Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Let $Q$ be an acyclic quiver and $w\ge 1$ be an integer. Let ${\mathsf{C}}_{-w}(\mathbf{k}Q)$ be the $(-w)$-cluster category of $$\mathbf{k}$ Q$\mathrm{.}Weshowthatthereisabijectionbetweensimple-mindedcollectionsin$$\mathsf{D}$^b ($\mathbf{k}$ Q)$lyinginafundamentaldomainof$$\mathsf{C}${-w} ($\mathbf{k}$ Q)$and$w$-simple-mindedsystemsin$$\mathsf{C}${-w} ($\mathbf{k}$ Q)$\mathrm{.}ThisgeneralisesthesameresultofIyama-Jininthecasethat$Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf{H}$ of a bounded t-structure in a Hom-finite, Krull-Schmidt, $\mathbf{k}$-linear saturated triangulated category $\mathsf{D}$ is functorially finite in $\mathsf{D}$ if and only if $\mathsf{H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in ${\mathsf{C}}_{-w}(\mathbf{k}Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.