Shifts on the Lamplighter Group

The paper’s Theorem 6.1 proves that for every effective H-system X, the induced L-system ι∘η(X) has an SFT cover, and moreover ι∘η(X) × Y is almost (Z/3)^2-to-1, with an explicit, effective construction via substitutive scaffolding, Wang towers, and an auxiliary SFT W that is a (Z/3)^2-cover of the lamp action Y; the AFT property is then obtained for ι∘η(X) × W, implying the almost (Z/3)^2-to-1 statement for the product with Y (see the statement and proof sketch of Theorem 6.1 and its follow-ups in the paper , and the detailed AFT/measurable uniqueness discussion and the role of W in Sections 5–6 ). By contrast, the candidate solution’s core “marker” layer posits two commuting base‑3 odometers along the t (height) and a (lamp) directions to form a Robinson-style grid on L. This cannot be realized: the lamplighter lamp-toggling generator a has order 2, so a local rule enforcing c_h(ga) = c_h(g) + 1 mod 3 for all g is incompatible with a = a^{-1} (it would force 0 ≡ 2 mod 3), and more broadly L’s Cayley graph is not a Z^2 grid; hence the proposed 2D hierarchical macro-tiles in the (t,a)-plane are ill-defined. The paper instead uses a carefully constructed SFT Xtree (and related structures) producing C0 = (Z/3)^2 labels and tree skeleta to implement substitutions and computations across S-cosets, together with a strictly ergodic auxiliary system W that covers Y with deck group (Z/3)^2, avoiding any need for an impossible “horizontal odometer along a” (see Section 5.7 and Corollary 5.16 for the C0-structure and its use as a finite deck group ). Therefore, the paper’s argument is correct, while the model’s construction hinges on an invalid local constraint on the order‑2 generator a.