  We give a complete classification of free and non-free multiplicities on the
$A_3$ braid arrangement. Namely, we show that all free multiplicities on $A_3$
fall into two families that have been identified by Abe-Terao-Wakefield (2007)
and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to
freeness derived via a connection to multivariate spline theory.
