  Building on work by Dan-Cohen--Wewers, Dan-Cohen [DC], and Brown, we push the
computational boundary of our explicit motivic version of Kim's method in the
case of the thrice punctured line over an open subscheme of Spec ZZ. To do so,
we develop a refined version of the algorithm of [DC] tailored specifically to
this case. We also commit ourselves fully to working with the polylogarithmic
quotient. This allows us to restrict our calculus with motivic iterated
integrals to the so-called depth-one part of the mixed Tate Galois group
studied extensively by Goncharov. An application was given in part one, where
we verified Kim's conjecture in an interesting new case.
