  We define a potential-weighted connective constant that measures the
effective strength of a repulsive pair potential of a Gibbs point process
modulated by the geometry of the underlying space. We then show that this
definition leads to improved bounds for Gibbs uniqueness for all non-trivial
repulsive pair potentials on $\mathbb R^d$ and other metric measure spaces. We
do this by constructing a tree-branching collection of densities associated to
the point process that captures the interplay between the potential and the
geometry of the space. When the activity is small as a function of the
potential-weighted connective constant this object exhibits an infinite volume
uniqueness property. On the other hand, we show that our uniqueness bound can
be tight for certain spaces: the same infinite volume object exhibits
non-uniqueness for activities above our bound in the case when the underlying
space has the geometry of a tree.
