Scalar rigidity, abstract prime systems, and the Riemann zeta function

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.