Cluster Morita theorem for negative cluster categories

Fix an integer $d\le -2$. We characterize Hom-finite algebraic triangulated categories admitting a $(-d)$-simple-minded system as stable categories $\underset{‾}{\mathrm{C}\mathrm{M}}(B)$ of proper $(-d)$-self-injective non-positive dg algebras; equivalently, each admits a $d$-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a $d$-Calabi–Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right $(d+1)$-Calabi–Yau structures on the finite-dimensional part with normalized right $d$-Calabi–Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav–Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left–right Calabi–Yau symmetry.