2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/188483The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (gamma-)trace constraints on gauge fields and parameters. When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent ``minimal'' local formulation is also possible. In this paper we complete the discussion of the ``minimal'' formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds. We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory. To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in exactly the same fashion in current exchanges.37 pages, Latex. Typos corrected, note and references added. Final version to appear in Nucl. Phys. BHigh Energy Physics - Theorytext