Pointed Gromov-Hausdorff Topological Stability for non-compact metric spaces

The paper’s main theorem (C-expansive + C-shadowing ⇒ topological pGH-stability) is stated as Theorem 4.3 and a proof sketch is provided that follows the classic shadowing/expansiveness template in the pointed GH0 setting. However, key steps rely on local pointed approximations that only control dynamics on a bounded ball around the basepoints, yet the construction of a bi-infinite pseudo-orbit {j ∘ g^n(q)} and the ensuing semiconjugacy are treated as if they held for all n, without ensuring that g^n(q) remains inside the controlled ball. This gap appears where the proof asserts dB(f(x),1/δ)(j(g^{n+1}(q)), f(j(g^n(q)))) < δ for all n, although Appδ-conditions only guarantee this when g^n(q) lies inside B(y, 1/δ) (Theorem 4.3 proof snippet and surrounding lines) . The model’s solution mirrors the same structure and introduces an additional circular choice (choosing η so that η ≤ ½ inf Σ on a ball whose radius is 1/η), and asserts a for-all-n pseudo-orbit bound by “iterating” local estimates that only hold while orbits stay in the pointed ball. It also appeals to “continuous dependence of the shadowing point” without supplying the needed lemma in the noncompact, gauge-dependent setting. Because both arguments hinge on local control but proceed as if it were global, both are incomplete. The definitions of dp_GH0 and Appε, and the theorem statement, are clear and consistent with the setup (Definitions 2.6, 3.1; Definition 4.4) , and the uniqueness-from-expansiveness and shadowing ingredients are correctly invoked (Definition 4.3, Remark 4.1) . But as written, neither proof closes the locality-to-globality gap needed for the semiconjugacy and the full pseudo-orbit construction in the pointed framework.