A Study of Directional Entropy Arising from Z × Z+ Semigroup Actions

The paper states and proves an upper bound for the directional measure-theoretic entropy of the Z^2-action generated by a 1D linear cellular automaton and the shift under a Bernoulli measure: 0 ≤ h_E(Φ) ≤ |H|(A−ℓ)·H(c), in the regime of “B not large” (i.e., for slopes with −A ≤ B/H ≤ −ℓ). This is Theorem 3 in the paper’s Section 3.1 on Bernoulli measures, which relies on the standard directional-window definition, the scaling property h_{U E}(Φ)=|U|h_E(Φ), and generator partitions . The candidate solution derives the same inequality by an explicit light-cone counting argument for the window W((ρ,1),L,C), using η (the one-site partition) as a generator, the independence under μ_c, and the monotone nesting of base-coordinate intervals when −A ≤ ρ ≤ −ℓ. It also correctly excludes the horizontal case H=0, where the inequality with factor |H| would be trivial but not informative; the paper’s proof framework similarly treats H≠0 through its window definition, and horizontal directions are handled separately via lattice directions Φ(B,0) (equal to f^B) . Thus, the model reproduces the same bound with a different but standard proof; both are correct, with a minor clarification needed in the paper to state H≠0 explicitly in Theorem 3.