie superalgebra. The central prediction is the Cabibbo angle: $\lambda = |V_{us}| = 73/324 = 0.22530864$ , obtained as $\lambda = (2/9)(1+\varepsilon_q^2/2)$ with $\varepsilon_q=1/6$ and the $\mathrm{SU}(3)$ quadratic Casimir $C_2=4/3$ , matching the experimental value $0.225300(700) $to$ +0.01\sigma $—a precision of 8 parts per million, with zero free parameters. The PMNS predictions—compared against current global fit data—are:$ \sin^2\theta_{12}=1/3-\lambda/9=0.30830 $($ -0.10\sigma $),$ \sin^2\theta_{23}=0.54609 $($ +0.00\sigma $),$ \sin^2\theta_{13}\in[1/46,1/44] $(within interval). The CP phases are$ \delta_{\rm CP}=240^\circ $($ +0.26\sigma $) and$ \delta_{\rm CKM}=65.3^\circ $. Projected future measurements by JUNO (targeting$ \sim0.3\% $precision on$ \theta_{12} $) and DUNE (mass ordering) will provide decisive tests of these predictions. The perturbation strengths—$ \varepsilon_{\nu_2}=1/36 $,$ \varepsilon_{\nu_3}=1/12 $,$ \varepsilon_q=1/6 $—are obtained from the algebra's Frobenius norms and the$ \mathfrak{u}(3)$ projection structure, with the Hybrid norm-filtered subclass rigorously proven to contain exactly 24 vectors. Every formula is presented with complete symbolic definitions and step-by-step derivations.

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