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 $w$ in the strip $\Re(w) \in(0,1)$. Under a mild integrability condition on the nonhomogeneous term, we establish a rigidity phenomenon: for every parameters $w, w'$ with $ \Re(w) \neq\Re(w)$ and $\Im(w) =\Im(w')$, the two solutions corresponding respectively to $w$ and $w'$, both taken with initial condition $1$, cannot be simultaneously bounded on $[1,+\infty)$. In particular when $w'=1-w$ and $\Re(w)\in (0,1)/\{\frac{1}{2}\}$. This result provides an answer to the Dynamical Conjecture formulated in \cite{Oukil}. Although this manuscript fits entirely within the theory of dynamical systems, it responds positively in particular to the Riemann Hypothesis and its generalizations.