Rigidity and Structural Asymmetry of Bounded Solutions

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\mathrm{\infty })$, depending on a complex parameter. We identify a “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 $1-\bar{s}$ lying in the critical strip, are both bounded on $[1,+\mathrm{\infty })$, then $\mathrm{\Re }(s)=\frac{1}{2}$.