  We compute the purely local-in-time (scale-free and logarithm-free) part of
the conservative dynamics of gravitationally interacting two-body systems at
the fourth post-Minkowskian order, and at the thirtiest order in velocity. The
gauge-invariant content of this fourth post-Minkowskian local dynamics is given
in two ways: (i) its contribution to the on-shell action (for both
hyperboliclike and ellipticlike motions); and (ii) its contribution to the
Effective One Body Hamiltonian (in energy gauge). Our computation capitalizes
on the Tutti Frutti approach [Phys. Rev. Lett. \textbf{123}, no.23, 231104
(2019)], and on recent post-Minkowskian advances [Phys. Rev. Lett.
\textbf{128}, no.16, 161103 (2022)], [Phys. Rev. Lett. \textbf{128}, no.16,
161104 (2022)], and [Phys. Rev. Lett. \textbf{132}, no.22, 221401 (2024)].
