Fermion Mixing Matrices and the Exceptional Jordan Algebra

Abstract

We extend the exceptional-Jordan spectral framework for fermion mass hierarchies to the problem of quark and lepton mixing. Following the companion mass paper~\cite{Teli:2026jgr}, each fermion sector is associated with a Hermitian element of $J_3(\mathbb{O}_{\mathbb{C}})$, where adjacent square-root mass ratios are obtained from cubic ladders in $\mathrm{Sym}^3(\mathbf 3)$. Here, these ratios are used as inputs to an adjacent-edge lift from spectral hierarchy data to two-generation mixing angles. The lift is derived from a Fritzsch-type two-state texture~\cite{Fritzsch:1977za, Fritzsch:1979zq} and should be regarded as an effective bridge ansatz rather than a theorem of the Jordan spectrum alone. The exact CP-transport input is supplied by the companion CP Letter~\cite{GuptaTeli:2026aqf}. In the quark sector, the octonionic ladder operator $α_2$ generates a real local rotor in the $(e_1,e_3)$ plane, and the up- and down-sector local Cabibbo-edge amplitudes are complex conjugates, giving the exact local law $ϕ_{12}=-2χ$. This is a transport-level Cabibbo-rung phase law, not by itself a prediction of the standard CKM Dirac phase. With the fitted companion mass ratios, the minimal two-angle extraction from the measured $|V_{us}|$ gives an effective Cabibbo-block phase $ϕ_{12}\simeq 105.7^\circ$; this number is a bridge diagnostic, while the balanced octonionic rotor remains the distinguished quadrature reference point. The $(2,3)$ sector requires a phenomenological normalization $κ_{23}\simeq0.56$, and the direct $(1,3)$ element remains a long-edge bridge problem. [Truncated]