Single Lie Brackets on Closed Manifolds

We prove that every smooth vector field on a closed smooth manifold is the Lie bracket of two smooth vector fields. The proof combines a finite transport construction, which allows prescribed data near a compact subset of one level set, with an exact extension across transverse graph disks. A loop of compactly supported transverse diffeomorphisms, followed by a time shear and an inverse shift, corrects an arbitrary scalar transition defect in any prescribed positive width. After arranging a canonical pair near an equator in a flow box, the global extension is completed on two disjoint graph caps. The argument applies in every dimension and requires no orientability hypothesis. All auxiliary extension statements are proved. As a consequence, every smooth function on the cotangent bundle that is linear on each fiber is a Poisson bracket of two such functions.