Every spectrum is the K-theory of a stable $\mathrm{\infty }$-category

We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\mathrm{\infty }$-category. We use these results to construct a stable $\mathrm{\infty }$-category $\mathcal{C}$ with a bounded t-structure such that $\mathrm{K}(\mathcal{C})$ is not equivalent to $\mathrm{K}({\mathcal{C}}^{\mathrm{♡}})$, disproving a conjecture of Antieau, Gepner, and Heller.