TAIL VARIATIONAL PRINCIPLE AND ASYMPTOTIC h-EXPANSIVENESS FOR AMENABLE GROUP ACTIONS

The paper proves the tail variational principle for actions of countable amenable groups (Theorem 2.1), namely h*(X,G) = max u1 = lim_k sup_μ θ_k, by (i) working first in the zero-dimensional case, where both the sup and inf in the definition of tail entropy are realized along a refining clopen sequence (equation (7.2)), and applying a conditional variational principle for clopen partitions (Proposition 5.1), and (ii) passing to general systems via principal zero-dimensional extensions and the preservation of tail entropy under principal extensions (Proposition 6.1). These steps and definitions are explicit in the paper’s Sections 3–7, including the formulation of h_G(U|W) via subadditivity and the Ornstein–Weiss lemma, the entropy-structure formalism with θ_k and u1, and the final assembly of the proof of Theorem 2.1 . By contrast, the model’s solution omits the necessary finite-entropy hypothesis, conflates topological conditional and relative entropy without invoking the topological Pinsker formula used in the paper to bridge them, and, most critically, argues circularly by appealing to the very tail variational principle it seeks to prove in the zero-dimensional step. The paper’s proof is coherent and complete; the model’s proof outline is incomplete and circular.