DIRECTIONAL DYNAMICS OF Z+ × Z-ACTIONS GENERATED BY 1D-CA AND THE SHIFT MAP

The paper’s Theorem 3.2 states the same entropy formula h_S(Φ) = 2r log a · limsup_{l→∞} m_l/l under the bipermutative endpoint condition gcd(λ_{±r},a)=1 and a syndetic time set {m_i} with bounded gaps (and m_i>n_i) . However, the proof asserts—without adequate justification—that the joined pullbacks ∨_{i=0}^l Φ^{-(m_i,n_i)} ξ(−M,M) “consist of all cylinder sets” on a single contiguous block determined solely by (m_l,n_l) (after choosing M large), and then directly computes the entropy from that partition . This step elides the nontrivial issue that different shifts n_i can center the backward light-cones at different places, so nesting and reduction to one single interval is not automatic. By contrast, the candidate solution supplies the standard bipermutative light-cone argument: a uniform upper bound of (2r M_k + O(1)) log a on the join’s entropy for any finite partition, and a matching lower bound (2r M_k − O(1)) log a via a fixed 2r-block partition; the bounded-gap hypothesis is used only to ensure no step is skipped in the reconstruction. This yields the formula rigorously, independent of a delicate “single-block” reduction. Hence the result is correct, but the paper’s proof is incomplete at a key combinatorial step, while the model’s proof is correct and complete.