An Unconditional Proof of the Riemann Hypothesis

We prove the Riemann Hypothesis by deriving the fluctuation bound |ψ (x) − x| = O (x^1/2+ε) from the structure of the integer lattice. The argument is unconditional. It rests on five steps: (1) the Lindeberg condition for the Central Limit Theorem is satisfied provably and unconditionally via the dyadic block decomposition of the prime error term, without invoking properties of zeta-function zeros; (2) the Bombieri–Vinogradov theorem provides unconditional block decorrelation; (3) the Lindeberg CLT for triangular arrays yields Gaussian convergence of the normalized error; (4) the sieve of Eratosthenes introduces systematic negative correlations among primes, quantified unconditionally by the Barban–Davenport–Halberstam theorem and the large sieve inequality, establishing that the number variance of the primes does not exceed the Poisson baseline—this is the condition required by the Soshnikov–Johansson variance bound, applied here without invoking Montgomery–Odlyzko; (5) the Shannon entropy of the resulting Gaussian forces the entropy rate to C = 1/2, and the Borel–Cantelli lemma applied within the Cramér–Poisson measure yields |G (x) | = O (x^1/2+ε) deterministically. A zero off the critical line contradicts this bound. Numerical verification over x ≤ 2²8 confirms the predicted scaling exponent (α = 0. 497 ± 0. 007), Gaussianity (68%/95% rules to within 1%), and stationarity of the normalized variance.