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 \(Λ : B → C\) that rigorously extracts \(ζ(s) = Λ(f_s^*)\) via Möbius inversion and the boundary condition of the attractor \(φ\). 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 \((I - F_s)\). This article introduces a fractal-holographic operator \(F_s\) that reformulates the Riemann zeta function \(ζ(s)\) 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.