  An outstanding problem in statistical mechanics is the determination of
whether prescribed functional forms of the pair correlation function $g_2(r)$
[or equivalently, structure factor $S(k)$] at some number density $\rho$ can be
achieved by $d$-dimensional many-body systems. The Zhang-Torquato conjecture
states that any realizable set of pair statistics, whether from a
nonequilibrium or equilibrium system, can be achieved by equilibrium systems
involving up to two-body interactions.
  In light of this conjecture, we study the realizability problem of the
nonequilibrium iso-$g_2$ process, i.e., the determination of density-dependent
effective potentials that yield equilibrium states in which $g_2$ remains
invariant for a positive range of densities.
  Using a precise inverse methodology that determines effective potentials that
match hypothesized functional forms of $g_2(r)$ for all $r$ and $S(k)$ for all
$k$, we show that the unit-step function $g_2$, which is the zero-density limit
of the hard-sphere potential, is remarkably numerically realizable up to the
packing fraction $\phi=0.49$ for $d=1$. For $d=2$ and 3, it is realizable up to
the maximum ``terminal'' packing fraction $\phi_c=1/2^d$, at which the systems
are hyperuniform, implying that the explicitly known necessary conditions for
realizability are sufficient up through $\phi_c$.
  For $\phi$ near but below $\phi_c$, the large-$r$ behaviors of the effective
potentials are given exactly by the functional forms $\exp[-\kappa(\phi) r]$
for $d=1$, $r^{-1/2}\exp[-\kappa(\phi) r]$ for $d=2$, and
$r^{-1}\exp[-\kappa(\phi) r]$ (Yukawa form) for $d=3$, where
$\kappa^{-1}(\phi)$ is a screening length, and for $\phi=\phi_c$, the
potentials at large $r$ are given by the pure Coulomb forms in the respective
dimensions, as predicted by Torquato and Stillinger [\textit{Phys. Rev. E}, 68,
041113 1-25 (2003)].
