Structural Asymmetry and Rigidity of Bounded Solutions for a Parametrized Complex Linear Differential Equation

We introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter within the critical strip. We establish a rigidity phenomenon: Two solutions with the same initial condition equal to $1$, corresponding to two different parameters lying on a horizontal line in the critical strip, cannot both be bounded on $[1,+\infty)$. In particular, when $\Re(w)\neq \tfrac{1}{2}$, the two solutions corresponding respectively to the parameters $w$ and $1-\bar{w}$, both taken with initial condition $1$, cannot be simultaneously bounded.