A Four-Prong Obstruction to (3+1)-Dimensional Emergence from Cl(6)

We prove that the Cl(6) bivector algebra so(6), viewed as the internal symmetry algebra of a Cl(6)-based emergence program, admits no clean assignment of spatial rotation generators compatible with the simultaneous preservation of su(3) color. The obstruction is dimension-theoretic and basis-independent: any designation of twointernal directions as spatial leaves a four-direction complement whose bivector algebra is so(4), of dimension six, which is strictly less than the eight required to house su(3). Combined with three further algebraic obstructions -- the one-dimensional centralizer of su(3) in so(6), the one-dimensional nature of the chain axis that supplies the 1+1 sector, and the parallel double-booking dilemma on the quaternion factor H -- the result constitutes a four-prong structural no-go for (3+1) emergence under the full Stance-1 hypothesis (no imported spacetime or gauge structure). All four prongs are computationally verified at machine precision. The no-go sharpens and completes earlier partial results. Under Stance 1 the only surviving escape is whether the emergent infrared theory voids the hypotheses of the Coleman-Mandula theorem bybeing genuinely gapless with no asymptotic particle states; this remains open. Under Stance 2, the centralizer theorem provides an algebraic justification for every import rather than a default assumption.