Abstract
<jats:p>We establish an explicit bridge between the scalar rigidity program for bounded solutions of linear differential equations and the theory of abstract prime systems. A scalar rigidity theorem is reformulated in terms of a limiting oscillatory profile $L_\eta(s,\theta)$, and we show how this profile can be systematically deformed using a holomorphic Wronskian, giving a family of rigidity conditions. We then demonstrate that for the particular choice $\eta(t)=\{t\}$ (the fractional part function) the corresponding functional $\mu_\eta$ encodes the Riemann zeta function, so that the rigidity condition directly controls the simultaneous vanishing of $\zeta(s)$ and $\zeta(1-\overline{s})$. Finally, we embed the whole framework into the language of abstract prime systems and system Euler products. This connection provides a structural interpretation of the rigidity functional and suggests new directions for explicit verification of the rigidity inequality. An exploratory numerical verification using a publicly available notebook illustrates the practical feasibility of the approach.</jats:p>