From Scalar Rigidity to System Solutions and Abstract Prime Systems

We develop 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 Banach spaces, and the construction of a functional calculus based on abstract prime systems. The scalar theorem states that under a Rotation Number Hypothesis and a specific rigidity inequality, the boundedness of two symmetric solutions forces a structural asymmetry, preventing simultaneous vanishing of a functional $\mu_\eta$ at conjugate parameters. This result is refined using a holomorphic Wronskian that yields a family of oscillatory rigidity profiles, and the whole programme is lifted to a system setting where the complex parameter is replaced by a bounded normal operator. Finally, we introduce abstract prime systems and show that for a piecewise‑linear profile derived from an arbitrary factorisation semi‑group, the functional $\mu_\eta$ factorises into a system Euler product. This provides a structural interpretation of the rigidity inequality and a direct link between the distribution of abstract primes and the asymmetry of bounded solutions. The article presents the full scalar proofs, the holomorphic Wronskian, the system generalisation, and the bridge to abstract prime systems in a self‑contained manner, concluding with a philosophical perspective on the possible role of the rigidity law as a consistency index for formal theories.