Fermion mass ratios from the exceptional Jordan algebra

Abstract

The origin of the three fermion generations and their highly hierarchical mass spectra remains one of the most profound puzzles in particle physics. We show that the complexified exceptional Jordan algebra $J_{3}(\mathbb{O}_{\mathbb{C}})$, the natural mathematical framework for the exceptional Lie group $E_{6}$, provides a unified explanation for both. The three generations arise from the three off-diagonal Peirce slots of $J_{3}(\mathbb{O}_{\mathbb{C}})$, each carrying an isomorphic $Cl(6,\mathbb C)$ minimal-ideal fiber and permuted cyclically by triality $S_3\subset\mathrm{Out}(\mathrm{Spin}(8))$; pre-breaking, the three families are identical by symmetry. After triality breaking the residual $SU(3)_F$ flavor symmetry organises the three generations of each family as a $\mathrm{Sym}^{3}(\mathbf{3})$ multiplet, the minimal $S_3$-symmetric degree-3 arena consistent with the cubic structure of the Jordan determinant and the unique $E_6$-invariant Yukawa. The mass-ratio formula follows from a one-line diagonal-action theorem: when $\langle X\rangle$ is Jordan-diagonalised to $\mathrm{diag}(a,b,c)$, the induced action $X^{\odot 3}$ on the $\mathrm{Sym}^{3}(\mathbf{3})$ monomial basis is diagonal with eigenvalues $a^pb^qc^r$, so a fermion identified with the weight state $|p,q,r\rangle$ has $\sqrt m\propto a^pb^qc^r$ and adjacent generations related by an edge move have $\sqrt m$-ratios that depend only on the edge type ($c/a$, $b/a$, $c/b$). We refer to this as $\textit{edge universality}$; it is monomial arithmetic, not a Clebsch-Gordan cancellation. The universal Jordan eigenvalue spectrum $(q-δ, q, q+δ)$ with $δ^{2}=3/8$ is fixed by the cubic on the coassociative slice of $J_3(\mathbb O_\mathbb C)$. [abstract truncated]