Introducing the framework of pseudo-motivic homology, the paper finishes the proof that the Brauer–Manin obstruction is the only obstruction to the local–global principle for zero-cycles on a Severi–Brauer fibration of squarefree index over a smooth projective curve over a number field, provided that the Tate–Shafarevich group of the Jacobian of the base curve is finite. More precisely, for such a variety the Chow group of global zero-cycles is dense in the subgroup of collections of local cycles that are orthogonal to the (cohomological) Brauer group of the variety.