Triality and the Three Generations: Beyond Duality on the Minimal Discrete Épure

The Standard Model's three fermion generations lack a structural origin. We argue that the binary duality ± underlying signature is only the first case (Z₂, square roots of unity); the cube roots of unity furnish a triality (Z₃), which in dimension 8 (octonions, the cascade's final level) takes the form unique in mathematics: the Spin(8) triality permuting the three representations 8v, 8s, 8c of equal dimension. We propose the three generations as these three representations. This basis is superior to prior candidates: it is an unbroken Z₃ (unlike the quaternionic i,j,k, whose symmetry the (1,2) costs break), intrinsic to 𝕆, and permutes three identical copies — exactly the nature of generations (identical but for mass). The mass hierarchy is the breaking of triality, posited, consistent with the programme's 'no fixed point' principle. The number 3 is thereby triply over-determined: Hurwitz, non-associativity, and triality. We state the open problem: the explicit triality-representation ↔ fermion-family correspondence remains to be constructed. The result reframes the question of limits — beyond positive/negative — as structurally fertile.