 GCH arithmetic at aleph omega

Put $\kappa ={\aleph }_{\omega }$. ZFC gives ${\kappa }^{\kappa }=2^{\kappa }$. König's theorem and the shift of product coordinates give $\kappa <{\prod }_n{\aleph }_n\le {\kappa }^{{\aleph }_0}\le 2^{\kappa }$. The single instance $2^{\kappa }={\kappa }^{+}$ of the generalized continuum hypothesis therefore fixes all three displayed values. The strict lower bound explains why countable exponentiation cannot leave this singular cardinal unchanged.