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Building on work by Dan-Cohen--Wewers, Dan-Cohen [DC], and Brown, we push thecomputational boundary of our explicit motivic version of Kim's method in thecase 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 tothis case. We also commit ourselves fully to working with the polylogarithmicquotient. This allows us to restrict our calculus with motivic iteratedintegrals to the so-called depth-one part of the mixed Tate Galois groupstudied extensively by Goncharov. An application was given in part one, wherewe verified Kim's conjecture in an interesting new case.

The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory II