We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Π1 + ¬Ω1has a proper end-extension to a model of Π1, and so Π1 + ¬Ω ⊢ BΣ1. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Π1 + ¬Exp ⊢ BΣ1. Both assumptions can be modified to versions which make it possible to replace Π1 by IΔ0as the base theory.We also show that any proof that IΔ0+ ¬Exp does not (...) prove a given finite fragment of BΣ1has to be “nonrelativizing”, in the sense that it will not work in the presence of an arbitrary oracle. (shrink)