Surfaces with Klein bottle topology occur in fusion reactor fields

The paper correctly invokes the Hopf–Poincaré index theorem to restrict compact magnetic surfaces for nowhere-vanishing fields to χ=0 and explicitly identifies torus or Klein bottle as the only options, then rules out the “usual” (self‑penetrating) Klein bottle immersion via a winding-number/degree argument after cutting along the self-intersection curve; both points match the model’s steps (1) and (2) and are sound as stated in the paper’s Section II and supporting discussion . For the “lemniscate” immersion, the paper explains that a reflection-hyperbolic fixed point (Tr<−2, det=1) can be viewed, via Iwasawa decomposition, as a squeeze composed with an integer‑and‑a‑half rotation; under a boundedness assumption for the time-dependent flow this leads to a lemniscate invariant set whose sweep produces an immersed Klein bottle, and it connects this to period-doubling bifurcations, with examples shown in Section III . The model’s step (3) mirrors this mechanism and adds standard dynamical-systems vocabulary (stable/unstable manifolds, area-preserving Poincaré map), but it implicitly assumes a pair of smooth homoclinic connections closing into a figure‑8; that existence is not guaranteed by reflection hyperbolicity plus boundedness alone. The paper’s own exposition of this part is heuristic rather than a complete proof. Hence, while parts (i)–(ii) are correct and aligned, both paper and model leave gaps in the general existence/structure claim for the lemniscate and its sweep to a Klein bottle; both are therefore incomplete on (iii).