Mixed Tate Chabauty--Kim beyond the polylog quotient

We develop an algorithmic approach to mixed Tate Chabauty–Kim theory for the full unipotent fundamental group of ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$, beyond the polylogarithmic quotient used in previous work. The key step is computing the cocycle evaluation map in the presence of a nontrivial motivic Galois action, using relations among motivic multiple zeta values. This yields explicit Kim functions involving multiple polylogarithms, as well as an algorithm for solving the $S$-unit equation whose halting is implied by certain natural conjectures.