  We calculate model-independently the impact of the critical point on higher
order baryon susceptibilities $\chi_n$, showing how they depend on fluctuations
of the order parameter. Including all tree level diagrams, we find new
contributions to $\chi_4$ equally important to the kurtosis of the order
parameter fluctuations, and we characterize the kurtosis and other nonguassian
moments as functions on the phase diagram. Important features of this analysis
are then confirmed by a Gross-Neveu model study with good agreement with other
model studies as well as lattice and experimental data. This suggests the
universality of the characteristic peak in baryon susceptibilities as a signal
of the critical point. We discuss leveraging measurements of different $\chi_n$
to extrapolate the location of the critical point.
