A Unified Zeta–MRI Operator Satisfying All Analytic Identities Required for the Riemann Hypothesis

This paper presents a fully self-contained mathematical framework reconstructing a unique geometry–operator–spectrum triple from a sufficiently large finite set of nontrivial zeros of the Riemann zeta function. The construction unifies three major components:(1) a nonlinear Zeta–MRI recursion acting on functions over a truncated integer manifold,(2) a multi-plane reconstruction system extracting geometric, spectral, symmetry, and stability information from the stabilized recursion field, and(3) a monotone rigidity lemma ensuring uniqueness of the stabilized configuration. The resulting operator possesses a spectrum coinciding with the nontrivial zeros, satisfies a global spectral determinant identity, and yields a trace formula reproducing the prime-power terms of the Riemann explicit formula. A finite-prefix closure principle establishes that this finite data already determines the global operator–geometry–spectrum triple and excludes all alternative extensions. In addition, a universality and classification theorem shows that any ξ-type spectral realization must coincide with the unique Zeta–MRI operator on its arithmetic slice. All analytic identities required for a spectral formulation of the Riemann Hypothesis arise internally. The reconstruction is rigid: perturbations of zeros off the critical line destabilize the multi-plane system, enforcing the critical-line configuration as the unique consistent solution. Consequently, the framework yields a complete operator-theoretic realization of the Riemann Hypothesis.