From scalar rigidity to rigid spaces 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, its extension to a "rigid spaces", and the construction of the "abstract prime systems". The scalar theorem states that under a Rotation Number Hypothesis, the boundedness of two symmetric solutions forces a structural asymmetry. A distinctive feature of the framework is that the complex variable $s$ of $\zeta(s)$ is regarded as the scalar realization of a "rigid operator" $W\in\mathcal X$, with $W=s$ when $\mathcal X=\mathbb C$.