This paper derives the complete set of representable spin quantum numbers (s = 0, 1/2, 1, 3/2, 2, 5/2) from the orientation structure of the 5-dimensional orthoplex using integer geometry. The distinction between fermions and bosons emerges geometrically. Force directionality is determined by sign-reversal parity. A 3+3+3+1 decomposition under S3 symmetry provides a geometric candidate for particle generations.
Noriaki Kihara (Fri,) studied this question.