  Polylogarithms are those multiple polylogarithms that factor through a
certain quotient of the de Rham fundamental group of the thrice punctured line
known as the polylogarithmic quotient. Building on work of Dan-Cohen, Wewers,
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 $\mathrm{Spec}\,\mathbb{Z}$. To do so, we develop a greatly refined version
of the algorithm of Dan-Cohen tailored specifically to this case, and we focus
attention on 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. We also discover an
interesting consequence of the symmetry-breaking nature of the polylog quotient
that forces us to symmetrize our polylogarithmic version of Kim's conjecture.
  In this first part of a two-part series, we focus on a specific example,
which allows us to verify an interesting new case of Kim's conjecture.
