  Over the past fifteen years or so, Minhyong Kim has developed a framework for
making effective use of the fundamental group to bound (or even compute)
integral points on hyperbolic curves. This is the third installment in a series
whose goal is to realize the potential effectivity of Kim's approach in the
case of the thrice punctured line. As envisioned in the last installment, we
construct an algorithm whose output upon halting is provably the set of
integral points, and whose halting would follow from conjectures. Our results
go a long way towards achieving our goals over the rationals, while broaching
the topic of higher number fields.
