Representation of measurable multidimensional flows by Lipschitz functions

The uploaded paper proves that for every k ∈ N the shift on Lip1(R^k) is a topological model for all free measurable R^k-flows (Theorem 1.2) and gives a complete proof based on (i) producing countably many orbital 1‑Lipschitz codes and an equivariant Borel embedding into (Lip1(R^k))^N, and (ii) an R^k-generalization of the Lipschitz Bebutov–Kakutani embedding to move from the product model into Lip1(R^k) itself, completing the isomorphism and hence the model property . The candidate solution outlines the same two-step strategy (Arzelà–Ascoli for compactness; a Borel, equivariant, a.e. injective coding; then an embedding step), and its conclusions match the paper’s main theorem and definitions of topological models . The only substantive discrepancy is bibliographic: the multidimensional Lipschitz embedding used for k ≥ 2 is proved in this paper (Theorem 1.4), not in GJT (2019), which covered only k = 1.