  We show that the map from $K({\mathbb S})$ to its chromatic completion is a
connective cover and identify the fiber in $K$-theoretic terms. We combine this
with recent work of Land-Mathew-Meier-Tamme to prove a form of "Waldhausen's
Chromatic Convergence Conjecture": we show that the map $K({\mathbb
S}_{(p)})_{(p)}\to \mathop{\rm holim} K(L^{f}_{n}{\mathbb S})_{(p)}$ is the
inclusion of a wedge summand.
