A PROOF OF THE Cr CLOSING LEMMA AND STABILITY CONJECTURE

The paper defines structural stability for flows via time-preserving topological conjugacy (φ_t ∘ h = h ∘ ψ_t) for all t, rather than the standard orbit-equivalence allowing continuous time reparametrization, see the introduction and definition in the first page of the PDF . It states the “stability conjecture” with condition (i) Ω(φ_t) = P(φ_t) and (ii) Ω(φ_t) hyperbolic, and then claims Theorem 1: if a flow is C^r structurally stable, it satisfies (i) and (ii) . A central step asserts Corollary 3.1: Ω(X) = P(X) for structurally stable systems, with a proof that mistakenly “coincides” an arbitrary homeomorphism h with the conjugacy h′ (this is not justified and is generally false) . Moreover, the paper purports to prove a C^r closing lemma for r ≥ 2 (Theorem 2.1) as a cornerstone of the argument, a result widely known to be open beyond C^1 and thus highly suspect; see their own positioning of Theorem 2.1 as the C^r closing lemma . The paper also claims Theorem 4.1 (hyperbolic splitting on Ω for structurally stable flows) , which would follow from their (i), but the preceding logic is unsound. By contrast, the candidate solution correctly notes that (a) Ω = P is false in general dimensions under the standard notion of structural stability (e.g., Anosov/geodesic flows have Ω strictly larger than the periodic set), and that (b) Ω is hyperbolic does hold for structurally stable flows (Axiom A), with the usual caveat that flows are considered up to time reparametrization. The candidate also correctly observes that with the paper’s stricter time-preserving conjugacy, structural stability is essentially vacuous (e.g., constant time rescalings cX are not conjugate), which undermines the paper’s framework.