  Most stars are born in the gaseous proto-cluster environment. The knowledge
of this intermediate stage gives more accurate constraints on star formation
characteristics. We demonstrate that a virialized globally supported structure,
in which star formation happens, is formed out of a collapsing molecular cloud,
and derive a mapping from the parent cloud parameters to the proto-cluster to
predict its properties, with a view to confront analytical calculations with
observations and simulations. The virial theorem is decomposed into two
dimensions to account for the rotation and the flattened geometry. Equilibrium
is found by balancing rotation, turbulence and self-gravity, while turbulence
is maintained by accretion driving and dissipates in one crossing time. The
angular momentum and the accretion rate of the proto-cluster are estimated from
the parent cloud properties. The two-dimensional virial model predicts the size
and velocity dispersion given the mass of the proto-cluster and that of the
parent cloud. The gaseous proto-clusters lie on a sequence of equilibrium with
the trend $R \sim M^{0.5}$, with limited variations depending on the
evolutionary stage, the parent cloud, and the parameters not well known like
turbulence driving efficiency by accretion and the turbulence anisotropy. The
model reproduces successfully observations and simulation results. The
properties of proto-clusters follow universal relations and they can be derived
from that of the parent cloud. Using simple estimates to infer the peak
position of the core mass function (CMF) we find a weak dependence on the
cluster mass suggesting that the physical conditions inside proto-clusters may
contribute to set a CMF, and by extension an IMF, that looks independent of the
environment.
