  We revisit the smallest non-physical singularity of the hard-sphere model in
one dimension, also known as Tonks gas. We give an explicit expression of the
free energy and reduced correlations at negative real fugacity and elaborate
the nature of the singularity: the free energy is right-continuous, but its
derivative diverges. We derive these results in several novel ways: First, by
scaling up the discrete solution. Second, by an inductive argument on the
partition function \`a la Dobrushin. Third, by a perfect cluster expansion
counting the Penrose trees in the Mayer expansion perfectly. Fourth, by an
explicit construction of Shearer's point process, the unique R-dependent point
process with an R-hard-core. The last connection yields explicit and optimal
lower bounds on the avoidance function of R-dependent point processes on the
real line.
