SS-3: Uniqueness of SU(3) from the Tetrahedral Cage

Abstract: SS-1 proved that the eight DI-bit hopping operators on the tetrahedral cage base equal the Gell-Mann generators and satisfy SU(3) commutation relations exactly. SS-3 proves the conditional converse: given that the cage operators are represented as traceless Hermitian matrices on C^3 with commutator bracket, su(3) is the unique Lie algebra they can generate. The argument has three steps — counting dimensions of traceless Hermitian operators on C^N gives N^2-1=8 for N=3, the matrix commutator on this space defines su(3) by definition, and the eight CPP operators span the full space by Gell-Mann orthogonality. Two corollaries follow within this representation: N=2 gives SU(2) and N=3 gives SU(3) with no other outcome possible, and no exotic gauge group can arise from three colour states. A physical interpretation maps the four bond modes and four junction modes of the full tetrahedron onto the same 8-dimensional space the Gell-Mann matrices span.