  We compute the 5PM order contributions to the scattering angle and impulse of
classical black hole scattering in the conservative sector at first self-force
order (1SF) using the worldline quantum field theory formalism. This
challenging four-loop computation required the use of advanced
integration-by-parts and differential equation technology implemented on
high-performance computing systems. Use of partial fraction identities allowed
us to render the complete integrand in a fully planar form. The resulting
function space is simpler than expected: in the scattering angle we see only
multiple polylogarithms up to weight three, and a total absence of the elliptic
integrals that appeared at 4PM order. All checks on our result, both internal -
cancellation of dimensional regularization poles, preservation of the on-shell
condition - and external - matching the slow-velocity limit with the
post-Newtonian (PN) literature up to 5PN order and matching the tail terms to
the 4PM loss of energy - are passed.
