Riemann hypothesis: Rigidity and Structural Asymmetry of Bounded Solutions

In this manuscript, we introduce a family of parameterized 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, assumed to belong to the closed unit ball of $L^{\infty}([1,\infty), \mathbb{C})$, which induces a structural asymmetry: if two solutions corresponding respectively to the parameters $s$ and $1-\overline{s}$ in the critical strip, and sharing the same initial condition, equal to $1$, are both bounded on $[1,+\infty)$, then necessarily $\Re(s)=\tfrac{1}{2}$.