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 \ (: B C\) that rigorously extracts \ ( (s) = (fₛ^*) \) via Möbius inversion and the boundary condition of the attractor \ (\). 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 - Fₛ) \). This article introduces a fractal-holographic operator \ (Fₛ\) that reformulates the Riemann zeta function \ ( (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.