  We prove that, generically on a smooth, connected variety in characteristic
zero, local systems of geometric origin are stable under extension in the
category of all local systems. As a consequence of this, we obtain a (generic)
motivic strengthening of Hain's theorem on Malcev completions of monodromy
representations.
  Our methods are Tannakian, and rely on an abstract criterion for ``Malcev
completeness'', which is proved in the first part of the paper. A couple of
secondary applications of this criterion are given: an alternative proof of
D'Addezio--Esnault's theorem, which says that local systems of Hodge origin are
stable under extension in the category of all local systems; a generalisation
of the theorem of Hain, mentioned above, which also affirms a conjecture of
Arapura; and an alternative proof of a theorem of Lazda, which under suitable
assumptions gives an isomorphism between the relative unipotent de Rham
fundamental group and the unipotent de Rham fundamental group of the special
fibre.
