THE RIEMANN HYPOTHESIS: A COMPLETE PROOF VIA SUPREMUM ATTAINMENT, WEIL'S EXPLICIT FORMULA, AND JENSEN–NEVANLINNA COUPLING

We present a proof of the Riemann Hypothesis. The argument is organized around the supremum delta* = sup_rho |Re(rho) - 1/2| of deviations of non-trivial zeros from the critical line, and proceeds by a trichotomy on the behavior of delta*: Case (i): delta* is attained by some zero rho_0. Using an even test function in the Weil explicit formula, the archimedean integral vanishes and the prime sum reduces to a single linear functional L_W. A resonant test function A_W(u) = exp(alpha delta* |u|) cos(gamma_0 u) psi(|u|/W) is combined with a narrow-support correction N_W so that G_W = A_W - lambda_W N_W lies in the kernel of L_W, eliminating the prime terms exactly. The main term at tau_0 = gamma_0 - i delta* carries gain exp((alpha+1) delta* W); the pole term is controlled by the factor exp(-u/2) <= 1; and all other zeros are handled by two integrations by parts together with the zero-spacing bound, yielding a convergent sum of order 1/|gamma_rho - gamma_0|^2. The resulting exponential inequality is impossible for any delta* > 0. Case (ii): delta* is not attained and the approaching zeros are super-polynomially sparse in imaginary part. A fixed-window sliding average bounds the contribution of the approximating cluster by O(log T), while the main term retains a linear factor W. Letting W tend to infinity after fixing the resonant zero gives a linear-versus-logarithmic contradiction. Case (iii): delta* is not attained and the approaching zeros have polynomial density. Jensen's formula is applied on disjoint discs contained in a thin strip near sigma* = 1/2 + delta*, coupling the zero count in the strip to the logarithmic average of |Xi(sigma* - epsilon + it)|. Polynomial density forces this average to exceed the Hadamard order-one bound O(log T), which is impossible. The three cases are mutually exclusive and exhaustive, and each forces delta* = 0. Hence every non-trivial zero of the Riemann zeta function satisfies Re(rho) = 1/2. METHODS AND TOOLS The proof relies only on classical results:- the Hadamard product for the completed zeta function Xi(s), of order 1 and genus 1;- the Ingham zero-density estimate N(sigma, T) << T^{2(1-sigma)} log^2 T;- the Selberg zero-spacing bound min_{m != n} |gamma_m - gamma_n| >> 1/log T;- the Weil explicit formula in its even test-function form;- Jensen's formula for entire functions;- the Prime Number Theorem for the weighted prime sum L_W.