The optimization of high-dimensional engineering systems presents fundamental mathematical challenges that classical integer-order optimization methods are frequently unable to resolve with adequate efficiency, robustness, or convergence reliability. Fractional calculus, extending differentiation and integration to non-integer orders, provides a mathematically richer framework for characterizing the non-local, memory-dependent, and anomalous diffusion phenomena that govern the behavior of complex engineering systems across domains including power systems, structural mechanics, control engineering, fluid dynamics, and signal processing. This paper develops a rigorous mathematical framework for hybrid fractional-order optimization in high-dimensional engineering systems, establishing formal convergence guarantees through Lyapunov stability analysis, fractional calculus functional analysis, and stochastic process theory. The hybrid methodology integrates fractional gradient descent with evolutionary computation, particle swarm dynamics, and adaptive memory operators to construct optimization algorithms whose convergence properties in high-dimensional, non-convex, and discontinuous search spaces are formally verifiable rather than empirically presumed. The fractional-order gradient operators employed in the framework exploit the non-local integration kernel of Riemann-Liouville and Caputo fractional derivatives to escape local optima traps that defeat integer-order gradient methods, while the evolutionary component provides global search capability with formal population diversity preservation guarantees. The adaptive memory mechanism exploits the hereditary property of fractional operators to modulate search behavior based on the historical trajectory of the optimization process, enabling self-calibrating adaptation to the local geometry of the objective function landscape without requiring explicit landscape characterization. Theoretical analysis establishes sufficient conditions for almost sure convergence of the hybrid algorithm in probability spaces defined over high-dimensional continuous search domains. Numerical validation across benchmark optimization problems and engineering applications demonstrates convergence reliability superior to competing methods with computational overhead less than thirty percent above integer-order equivalents. The framework provides a mathematically rigorous foundation for applying fractional-order optimization methodology to real engineering design and control problems where convergence guarantees are operationally essential.