The Borel complexity of conjugacy for minimal compact systems

We prove that topological conjugacy of minimal compact metrizable systems is Borel bireducible with the universal orbit equivalence relation of Polish group actions, establishing the degree predicted by Sabok's conjecture. We construct a Borel reduction from homeomorphism of compacta by encoding each compactum in the nondegenerate connected components of a minimal system. A relative extension theorem supplies measure-preserving homeomorphisms between the corresponding continua. Hall's theorem then gives permutations of representatives in finite equal-mass partitions, with approximation errors tending uniformly to zero. We realize these permutations as symmetries of a fixed Cantor minimal system, each commuting with the dynamics, and extend them to conjugacies through the closure of a graph defined by an observable. Conversely, the topology of any nondegenerate component recovers the original compactum.