GIBBS PROPERTIES OF EQUILIBRIUM STATES

The paper proves two main results: (A) if a continuous potential φ on Ω = E^Z has the extensibility property, then every equilibrium state is both DLR and weak Bowen–Gibbs; (B) conversely, any translation-invariant DLR Gibbs measure is an equilibrium state for some extensible potential. The statements and their proof strategy are explicit in the PDF (Theorem A and B) and rely on constructing a cocycle ρ from two-sided Birkhoff differences (Proposition 4.1), building a specification γ_φ from ρ (Theorem 4.2), and transferring estimates via a weak cohomology link between φ and the one-sided φ_{γ_φ}, culminating in subexponential cylinder distortion and a relative-entropy variational argument (e.g., equations (21), (22), (34), (35)–(42), (47), (48)) . By contrast, the candidate solution incorrectly identifies the paper’s extensibility hypothesis with Walters’ condition and therefore invokes Walters’ one-sided cohomology and the Ruelle–Perron–Frobenius theorem to get cylinder estimates. The paper explicitly states that extensibility strictly covers Walters’ class and does not imply uniqueness (see Section 3, condition (4) and surrounding discussion) . Furthermore, the candidate’s converse direction (Part B) is proved only for translation-invariant uniformly absolutely summable (UAC) interactions, whereas the paper treats all translation-invariant DLR specifications, including those not representable by a TI UAC interaction . Hence the paper’s argument is correct and more general, while the candidate’s proof relies on stronger, unjustified assumptions and does not cover the full scope.