Euler products from abstract prime systems and rigidity

Abstract

We introduce an abstract notion of prime system (a purely structural concept that captures factorization without reference to ordinary integers) and construct an associated system Euler product. The scalar differential equation of the rigidity program is lifted to a Hilbert‑space setting, and the role of the non‑homogeneous term $\eta$ is played by a piece-wise 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.