Nonvanishing derived limits without scales

The derived functors ${\lim }^n$ of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors, for instance in the case of the inverse system $\mathbf{A}$ of abelian groups indexed by ${}^{\omega }\omega$. So far, consistency results for nonvanishing derived limits of $\mathbf{A}$ have always assumed the existence of a scale (i.e. a linear cofinal subset of $({}^{\omega }\omega ,{\le }^{\ast })$, or equivalently that $\mathfrak{b}=\mathfrak{d}$). Here we do away with that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of ${\mathrm{\aleph }}_1\le \mathfrak{b}\le \mathfrak{d}<{\mathrm{\aleph }}_{\omega }$, thus giving a partial answer to a question of Bannister.