Finite type as fundamental objects even non-single-valued and non-continuous

The paper’s main equivalence is explicitly stated and proved: for a compact totally disconnected Hausdorff space X and a (closed-valued, upper semicontinuous) set-valued map F, shadowing holds if and only if (X,F) is conjugate to an inverse limit of finite-type set-valued systems satisfying a Mittag–Leffler condition, with the induced orbit systems also satisfying the Mittag–Leffler condition (Theorem 6.8, derived from Theorem 4.12 and the orbit/partition machinery in Section 6) . The construction via clopen partitions (O(U), PO(U)), the conjugacy of orbit systems, and the ML property are handled carefully and non-metrically (Theorem 6.6, Lemma 6.5) , with the structural representation by simplest finite-type systems supplied earlier (Theorem 5.8) . By contrast, the model’s proof outline contains a critical flaw: its “uniform δ-adjacency” rule is too strong to be guaranteed by a single shadowing orbit, and its verification of the bonding compatibility (Condition (2.1)) incorrectly uses a symmetric Hausdorff bound that does not follow from mere upper semicontinuity; the constants also do not close (δ_m/3 + δ_n/3 ≤ δ_n/3 is impossible). Thus, the paper’s argument stands, while the model’s proposed proof is unsound.