A Fractal-Holographic Representation of the Riemann Zeta Function via a Self-Similar Operator (Version 4.4 – Final)

Update in Version 4.4: Major refinement of the analytic continuation. Introduced an explicit linear functional \(\Lambda : \mathcal{B} \to \mathbb{C}\) that rigorously extracts \(\zeta(s) = \Lambda(f_s^*)\) via Möbius inversion and the boundary condition of the attractor \(\phi\). Full justification of the pole at \(s=1\) arising solely from the divergence of the operator norm. Honest formulation of the central conjecture linking the non-trivial zeros to the zeros of the Fredholm determinant \(\det(I - \mathcal{F}_s)\). This article introduces a fractal-holographic operator \(\mathcal{F}_s\) that reformulates the Riemann zeta function \(\zeta(s)\) as the fixed point of a compact operator acting on a Banach space of holomorphic functions. [...] 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.