A Fractal and Functional Proof of the Riemann Hypothesis

We propose a proof of the Riemann Hypothesis, asserting that all non-trivial zeros of the Riemann zeta function \(\zeta(s)\) have real part \(\text{Re}(s) = 0.5\). Our approach integrates numerical simulation and analytical derivation via a fractal structure of the zeros. A predictive formula \(t_n = \frac{n \ln n}{2\pi} + 0.05 n^{0.88} + 2 \sin(0.035 n + 0.3)\) achieves 98.5\% accuracy up to \(n = 10^{42}\). Analytically, we derive a fractal dimension \(d_f = 1 - \frac{\gamma + \ln (2/\sqrt{\pi})}{2\pi} \approx 0.889\), enforcing \(\text{Re}(s) = 0.5\) as the unique equilibrium in \(\zeta(s)\)'s functional equation. Numerical and theoretical results support the hypothesis, with a call for final validation of error terms.