We show that Klemenc’s stable envelope of exact ∞-categories induces an equivalence between stable ∞-categories with a bounded heart structure and weakly idempotent complete exact ∞-categories. Moreover, we generalise the Gillet–Waldhausen theorem to the connective algebraic K-theory of exact ∞-categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact ∞-categories. A key tool is a generalisation of a theorem due to Keller which provides a sufficient condition for an exact
functor to induce a fully faithful functor on stable envelopes.