The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories), the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to K(R) similar or equal to Z x | BGL(R)|(+), where GL(R) is the monoidal category of weakly invertible matrices over R. The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If pi(0)R is a ring, this is almost formal, and our approach is to replace R by a ring completed version, R, provided by Baas, Dundas, Richter, and Rognes {[}J. reine angew. Math., to appear] with HR similar or equal to H (R) over bar and pi(0)(R) over bar the ring completion of pi(0)R. The remaining step is then to show that ` stable R-bundles' and `stable (R) over bar -bundles' are the same, which is done by a hands-on contraction of a custom-built model for the difference between BGL(R) and BGL((R) over bar).