Fractal Riemann Hypothesis PART G, Intersection of Fractal Geometry and the Riemann Hypothesis

This paper proposes a new paradigm of the ”Fractal Riemann Hypothesis”: forfractal sets satisfying conditions of compact self-similarity, finite ramification, andspectral gap, the real parts of all non-trivial zeros of their zeta function are constantly equal to half of the Hausdorff dimension. The article constructs a universalasymptotic expansion for the fractal heat kernel, proves that the entire functionorder of the zeta function is 1 and gives the Riemann–Hadamard infinite productrepresentation; argues using the spectral geometric version of the Selberg trace formula that the zeros must accumulate on the critical line; and designs a quantumphase estimation + Monte Carlo hybrid algorithm to locate zeros with < 0.15%error on the Sierpiński carpet. The work establishes new bridges among the fractal prime number theorem, quantum chaotic energy level statistics, and AdS/CFTduality, providing a cross-disciplinary paradigm for geometric number theory andquantum gravity.