Negative Mass Is Not the Hypothesis. It Is the Theorem.

We propose a minimal extension of the bimetric Clifford framework Cl(10, 2) [1] to theˆ Cl(8, 0). The real algebra Cl(12, 2), constructed as the graded tensor product Cl(4, 2) ⊗ internal symmetry group enlarges from Spin(6) ∼= SU(4)PS (Pati–Salam) to Spin(8), which is the unique Spin group possessing an outer automorphism of order 3: the triality of D4 . This automorphism cyclically permutes the three inequivalent 8-dimensional representations 8v ↔ 8s ↔ 8c , providing a geometric origin for the three observed fermion generations. The conformal mass-inversion operator Ê0 = e+ e− and the bimetric projectors P± = (1 ±Ê0 )/2 are structurally unaffected, ensuring that each generation possesses its negative-mass counterpart. The 91 bivectors of so(12, 2) decompose as 15grav ⊕ 28int ⊕ 48mix , where the 13 new internal generators (relative to [1]) naturally accommodate the CKM/PMNS inter-family mixing and a conserved family number.