Torsion-free S-adic shifts and their spectrum

The paper’s Theorem 13 proves that if a minimal shift is somewhere finite-to-one over an odometer, then its maximal equicontinuous factor (MEF) is itself an odometer given by a rotation on a compact group that is a finite cyclic extension of the given odometer; the proof explicitly factors the map through the MEF, shows the induced map is a homomorphism, uses fiber cardinalities to force a finite kernel, and concludes total disconnectedness and procyclicity of the MEF group . The candidate solution follows the same structure: factor through the MEF, prove the factor map is affine (hence a homomorphism after translation), use a somewhere finite fiber to deduce a finite kernel, infer total disconnectedness, and conclude that the MEF group is procyclic (so an odometer), with a cyclic kernel. The logical steps, including the continuity/density argument to prove homomorphism and the kernel/fiber-size argument, align closely with the paper’s proof sketch; no substantive discrepancies were found.