On the Garden of Eden theorem for B-free subshifts

The paper proves, for Erdős B-free subshifts, that every cellular automaton (CA) is a monotone sliding block code modulo a power of the shift, the Garden of Eden theorem holds, and any surjective CA is a power of the shift (Theorem 1.1; proved via Theorem 3.12 together with Propositions 3.2, 3.3, 3.8) . The candidate solution reaches the same conclusions but hinges on an unproven Step 3: that any CA induces a translation by a uniform k on the “odometer of avoided residue classes.” This odometer-translation claim is neither stated nor used in the paper’s proof strategy, and it would in fact imply nontrivial measure-theoretic consequences that the authors explicitly leave open (e.g., preservation of the Mirsky measure by automorphisms) . The remainder of the candidate’s argument (monotone-after-alignment, GoE, onto ⇒ power of shift) matches the paper’s outcomes, but the key alignment-by-k step is not justified and mis-cited. In contrast, the paper obtains the needed alignment t via a direct CRT-based combinatorial construction (Proposition 3.2), from which S^{-t}∘τ is monotone, and then deduces Moore/Myhill and the onto classification without any odometer action hypothesis .