Stable arcs connecting polar cascades on a torus

The paper proves that any two polar gradient‑like diffeomorphisms on T^2 (with all non‑wandering points fixed and of positive orientation type) can be connected by a stable arc that has only finitely many generically unfolded, non‑critical saddle‑node bifurcations, and it explains why in general one cannot connect arbitrary such maps by a bifurcation‑free arc because the closures of invariant manifolds may lie in different homotopy classes; the authors introduce an integer invariant J capturing this obstruction and show how to change it via controlled saddle‑nodes. The candidate solution claims a global arc with zero bifurcations by ambiently isotoping the ‘skeleton’ in an annulus and then invoking structural stability; this ignores the homotopy‑class obstruction encoded by J and misuses isotopy in an annulus (relative homotopy/winding number cannot be altered without bifurcation). Hence the model’s proof fails in the generic case, while the paper’s argument is coherent and complete.