Equal Sigma and Xi hyperfine splittings ($M_{qqs,3/2}-M_{qqs,1/2}=M_{qss,3/2}-M_{qss,1/2}=3A^{\prime }/(2m_q{m}_s)$)

Use dimensionless spin angular momentum operators with $\hslash =1$ and the common-coefficient baryon hyperfine mass formula. The identical pair has total spin one, so its pair expectation is $1/4$. For total baryon spin $1/2$, the two mixed-flavour pair expectations sum to $-1$; for total spin $3/2$, they sum to $1/2$. Their difference is $3/2$, giving the title formula for either repeated-flavour choice. The equality assumes the same effective coefficient $A^{\prime }$ in the two flavour configurations; different spatial overlaps can change it.