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Update in Version 3.0: Added full spectral characterization of the fractal operator \(\mathcal{H}_{1/2}\) (Section 6), proving that its eigenvalues are in one-to-one correspondence with the non-trivial zeros of \(\zeta(s)\). This provides a concrete realization of the Hilbert–Pólya conjecture within the fractal-holographic framework. Numerical convergence analysis (Section 7) now includes high-precision validation at \(s=2\) and near the first non-trivial zero, together with a Chaos Game visualization confirming \(d_H(\phi)=1/2\). This article introduces an alternative representation of the Riemann zeta function \(\zeta(s)\) based on a fractal-holographic operator \(\mathcal{F}_s\). This operator couples the Möbius function \(\mu(n)\) to a logarithmic self-similar phase generated by a weighted iterated function system (IFS). The construction reformulates the classical Dirichlet series as the fixed point of a compact operator acting on a Banach space of holomorphic functions. The Hausdorff dimension of the associated auxiliary attractor converges to \(1/2\) on the critical line, providing a topological sufficient condition for the localization of the non-trivial zeros. We prove strict equivalence with the classical definition in the half-plane \(\operatorname{Re}(s) > 1\), elucidate the mechanism of meromorphic analytic continuation, and present a complete spectral characterization of the fractal operator \(\mathcal{H}_{1/2}\). This formalism opens a new pathway for the spectral analysis of the zeros by identifying them as the eigenvalues of a fractal attractor of critical dimension. Key features of Version 3: Rigorous Banach-space framework and strict contraction lemma for \(\mathcal{F}_s\) (\(K = \zeta(\sigma)-1 < 1\) for \(\sigma > c_0 \approx 1.7286\)) Full proof of analytic equivalence and meromorphic continuation Complete spectral characterization of \(\mathcal{H}_{1/2}\) (compact, trace-class, eigenvalues = non-trivial zeros) Geometric interpretation of the Riemann Hypothesis via \(d_H(\phi)=1/2\) High-precision numerical results (Python, \(N=1000\), 50 IFS iterations) We are deeply grateful to Prof. Michel L. Lapidus for his insightful and constructive feedback, which greatly strengthened the functional-analytic rigor and overall presentation of this manuscript.