Rigidity and Structural Asymmetry of Bounded Solutions

We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a {\it {Rotation number hypothesis}} on the non-homogeneous term, which a structural asymmetry between the solutions corresponding to the parameters \(s\) and \(1-\overline{s}\). More precisely, under a specific rigidity inequality, if both solutions with initial value \(1\) are bounded on \([1,+\infty)\), then necessarily \(\Re(s)=\tfrac12\).