This paper develops a twin-orchestrated framework for fractional battery trajectories in which delayed state evolution, fractional memory, and spectral representations are treated within a dual-topology FKF setting. The construction is motivated by the battery sizing, ageing, mobility, solar-assistance, and intelligent-transport perspectives represented in the supplied literature. A countably multinormed test-function structure is used to keep delay shifts continuous while its dual admits impulsive, incomplete, and disruption-driven observations. Fractional state-of-charge and state-of-health trajectories are then coupled to a synchronized digital-twin layer so that the physical battery state, its delayed spectral representation, and the computational replica remain mutually consistent. The resulting architecture also provides a controlled interface to resilience analytics, artificial intelligence, blockchain traceability, and hybrid quantum optimization. The emphasis is theoretical: the FKF topology supplies admissible growth, sequential completeness, shift continuity, and recovery bounds, while the twin supplies an operational mechanism for replay, prediction, and uncertainty management. The framework therefore connects battery trajectory mathematics with intelligent mobility and resilient logistics without treating any individual computational technology as a replacement for the underlying physical or functional-analytic constraints.

Keywords—Fractional battery dynamics; FKF transform; dual topology; digital twin; state of charge; state of health; delay systems; spectral analysis; smart mobility; supply chain resilience; quantum logistics; sustainable mobility.