Obstacles to Topological Factoring of Toeplitz Shifts

The paper’s Theorem 1.1 proves that for Toeplitz shifts with constructive period structures, any factor map π mapping x to y forces the supernatural divisibility q | p, and under conjugacy p = q, via a Bratteli–Vershik and local-morphism argument that constrains levels and yields divisibility relations (|θ(0,i0+i]| divides |ξ(0,ni0+i]|) leading to a contradiction if q ∤ p . A preliminary result (Proposition 3.9) already gives an index shift i0 with qi | pi0+i . The candidate solution claims a direct “phase map” proof that for each i, qi | pi, and that constructive hypotheses are unnecessary. This is flawed: it implicitly assumes that the composed map Θy,qi ∘ π factors through a finite cyclic phase of X (i.e., through Θx,pi), which is neither proved nor generally true without invoking the maximal equicontinuous factor (or the paper’s constructive Bratteli–Vershik framework). It also defines a map Fi: Z/piZ → Z/qiZ by Fi([k]) = k mod qi without first proving well-definedness; that well-definedness is equivalent to the desired divisibility and hence cannot be assumed. In contrast, the paper rigorously supplies the missing factorization via ordered Bratteli diagrams and a zero-dimensional factoring theorem .