From Scalar Rigidity to System Solutions and Abstract Prime Systems

We propose a unified framework for the Prime Rigidity Theory (PR), integrating three pillars: a scalar rigidity theorem for bounded solutions of non-homogeneous complex linear differential equations, its extension to system solutions in Hilbert spaces, and the construction of a functional calculus based on abstract prime systems. The scalar theorem states that under a Rotation Number Hypothesis, the boundedness of two symmetric solutions forces a structural asymmetry, preventing simultaneous vanishing of a functional $\mu_\eta$ at conjugate parameters. We introduce abstract prime systems and show that for a piece-wise linear profile derived from an arbitrary factorization semi-group, the functional $\mu_\eta$ factorizes into a system Euler product.