Automorphisms of B-free Toeplitz systems

The paper’s Theorem 7.9 establishes, under General Hypothesis 3.12, a finiteness hypothesis sup_n |A^∞,p_{S_n}| < ∞, and the unbounded gcd condition (93), that there exists k0 with BW + (yF − Δ(k0)) ⊆ BW and, for all n and all a_n ∈ A^∞,p_{S_n}, lcm((S̃^{a_n}_n)_prim) ⋅ a_n | (yF − Δ(k0))_n (). The proof depends on: (i) producing local invariance cylinders and a finite set K via Proposition 7.1 and Proposition 7.4 (; ), (ii) the κ(a_n)-labeling and its uniqueness for a_n > 2m from Corollary 7.5 (), (iii) the key lcm-divisibility from Theorem 7.6 (), and (iv) collapsing K to a singleton using (93), then deducing the global divisibility via Theorem 4.2 (). The candidate solution correctly reconstructs the κ(a_n)-argument and the lcm-divisibility mechanism, but it omits the paper’s explicit hypothesis sup_n |A^∞,p_{S_n}| < ∞ required to apply Theorem 7.6(b) uniformly (see footnote to (76) in the paper) and to run the (93)-based collapse of K. More seriously, its Step 3 attempts to deduce BW + (yF − Δ(k0)) ⊆ BW from the divisibility alone by passing from a non-primitive gcd-class to a primitive one while claiming the intersection with the zero-coordinate set remains infinite; this implication is not justified and can be false. The paper avoids this by first proving K is a singleton and then invoking the local invariance cylinders to get the global inclusion (; ). With these corrections (explicitly adding sup_n |A^∞,p_{S_n}| < ∞ and replacing Step 3 by the Proposition 7.1 + 7.4 argument), the model’s outline aligns with Theorem 7.9.