Seminorm Duals for Constrained Resilience: An FKF–FKL Operational Calculus of Digital Twins, Green Reverse Flows, and Quantum Unit-Load Planning

Abstract

Smart logistics couples delayed replenishment, incomplete Internet-of-Things packets, fractional battery memory, and discrete unit-load-device decisions inside a single planning horizon. Classical Lebesgue and Sobolev spaces lose the time-shift property on that mixture, so continuity of delay operators cannot be certified by a single norm. This manuscript constructs a two-sided Fourier–kernel fractional (FKF) transform and a causal Fourier–kernel Laplace (FKL) transform on countably multinormed testing-function spaces of Roumieu and Beurling type. Sequential convergence makes delay operators continuous and places singular disruptions, stochastic lead-time laws, and residue-based resonances in the corresponding duals. The same topology is used as a certified interface among classical simulators, digital twins, hash-chain provenance, and hybrid quantum-classical solvers. Delayed multi-echelon inventory, bullwhip amplification, constrained resilience, fractional state-of-charge and state-of-health of plug-in and solar-assisted packs, QUBO encodings of unit-load-device configuration, live twin orchestration, graph delay systems, and closed green reverse flows become algebraic identities in the transform domain. One pair of spaces and one pair of transforms therefore certify stock, charge, configuration, and custody without a change of seminorm language [3], [5], [16], [27], [31]. Keywords— distributional FKF/FKL transforms; sequential convergence; testing spaces; shift/modulation and spectral properties; resonance analysis; multi-echelon lead time; supply-chain resilience; digital twins; green/sustainable logistics; blockchain traceability; quantum logistics, annealing and QUBO; ULD optimization; IoT/ITS; Industry 5.0; hybrid/plug-in EVs; battery sizing; solar mobility; swipe control; PLC anti-theft..