Electromagnetic tracking \added[]{is promising for} remote navigation of endoluminal \added[]{medical tools}. However, existing six-degree-of-freedom (6-DoF) pose recovery approaches often require pre-calibrated workspace field maps or iterative nonlinear optimization. This letter presents a Gradiometer-Based Electromagnetic Localization System (GELS), a closed-form tracking framework that uses a compact magnetometer array as an embedded quasi-gradiometer to estimate local magnetic fields and gradient tensors. These quantities are mapped by the Euler homogeneous relation to displacements between source and array, from which Procrustes registration over multiple sources recovers the array orientation and position using at least three noncollinear sources. The algorithm requires known source positions and array geometry, but no pre-calibrated workspace field maps, initial pose guesses, or calibrated source magnetic moments. The recovered \added[]{array} pose also enables a demonstration of sublevel dipole localization by serving as a mobile magnetic reference frame. Benchtop experiments across sensor array configurations and excitation modes validate the proposed closed-form 6-DoF localization pipeline, with a median solver runtime of \SI{172.00}{\micro\second}. A perturbation analysis identifies inter-sensor inconsistency and dipole model mismatch as dominant accuracy limits, informing future sensor array and source design to improve pose accuracy.

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