Operational streams from packs, fleets, and multi-echelon networks arrive as delayed, incomplete, and scale-bearing records rather than as a single smooth series. The paper treats that arrival law as a mathematical fact: time is an oscillatory coordinate, unsigned magnitude is a separate positive coordinate, and sparse marks live in a dual that does not pre-classify events. A mixed product kernel is formed from a two-sided Fourier factor in time and an imaginary-order Macdonald factor on a strictly positive scale. The Fourier coordinate carries oscillation and lag; the KL coordinate carries energy, inventory, payload, and other unsigned magnitudes. Generalized observations are admitted so that switching pulses, theft signatures, demand shocks, and missing telemetry remain distinct until a later classification map is chosen. Quantum annealing is not treated as a front-end estimator. It is placed after spectral consistency, energy feasibility, and provenance checks so that prediction never substitutes for authorization. The same operator language is then used for battery envelopes, mobility cycles, residual twins, and constrained binary dispatch.

The kernel is not offered as an ornamental spectrum. It is the device that keeps lag, unsigned magnitude, and sparse events in distinct slots until a later map is justified. Inversion, when it exists in a chosen weighted space, is treated as a reconstruction of admissible fields rather than as a license to command hardware. The same chart therefore supports pack telemetry, fleet cycles, warehouse flow, and twin residuals without flattening them into one classical oscillator.

Keywords: FKF transform, FKL transform, testing-function space, Macdonald function, distributional transform, sequential convergence, time-shift operator, spectral resonance, digital twin, electric-vehicle battery, smart mobility, supply-chain resilience, blockchain provenance, quantum annealing, QUBO