Shadowing for Families of Endomorphisms of Generalized Group Shifts

The paper’s Theorem 8.1 is established via column factorizations, a finite-type reduction, and a Lipschitz estimate that turns small local errors into exact window-consistency and then a global realization; the chain of lemmas and constructions is coherent and complete. In contrast, the model’s proof invokes an “admissible Artinian” structure on Σ^Λ and a Mittag–Leffler stabilization on projections Σ^{Λ′}→Σ^Λ where such admissibility is not defined. The ML step is unnecessary (the pseudo-orbit itself provides a compatible family) and, as stated, not justified. The model also does not address metric-independence (Definition 3.3). Hence the paper’s argument stands, but the model’s is flawed in key logical steps.