Sharp boundaries for the swampland

We reconsider the problem of bounding higher derivative couplings in consistent weakly coupled gravitational theories, starting from general assumptions about analyticity and Regge growth S-matrix. Higher are expected to be order one units UV cutoff. Our approach justifies this expectation allows prove precise bounds on coefficients. main tool dispersive sum rules for overcome difficulties presented by graviton pole measuring at small impact parameter, rather than forward limit. illustrate method theories containing a massless scalar gravity, with maximal supersymmetry.