Bivector Classification of Spin from the Orientation Structure of the 5-Dimensional Hypercube

This paper derives the geometric classification of spin from the orientation structure of the 5-dimensional hypercube (5-cube), the dual polytope of the 5-dimensional orthoplex. Under the assumption that five coordinate axes are divided into three observable spatial axes and two unobservable axes (Assumption A), the 40 cubic 3-cells of the 5-cube are naturally indexed by 10 bivectors, providing a more physically transparent derivation of spin as a bivector quantity. The same six spin values, fermion-boson distinction, force directionality, generation structure, and matter-antimatter asymmetry are derived, establishing a duality theorem with the orthoplex formulation.