PROJECTIONAL ENTROPY FOR ACTIONS OF AMENABLE GROUPS

The paper proves exactly the statement at issue: for a countable amenable group G, a normal subgroup H, and a D-strongly irreducible G-subshift X with h(X)=h(X_H), one has X = X_H^{G/H}. The proof proceeds via: (i) defining the cosetwise product lift X_H^{G/H} (independent of transversal) and noting X ⊂ X_H^{G/H} (Theorem 3.5 and Fact 3.6), (ii) establishing h(X_H^{G/H}) = h(X_H) (Theorem 3.8, using the infimum rule), (iii) showing D-strong irreducibility passes to the product lift (Theorem 3.13), and (iv) strict entropy monotonicity for strongly irreducible subshifts (Theorem 3.12), yielding the conclusion (Theorem 3.14) . The candidate solution mirrors these steps and cites the same results. The only minor imprecision is in Step 3, where the separation condition is informally expressed via D∩H inside H; the paper handles this precisely by translating patterns back to H via coset representatives before applying strong irreducibility. This does not affect correctness.