Subshifts on groups and computable analysis

The paper’s Theorem 4.2 is proved by a standard, carefully effectivized inverse-limit construction of effective subshift covers and a quotient to enforce relators, culminating in a computable action on an effectively closed zero-dimensional subset of Cantor space that factors onto the given EDS; the key steps are explicit and internally referenced (effective covers via Lemma 4.17, effective subshifts via Proposition 4.14, the inverse limit Z and factor map φ, and the final conjugacy to Cantor space via Proposition 4.20), yielding a complete, correct proof . By contrast, the candidate solution’s direct construction hinges on two unjustified (and generally false) steps: (i) the existence, for each cover element B and generator s, of a single next-level ball D that contains f_s(K∩B); a finite cover by small-radius balls does not ensure such “image selection” containment, so the search for L_s(n,B) may not terminate; and (ii) a non-emptiness argument for the constraint set Z that attempts to satisfy y(wr)=y(w) for all relators using “disjoint level tracks.” This would require every p∈P to be a fixed point of T_r for all relators r, which need not hold, and the finite-consistency argument offered does not resolve the resulting global equalities across all tracks. Hence the model’s proof is incomplete/incorrect on essential points, while the paper’s proof is sound.