A central ingredient in Kim's work on integral points of hyperbolic curves is the "unipotent Kummer map" which goes from integral points to certain torsors for the prounipotent completion of the fundamental group, and which, roughly speaking, sends an integral point to the torsor of homotopy classes of paths connecting it to a fixed base-point. In joint work with Tomer Schlank, we introduce a space Ω of rational motivic loops, and we construct a double factorization of the unipotent Kummer map which may be summarized schematically as points → rational motivic points → Ω-torsors → π1-torsors. Our "connectedness theorem" says that any two motivic points are connected by a non-empty torsor. Our "concentration theorem" says that for an affine curve, Ω is actually equal to π1.