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Following Elliott's earlier work, we show that the Elliott invariant of anyfinite separable simple $C^*$-algebra with finite nuclear dimension can alwaysbe described as a scaled simple ordered group pairing together with a countableabelian group which unifies the unital and nonunital, as well as stablyprojectionless cases. We also show that, for any given such invariant set,there is a finite separable simple $C^*$-algebra, whose Elliott invariant isthe given set, a refinement of the range theorem of Elliott in the stable case.  In the stably projectionless case, modified model $C^*$-algebras areconstructed in such a way that they are of generalized tracial rank one andhave other technical features.  We also show that every stably projectionless separable simple amenable$C^*$-algebra in the UCT class has rationally generalized tracial rank one.

On classification of non-unital amenable simple C*-algebras, III, the range and the reduction