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Perturbative regimes of gravitational d=4 N=1 effective field theories are characterised by special fundamental axionic strings, dubbed EFT strings. I will discuss how, applying the completeness hypothesis to these strings and demanding their quantum consistency, one can derive bounds on four-derivative gravitational couplings and on the rank of the gauge sector in terms of the these gravitational couplings. I will then discuss some UV tests of these bottom-up predictions in string theory.