Fixed-point topology meets fractal memory: a Kutumba-stabilized framework for nonlocal fractal-fractional dynamics

This paper presents an innovative synthesis of generalised fixed-point theory, advanced topological degree methodologies, and high-order computational frameworks for memory-driven dynamical systems. We expand contraction principles in complete metric and ANR spaces, enhancing existence results through the Leray-Schauder degree in the context of non-compact perturbations. We develop a high-order numerical framework for fractal-fractional differential equations employing the Taylor Operational Matrix Method (TOMM) and an adaptive Adams-Bashforth-Moulton (ABM) scheme. This framework leverages global basis approximations to achieve high precision and has strong stability guarantees thanks to the Ulam-Hyers-Rassias criteria and Lyapunov-Razumikhin functionals. This framework has strong stability guarantees thanks to the Ulam-Hyers-Rassias criteria and Lyapunov-Razumikhin functionals. We incorporate sociomathematical structures, such as Kutumba-inspired familial support, into epidemic models, illustrating how topological and fractional tools collectively encapsulate memory, heterogeneity, and resilience. Applications encompass fractional epidemiology, biomedical hysteresis, and cyber-virus dynamics, demonstrating interdisciplinary effectiveness. The work connects abstract analysis with real-world complexity, giving a single method for nonlocal, socially embedded systems.