  If Z is an open subscheme of Spec ZZ, X is a sufficiently nice Z-model of a
smooth curve over QQ, and p is a closed point of Z, the Chabauty-Kim method
leads to the construction of locally analytic functions on X(ZZ_p) which vanish
on X(Z); we call such functions "Kim functions". At least in broad outline, the
method generalizes readily to higher dimensions. In fact, in some sense, the
surface M_{0,5} should be easier than the previously studied curve M_{0,4}
since its points are closely related to those of M_{0,4}, yet they face a
further condition to integrality. This is mirrored by a certain "weight
advantage" we encounter, because of which, M_{0,5} possesses new Kim functions
not coming from M_{0,4}. Here we focus on the case "ZZ[1/6] in half-weight 4",
where we provide a first nontrivial example of a Kim function on a surface.
  Central to our approach to Chabauty-Kim theory (as developed in works by S.
Wewers, D. Corwin, and the first author) is the possibility of separating the
geometric part of the computation from its arithmetic context. However, we find
that in this case the geometric step grows beyond the bounds of standard
algorithms running on current computers. Therefore, some ingenuity is needed to
solve this seemingly straightforward problem, and our new Kim function is huge.
