  When is a topological branched self-cover of the sphere equivalent to a
rational map on CP^1? William Thurston gave one answer in 1982, giving a
negative criterion (an obstruction to a map being rational). We give a
complementary, positive criterion: the branched self-cover is equivalent to a
rational map if and only if there is an elastic spine that gets "looser" under
backwards iteration.
  This completes a series announced in arXiv:1502.02561 and started in
arXiv:1507.05294 and arXiv:1607.00340.
