  We study the Gibbs statistics of high-density hard-core configurations on a
unit square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance
$D$. As a by-product, we solve the disk-packing problem on $\mathbb{Z}^2$ for
disks of diameter $D$. The key point is an analysis of solutions to norm
equations in $\mathbb{Z}[{\sqrt[6]{-1}}]$. We describe the ground states in
terms of M-triangles, i.e., non-obtuse $\mathbb{Z}^2$-triangles of a minimal
area with the side-lengths $\geq D$.
  There is a finite class (Class S) formed by values $D^2$ generating sliding,
a phenomenon leading to countable families of periodic ground states. We
identify all $D^2$ with sliding. Each of the remaining classes is proven to be
infinite; they are characterized by uniqueness or non-uniqueness of a minimal
triangle for a given $D^2$, up to $\mathbb{Z}^2$-congruencies. For values of
$D^2$ with uniqueness (Class A) we describe the periodic ground states as
admissible sub-lattices in $\mathbb{Z}^2$ of maximum density. By using the
Pirogov-Sinai theory, it allows us to identify the extreme Gibbs measures (pure
phases) for large values of fugacity and describe symmetries between them.
  Next, we analyze the values $D^2$ with non-uniqueness. For some $D^2$ all
M-triangles are ${\mathbb{R}}^2$-congruent but not $\mathbb{Z}^2$-congruent
(Class B0). For other values of $D^2$ there exist
non-${\mathbb{R}}^2$-congruent M-triangles, with different collections of
side-lengths (Class B1). Moreover, there are values $D^2$ for which both cases
occur (Class B2). The large-fugacity phase diagram for Classes B0, B1, B2 is
determined by dominant ground states.
  Classes A, B0-B2 are described in terms of cosets in
$\mathbb{Z}[{\sqrt[6]{-1}}]$ by the group of units.
