  We study the budget aggregation problem in which a set of strategic voters
must split a finite divisible resource (such as money or time) among a set of
competing projects. Our goal is twofold: We seek truthful mechanisms that
provide fairness guarantees to the projects. For the first objective, we focus
on the class of moving phantom mechanisms [Freeman et al., 2021], which are --
to this day -- essentially the only known truthful mechanisms in this setting.
For project fairness, we consider the mean division as a fair baseline, and
bound the maximum difference between the funding received by any project and
this baseline. We propose a novel and simple moving phantom mechanism that
provides optimal project fairness guarantees. As a corollary of our results, we
show that our new mechanism minimizes the $\ell_1$ distance to the mean (a
measure suggested by Caragiannis et al. [2022]) for three projects and gives
the first non-trivial bounds on this quantity for more than three projects.
