This study examines on the convergence analysis of a cohesive three-step iteration scheme for estimating fixed points of mean nonexpansive mappings in CAT(0) spaces, which are important in nonlinear analysis due to their rich geometric structure. The significance lies in generalizing and extending existing iterative processes within a cohesive framework that operates effectively in spaces lacking linearity. Using the geometric properties of CAT(0) spaces, we develop and analyze a cohesive iteration method and establish Λ – convergence results under minimal assumptions. The findings presented herein extend and refine several recent advancements while providing substantial contributions to the existing body of knowledge in the field. The developed framework not only improves existing results but also unifies them under a single structure.