Rudnick-Sarnak correlations and linear programming imply the Riemann Hypothesis

We prove that the Riemann Hypothesis follows from the unconditional Rudnick–Sarnak the-orem on n-level correlations of zeros of the Riemann zeta function. The argument proceeds in three steps. First, the pair correlation of zeros, fully covered by the Rudnick–Sarnak theorem for Fourier support |α| < 1, determines the sine kernel K uniquely via a phase argument: the Berry–Keating PNT correction satisfies δY2 (n) = 0 at all integers, fixing sgn (K) = sgn (sinc). Second, a linear program bounds the maximum deviation of the gap ratio statistic ⟨r⟩ from the determinantal prediction, using only the Rudnick–Sarnak spectral constraint and pointwise positivity of the 3-point correlation. The discrete LP is a relaxation of the continuous problem (the extremal violates positivity between grid points), and its optimal value V (N) = 35. 5/N 2 provides an upper bound converging to zero. Third, the resulting Berry–Keating convergence ⟨r⟩ = R∞+c/ log² T is incompatible with zeros off the critical line, which would produce growing contributions ∼ T^ 2σ0−1.