Leptonic CP Conservation and the Quark CP Phase from Octonionic Flavor Structure

Abstract

One generation of standard-model fermions can be realized on the complexified octonions through the Clifford algebra $\mathcal{C}l(6)$; the octonionic unification programme extends this to three generations, with generation transport implemented by $G_2$ automorphisms or by rotors built from the ladder operators. We prove a localization theorem for the CP-violating phases of this structure, using only the $\mathcal{C}l(6)$ construction and the stated three-generation representatives, independently of the wider programme. For quarks, the first-to-second generation step is the occupation flip of one ladder mode, with the up and down species coupling to conjugate ladder directions; a conjugation theorem forces $A_d=A_u^*$ for every real transport, and the most general rung-generated rotor yields the exact one-parameter law $ϕ_{12}=-2χ$: the $(1,2)$ transport phase is twice one Yukawa orientation angle. The programme's geometric rotor sits exactly at the quadrature-balanced point $|ϕ_{12}|=π/2$; the companion analysis reproduces the Cabibbo \emph{magnitude} $|V_{us}|$ with a single real tilt, leaving the rung near quadrature, but it does not extract a CKM CP phase, so the quark Dirac phase is fixed only once the underlying Yukawa orientation is computed. For leptons we prove a reality theorem: every charged-lepton and every neutrino transport amplitude is exactly real for every $G_2$ automorphism and every rotor that does not mix the identity line $\mathbb C\cdot1$ with the lepton--flavor plane $\mathrm{span}(e_7,e_5,e_2)$ a class that contains the entire quark-rung family--and identity--flavor mixing across that plane is the unique possible source of a leptonic phase. [Truncated]