Typical ground states for large sets of interactions

The paper works in the interaction space B0, canonically identified with C(Ω), and uses Morris’ theorem to pull generic properties of invariant measures (weakly mixing, not mixing, singular spectrum, non-Gibbs) to a dense Gδ set of potentials. This is explicitly stated in Section 3 and Corollary 2, with discussion in the abstract and introduction . The candidate’s contradiction hinges on transferring a residual periodic-maximizer statement from the Lipschitz class on the one-sided shift to an open dense set in C(Ω) for the two-sided shift. That density step fails: functions depending only on nonnegative coordinates (lifted via π) are not dense in C(Ω), so the proposed approximation of an arbitrary U ∈ C(Ω) by V = W∘π is generally impossible. Hence the claimed open dense subset O ⊂ C(Ω) with periodic ground states cannot be established. The paper itself even acknowledges that in smaller/regularity-restricted spaces (e.g., Lipschitz in d=1) periodic ground states can be generic, but stresses that its generic results are for C(Ω) (B0) .