Around Smale’s 14th problem

The uploaded paper (arXiv:2202.01687) explicitly proves two key claims the model states were likely open: (1) existence of a parameter value for the classical Lorenz ODE at which there is a trefoil-shaped heteroclinic connection, and (2) a deformation (“isotopy”) from the Lorenz flow at that parameter to an extended geometric Lorenz model. Theorem 1.1 states both claims, and Section 2 provides a concrete proof of existence of a heteroclinic trefoil (Theorem 2.4) using a global cross section, a linking/winding argument between two homoclinic parameters, and a careful analysis of the stable manifolds of the wing centers (Proposition 2.3) . The linking-count input is supplied by Chen’s theorem on homoclinic loops with prescribed rotations (Theorem 2.5 as cited in the paper) and is built into the existence proof . Section 3 then derives a two-symbol return-map picture at the T-point and proves that every periodic orbit of the extended geometric model is realized by the Lorenz flow, supporting the stated isotopy at the level of periodic orbits (Theorem 1.2) . While the paper’s use of the word “isotopic” would benefit from a definition and a brief justification of how the surface-dynamics tools ensure no spurious periodic orbits are created during the deformation, the core existence theorem for the trefoil T-point is rigorously argued. Consequently, the model’s claim that these points were not yet proven is refuted by this paper.