The paper explores the computation of gravitational bound waveforms from scattering amplitudes, focusing on the two-body dynamics using Schwinger-Dyson equations and Bethe-Salpeter recursion. The authors show that the tree-level scattering waveform can be naturally analyticly continued to the bound waveform, which is confirmed through an independent calculation of time-domain multipoles at large eccentricity in the Post-Newtonian (PN) expansion. They demonstrate the consistency of this scattering-to-bound map with the Damour-Deruelle prescription for orbital elements in the quasi-Keplerian parametrization and with the analytic continuation of radiated energy and angular momentum at 3Post-Minkowskian (3PM) order. The paper also discusses the classical recursion relations for amplitudes and the 2-massive-particle-irreducible (2MPI) kernels, providing a detailed derivation of the classical Bethe-Salpeter equation for the two-massive-particle reducible amplitude.The paper explores the computation of gravitational bound waveforms from scattering amplitudes, focusing on the two-body dynamics using Schwinger-Dyson equations and Bethe-Salpeter recursion. The authors show that the tree-level scattering waveform can be naturally analyticly continued to the bound waveform, which is confirmed through an independent calculation of time-domain multipoles at large eccentricity in the Post-Newtonian (PN) expansion. They demonstrate the consistency of this scattering-to-bound map with the Damour-Deruelle prescription for orbital elements in the quasi-Keplerian parametrization and with the analytic continuation of radiated energy and angular momentum at 3Post-Minkowskian (3PM) order. The paper also discusses the classical recursion relations for amplitudes and the 2-massive-particle-irreducible (2MPI) kernels, providing a detailed derivation of the classical Bethe-Salpeter equation for the two-massive-particle reducible amplitude.