Topological isomorphism of procountable groups is universal analytic

We prove that topological isomorphism of procountable groups is a universal analytic equivalence relation. This answers a question of Gao, Nies, and Paolini and determines the complexity of topological isomorphism of non-Archimedean Polish groups. We encode permutative equivalence of unconditional basic sequences by integer weights on a countable structure. A tree construction turns bounded changes in these weights into uniform homeomorphisms, while a finite-level counting argument recovers the weights up to bounded error. A graph-to-group functor whose nontrivial homomorphisms, modulo inner automorphisms of the target, come from unique graph maps transfers this equivalence to inverse systems of countable groups. We then obtain a Borel realization of their limits as closed permutation groups.