  Networks of particles connected by springs model many condensed-matter
systems, from colloids interacting with a short-range potential, to complex
fluids near jamming, to self-assembled lattices, to origami-inspired materials.
Under small thermal fluctuations the vibrational entropy of a ground state is
given by the harmonic approximation if it has no zero-frequency vibrational
modes, yet such singular modes are at the epicenter of many interesting
behaviors in the systems above. We consider a system of $N$ spherical
particles, and directly account for the singularities that arise in the sticky
limit where the pairwise interaction is strong and short ranged. Although the
contribution to the partition function from singular clusters diverges in the
limit, its asymptotic value can be calculated and depends on only two
parameters, characterizing the depth and range of the potential. The result
holds for systems that are second-order rigid, a geometric characterization
that describes all known ground-state (rigid) sticky clusters.
  To illustrate our theory we address the question of emergence: how does
crystalline order arise in large systems when it is strongly disfavored in
small ones? We calculate the partition functions of all known rigid clusters up
to $N\leq 19$, and show the cluster landscape is dominated by hyperstatic
clusters (those with more than $3N-6$ contacts) -- singular and isostatic
clusters are far less frequent, despite their extra vibrational and
configurational entropies. Since the most hyperstatic clusters are close to
fragments of a close-packed lattice, this underlies the emergence of order in
sticky-sphere systems, even those as small as $N=10$.
