G-INDEX, TOPOLOGICAL DYNAMICS AND MARKER PROPERTY

The paper proves the existence of a free Z-system without the marker property via a precise coindex obstruction that uses two test systems Y and Z, a universality construction X(N,δ), a finite-reduction lemma from infinite to finite products, and coindex calculus. Concretely: (i) if a system has the marker property, there is an equivariant map to Y (Lemma 5.3), and any map from an infinite product to Y reduces to a map from some finite subproduct to Z (Lemma 6.1) ; (ii) one chooses N,δ so that for all primes p, coind_p P_p(X(N,δ)) ≥ coind_p P_p(Z) + 1 (equation (6.3)) ; (iii) using variants X_m(N,δ) with no m-periodic points and coind_p P_p(X_m(N,δ)) = coind_p P_p(X(N,δ)) for p>m (Lemma 4.3) , the product X = ∏_m X_m is free and, if it had the marker property, would yield a Z_p-map from P_p(∏_{m=1}^M X_m) to P_p(Z). By monotonicity and the product-min rule for coindex (Proposition 3.1(1)(2)), this would force coind_p P_p(X(N,δ)) ≤ coind_p P_p(Z) (inequality (6.4)), contradicting (6.3) . The model follows this general outline but introduces a critical, unsupported claim: a uniform bound K with coind_p P_p(Z) ≤ K for all primes p (attributed to (6.3)), and then argues with that fixed K. The paper does not establish any such uniform bound—in fact, whether coind_p P_p(Z) is unbounded is posed as an open question (Problem 7.2) . The paper’s proof succeeds without a uniform K, using pointwise inequalities in p, whereas the model’s writeup as stated depends on the non-existent uniform K and is therefore incorrect as written. The paper’s argument is correct and complete (Theorem 1.1) .