The modern S-matrix program offers a mechanical approach to bootstrapping quantum field theories without the aid of an action. After reviewing progress in gravity and gauge theory, I show how EFTs are similarly constructible by carving out a theory space of consistent EFTs from first principles. I then describe how gravity serves as the “mother of all theories” whose amplitudes secretly encode, among others, all gluon and pion amplitudes, revealing hidden dualities relating color, flavor, and kinematics. I also demonstrate how these formal structures, together with classical tools from non-relativistic EFT, can be harnessed to derive new results directly relevant to gravitational wave physics. In particular I present the first ever calculation of the conservative Hamiltonian for a black hole binary system at third post-Minkowskian order.