couplings to $SU(2)$ doublets at dimension three or four. These two twisted sector fields had a selection-rule allowed coupling to a non-Abelian group singlet at the leading (dimension three) order, which could therefore play the role of $S_3$ in (\[Wfull\]). Requiring these two fields to carry hypercharge $Y= \pm 1$ (and thus automatically ensuring that the candidate $S_3$ have vanishing hypercharge) placed two constraints on the allowed hypercharge embedding.

We then proceed to the coupling (\[WT\]), or $\lambda_T$ in (\[Wfull\]) for the $Y=+1$ species. Each bifundamental had several couplings of this form to various pairs of $SU(2)$ doublets, at both dimension three and dimension four. By considering all possible pairs and requiring that the doublets involved be assigned $Y= -1/2$ places two more constraints on the allowed hypercharge embedding. Finally, we proceed to the equally critical $\lambda_2$ coupling in (\[Wfull\]) for the oppositely charged $Y=-1$ species. Again by considering all possible pairs of doublets with this coupling, and requiring that both have hypercharge $Y=+1/2$ we typically constrained the hypercharge embedding to a unique embedding. The hypercharges of all the states in the theory are then determined. Not all will have Standard Model hypercharges, and thus most will have fractional or 