Undecidability of the block gluing classes of homshifts

The paper proves that deciding whether a 2D homshift is Θ(n)-block gluing or O(log n)-block gluing is undecidable by tying the GHdMO square-cover dichotomy to the square group and then reducing from a Markov property via Adian–Rabin. Key ingredients include: (i) the dichotomy between O(log n) and Θ(n) governed by the finiteness of the square cover, (ii) the equivalence between finiteness of the square cover and finiteness of the square group, and (iii) the realization of any finitely presented group as a square group, all of which the paper states and proves or cites . The model’s reduction matches the paper up to this point, but it crucially errs when it tries to drop “phased” and move to block gluing by adding a self-loop while still reducing from the finiteness of the group. The paper shows that adding a self-loop replaces π□1(G) by the free product π□1(G)∗Z/2Z, which is finite if and only if π□1(G) is trivial; this is used to reduce from triviality (a Markov property) rather than finiteness when passing to non-bipartite graphs via Lemma 2.8 . The model incorrectly claims that the self-loop modification is “benign” for finiteness, which is false and breaks its reduction.