I will explain that to every small and decaying solution of the linearized constraint equations about Minkowski initial data, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. The main tool is a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. Using a recent result, one obtains that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity.