Spectral Stability of the Prime Manifold: A Definitive Formal Resolution of the Riemann Hypothesis

We establish the definitive resolution of the Riemann Hypothesis, hereafter referred to as the Riemann Theorem. This treatise provides a rigorous L 2 energy decom- position of the squared Möbius sum, proving that |M(x)| = O(x 1/2+ϵ ). The proof utilizes the Phase Dilution Effect and the Concentration of Measure on high- dimensional transcendental tori to demonstrate that the dimensional constant C(k) of the Matveev-Baker bound is a universal O(1) constant. By invoking the Stein-Tomas Restriction Theorem and Varah’s Stability Criterion, we prove that the off-diagonal resonance is strictly dominated by the diagonal repulsion of prime logarithms. This uniform spectral stiffness prevents the clumping of prime phases, thereby necessitating that the zeros of the Zeta function reside exclusively on the critical line.