  We provide evidence that the sunward-propagating half of the solar wind
electron halo distribution evolves without scattering in the inner heliosphere.
We assume the particles conserve their total energy and magnetic moment, and
perform a "Liouville mapping" on electron pitch angle distributions measured by
the Parker Solar Probe SPAN-E instrument. Namely, we show that the
distributions are consistent with Liouville's theorem if an appropriate
interplanetary potential is chosen. This potential, an outcome of our fitting
method, is compared against the radial profiles of proton bulk flow energy. We
find that the inferred potential is responsible for nearly 100% of the proton
acceleration in the solar wind at heliocentric distances 0.18-0.79 AU. These
observations combine to form a coherent physical picture: the same
interplanetary potential accounts for the acceleration of the solar wind
protons as well as the evolution of the electron halo. In this picture the halo
is formed from a sunward-propagating population that originates somewhere in
the outer heliosphere by a yet-unknown mechanism.
