A mean–square stability functional and a foundational reformulation of the Riemann Hypothesis

We propose a structural reformulation of the Riemann Hypothesis based on the global analysis of the residual oscillations in the distribution of prime numbers. We introduce a mean–square stability functional associated with the Chebyshev function normalized in logarithmic variable and interpret the Riemann Hypothesis as a condition of global stability of the resulting arithmetic signal. We show that the invariant introduced is intrinsic, in the sense that it is independent of the choice of coarse–graining within a natural structural class of scale operators, and we prove that its finiteness implies the Riemann Hypothesis. Conversely, assuming the Riemann Hypothesis to hold, we reduce the existence of the global invariant to a single mean–square estimate of harmonic–analytic nature. This point of view clarifies which aspects of the problem are already rigidly controlled and isolates precisely the remaining analytic bottleneck. No proof of the Riemann Hypothesis is claimed; the present work is structural and preparatory in nature.