  Leveraging scattering information to describe binary systems in generic
orbits requires identifying local- and nonlocal-in-time tail effects. We report
here the derivation of the universal (non-spinning) local-in-time conservative
dynamics at fourth Post-Minkowskian order, i.e. ${\cal O}(G^4)$. This is
achieved by computing the nonlocal-in-time contribution to the deflection
angle, and removing it from the full conservative value in
[2112.11296,2210.05541]. Unlike the total result, the integration problem
involves two scales, velocity and mass ratio, and features multiple
polylogarithms, complete elliptic and iterated elliptic integrals, notably in
the mass ratio. We reconstruct the local radial action, center-of-mass momentum
and Hamiltonian, as well as the exact logarithmic-dependent part(s), all valid
for generic orbits. We incorporate the remaining nonlocal terms for
elliptic-like motion to sixth Post-Newtonian order. The combined Hamiltonian is
in perfect agreement in the overlap with the Post-Newtonian state of the art.
The results presented here provide the most accurate description of
gravitationally-bound binaries harnessing scattering data to date, readily
applicable to waveform modelling.
