Eigenvalues and the Stabilized Automorphism Group

The paper proves that for minimal systems with at least one nontrivial rational eigenvalue, an isomorphism Aut(∞)(X,T) ≅ Aut(∞)(Y,S) forces the same rational eigenvalues (Theorem 1.1), via a careful analysis of centralizers and wreath-product decompositions of stabilized automorphism groups, combined with the equivalence between rational eigenvalues and clopen cyclic partitions in the transitive/minimal setting . By contrast, the candidate solution hinges on an unproven “mod-q index” epimorphism Θq: Aut(X,Tq) → Z/qZ whose existence is asserted to be equivalent to having a rational eigenvalue and to be functorially recognizable from Aut(∞). This equivalence is neither established nor used in the paper’s argument; the paper instead detects the integers q via Sym(q) factors in centralizer/wreath-product structures and uniqueness results for wreath products (see the proof of Theorem 1.1: equations (5.1)–(5.5) and the associated centralizer isomorphisms) . Hence, while the candidate’s final conclusion matches the paper’s theorem, key steps of the candidate’s proof are unsupported or incorrect.