Extended Cellular Automata

Definition 3.3 explicitly builds “local commutation” into the notion of Z‑SBC/Zip‑CA, and this includes both the functional equality R ∘ σ_τ^i = σ_τ^i ∘ R and the set-theoretic equality R(σ_τ^{-i}(x̄)) = σ_τ^{-i}(R(x̄)) (for i ≥ 0) as part of the definition, not as a consequence to be proved from locality alone. The Extended CHL Theorem 3.5 then assumes this same local commutation in the hypothesis of the (⇐) direction and only needs to produce a local rule satisfying (1)–(2), which the paper sketches via uniform continuity and the usual sliding-block argument. The candidate solution weakens “Zip‑CA” by dropping the set-equality clause from Definition 3.3, constructs a counterexample to that clause, and then declares the theorem false; but this attacks a different statement than the paper’s theorem. The paper’s equivalence, as stated with its definitions, is consistent; the model misinterprets the definitions and is therefore incorrect. See the statement of local commutation containing both equalities in Definition 3.3 and the theorem statement/proof of Theorem 3.5.