Weak pure infiniteness and Cuntz-algebra absorption

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C^∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $\mathcal O_\infty$, without unitality or simplicity assumptions.