Specification in Mahavier Systems via Closed Relations

The paper proves the equivalence of (1)–(4) for Mahavier systems under p1(F)=p2(F)=X by passing between the forward system and its inverse limit using Theorem 3.18, plus surjectivity-based equivalence of specification and initial specification (Theorem 2.12), and a transfer result for inverse limits (Theorem 2.20). This is stated as Theorem 3.19 and its proof is complete and correct . The candidate solution reaches the same equivalence via a different route: natural extension and a direct “lift” of initial specification to the inverse limit. Two issues need correction: (i) it claims specification ⇔ initial specification for arbitrary compact systems (false in general; see Example 2.11 and Theorem 2.12, which require surjectivity) , and (ii) it asserts that specification descends through any factor map, which is not generally valid and is unnecessary here. After restricting Lemma A to surjective dynamics (which holds here since σ+F is surjective and σF is a homeomorphism by Observation 3.16) and relying on the natural extension equivalence (Theorem 3.18) along with inverse-limit transfer (Theorem 2.20) , the model’s proof aligns with the paper’s conclusion.