Euler products from abstract prime systems and rigidity

We introduce an abstract notion of prime system (a purely structural concept that captures factorisation without reference to ordinary integers) and construct an associated system Euler product. The scalar differential equation of the rigidity programme is lifted to a Banach‑space setting, and the role of the non‑homogeneous term $\eta$ is played by a piecewise‑linear profile derived from the counting function of the abstract integers. For a normal operator $W$ with positive real part, the functional $\mu_\eta(W)$ is expressed through an system zeta function, which admits an Euler product indexed by the abstract primes. Inverting this relation yields a direct link between the Euler product and $\mu_\eta(W)$, providing a tool for analysing the rigidity criterion $\mathcal{L}_\eta(W,\theta)\neq 0$. The framework is general enough to encompass classical Beurling primes, graph‑theoretic prime nodes, or any factorisation semi‑group.