Spectral Syzygies, Arithmetic Selection, and Galois Rigidity in Octonionic Flavor Structure

We establish exact spectral laws (syzygies) for the two physical Yukawa channels of the octonionic flavor framework. In the protected channel T_u = (2,4,6), the product of eigenvalues is affine in their sum. In the charged channel T_d = (3,5,6), a factorized characteristic polynomial selects the unique rational interior point t = 6/7 with spectrum in Q(√3). A three-entry Gram lemma classifies the seven Fano lines into five affine and two radical. The Casimir commutator satisfies the cubic law C₂³ = −ω²C₂ with ω² ∈ Q(√6). The four-field arithmetic constellation Q(√2), Q(√3), Q(√5), Q(√6) forms a Galois sub-complex stable under order-2 corrections. The principal branch predicts δ_PMNS ≈ +45° (sin δ > 0), experimentally distinguishable from entanglement minimization (~180°) and vacuum-phase 2HDM models (~293°).