Prime Cliord Algebra over Number Fields: The Class Group Obstruction

We extend the Prime Cliord Algebra (PCA) framework from the rational numbers Qto arbitrary algebraic number elds K. By replacing the rational prime basis with a basisindexed by prime ideals p of the ring of integers OK, we generalize the Prime Vector Spaceand the associated Cliord algebra. We rigorously prove that the Fundamental Barrierto linear addition persists in this setting, manifesting as the non-linearity of the ideal sum.Furthermore, we identify a novel topological feature absent in the rational case: the ClassGroup Obstruction. We show that the Inverse Sum Operator is well-dened only onthe sublattice of principal ideals, and that the Geometric Gap between the full Cliordmanifold and the arithmetic of K is isomorphic to the Ideal Class Group Cl(K). This resultprovides a geometric characterization of the failure of unique factorization in terms of thetopology of the PCA phase space.