On the regularity of time-delayed embeddings with self-intersections

The paper proves the two claims via a polynomial-perturbation prevalence scheme and two key ingredients: (i) an almost-sure inverse-Lipschitz estimate at µ-a.e. x that is global in y (inequality (1.5) in Theorem 1.1), and (ii) an almost-sure local immersion that, together with (i), yields a local C^r-diffeomorphism onto a submanifold when k > dim M. These appear explicitly in Theorem 1.1 and its general version Theorem 4.1, with the periodic-points measure-zero assumption µ(⋃_{p=1}^{k−1}Per_p(T))=0 built in. See Theorem 1.1 for the precise statements, including inequality (1.5), and the explanation that V ∩ φ_{h,k}(M) is an embedded C^r submanifold near φ_{h,k}(x) . The technical backbone is Proposition 4.2 (global almost sure continuity of the inverse) and Proposition 4.5 (almost-sure local immersion), assembled into Theorem 4.1, the polynomial-perturbation version from which Theorem 1.1 follows . The model’s solution cites exactly this theorem to conclude both parts and mentions the same prevalence mechanism (finite-dimensional probe with random polynomial perturbations) and the same patching step that upgrades pointwise inverse-Lipschitz to the global-in-y inequality; this patching matches the argument explained in the non-dynamical Theorem 3.1 proof (Fact 1 + Fact 2 + a δ–ε separation), which the paper uses verbatim in the dynamical setting via Theorem 4.1 . Assumptions, thresholds (k > dim_H X with k ≥ dim M for (1), and k > dim M for (2)), and the periodic-points condition align exactly with the paper. Hence, both are correct; the model’s reasoning is essentially a synopsis of the paper’s proof strategy and results.