A Rigidity Hypothesis for the Structural Asymmetry of the Bounded Solutions of a Parametric Complex Linear Differential Equation

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 \textit{rigidity hypothesis} along the critical line, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\bar{s}$, are both bounded on $[1,+\infty)$, then $\Re(s)=\frac{1}{2}$. This rigidity hypothesis is satisfied when the non-homogeneous term is the fractional part function.