Equilibrium states which are not Gibbs measure on hereditary subshifts ∗

The paper’s main theorem (Theorem 1.1) is proved via a clear and correct cylinder-counting argument combined with a Gibbs lower bound and an explicit formula for the multiplicative convolution κ = ν ∗ Bq,1−q on hereditary closures; it culminates in a contradiction with non-atomicity by forcing a uniform positive lower bound on ν of a sequence of cylinders, invoking Lemma 4.2 (atomicity criterion). The candidate solution, while capturing a related intuition (an upper bound on κ([W]) and a Gibbs lower bound), replaces the paper’s quantitative comparison by an unproven ‘tilted’ optimization step P̃ ≥ D^κ_{φ̃ + tζ} and a flawed sign analysis. Key steps (a replacement lemma controlling sup S_n φ̃ under 0→1 flips, and the derivation of P̃ ≥ D_{φ̃} + tD) are asserted but not established, and the final inequality comparing the lower and upper bounds on P̃ is not rigorously justified.