On exact categories and their stable envelopes

We show that Klemenc’s stable envelope of exact $\mathrm{\infty }$-categories induces an equivalence between stable $\mathrm{\infty }$-categories with a bounded heart structure and weakly idempotent complete exact $\mathrm{\infty }$-categories. Moreover, we generalise the Gillet-Waldhausen theorem to the connective algebraic K-theory of exact $\mathrm{\infty }$-categories and deduce a universal property of connective algebraic K-theory as an additive invariant on exact $\mathrm{\infty }$-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.