Note on the failure of the strong rotation property for smooth fields with infinite spectrum

The present note identifies the fundamental mechanism governing the extension of the contraction method. Although every periodic vector field defined by a finite trigonometric polynomial admits a bounded deviation from a linear drift, this property fails in general for smooth periodic vector fields. We show that the contraction argument underlying the original proof extends to any field satisfying a natural uniform summability condition, and that the counterexample violates this condition, thereby revealing the obstruction that prevents the method from extending beyond the finite-spectrum setting.