On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice Z2

The paper’s main theorem explicitly establishes a finite-range interaction on Z^2 for which no continuous-parameter selection of equilibrium states converges as temperature goes to zero, i.e., for any one-parameter family (μβ) the limit does not exist, thus resolving the d=2 case left open in [CH10] (Theorem 1.1; see the statement and surrounding discussion) . The proof uses a 2D SFT built from a specially crafted 1D effective subshift (via DRS12), plus a locally constant penalty potential that vanishes on the SFT and entropic counting to force oscillation between two disjoint subsets, formalized in Proposition 6.1 (parity-based concentration) and underpinned by the embedding theorem and Proposition 3.2 on preimage counts . By contrast, the candidate solution relies on an unsubstantiated claim that every 2D equilibrium state’s pushforward under a fixed factor map equals the unique 1D equilibrium state at an “effective” inverse temperature proportional to β. The paper does not prove such a uniqueness/pushforward identification—indeed it remarks it does not analyze uniqueness at low temperature—and instead proceeds by direct entropy and pattern-counting arguments, avoiding any need for that claim . Because the candidate’s conclusion depends on the unproven pushforward-uniqueness property, the proposed proof is not valid, even though the high-level idea (embedding a 1D engine via DRS12) overlaps with the paper’s approach.