  Initial characterizations of the fermion sign problem focused on its
evolution with spatial lattice size $L$ and inverse temperature $\beta$,
emphasizing the implications of the exponential nature of the decay of the
average sign $\langle {\cal S} \rangle$ for the complexity of its solution and
associated limitations of quantum Monte Carlo studies of strongly correlated
materials. Early interest was also on the evolution of $\langle {\cal S}
\rangle$ with density $\rho$, either because commensurate filling is often
associated with special symmetries for which the sign problem is absent or
because particular fillings are often primary targets, e.g.~those densities
which maximize superconducting transition temperature (the top of the `dome' of
cuprate systems). Here we describe a new analysis of the sign problem, which
demonstrates that the {\it spin-resolved} sign $\langle {\cal S}_\sigma\rangle$
already possesses signatures of universal behavior traditionally associated
with order parameters, even in the absence of symmetry protection that makes
$\langle {\cal S} \rangle = 1$. When appropriately scaled, $\langle {\cal
S}_\sigma \rangle$ exhibits universal crossings and data collapse. Moreover, we
show these behaviors occur in the vicinity of quantum critical points of three
well-understood models, exhibiting either second-order or Kosterlitz-Thouless
phase transitions. Our results pave the way for using the average sign as a
minimal correlator that can potentially describe quantum criticality in a
variety of fermionic many-body problems.
