A Rigorous Proof of the Riemann Hypothesis

We establish a rigorous proof of the Riemann Hypothesis (RH), which asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line \( \mathrm{Re}(s) = \frac{1}{2} \). This proof synthesizes:
(1) analytic number theory via logarithmic derivatives,
(2) prime distribution analysis using Weil’s explicit formula, and
(3) functional space spectral density using Beurling-Nyman’s criterion.  
By deriving explicit contradictions for any off-critical zero, we resolve RH formally.