Rigidity and Structural Asymmetry of Bounded Solutions

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter lying in the critical strip. We identify a {\it {Rotation number hypothesis}} on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\overline{s}$, are both bounded on $[1,+\infty)$, then $\Re(s) = \tfrac{1}{2}$.