  The process referred to as "semi-convection" in astrophysics and
"double-diffusive convection in the diffusive regime" in Earth and planetary
sciences, occurs in stellar and planetary interiors in regions which are stable
according to the Ledoux criterion but unstable according to the Schwarzschild
criterion. In this series of papers, we analyze the results of an extensive
suite of 3D numerical simulations of the process, and ultimately propose a new
1D prescription for heat and compositional transport in this regime which can
be used in stellar or planetary structure and evolution models.
  In a preliminary study of the phenomenon, Rosenblum et al. (2011) showed
that, after saturation of the primary instability, a system can evolve in one
of two possible ways: the induced turbulence either remains homogeneous, with
very weak transport properties, or transitions into a thermo-compositional
staircase where the transport rate is much larger (albeit still smaller than in
standard convection).
  In this paper, we show that this dichotomous behavior is a robust property of
semi-convection across a wide region of parameter space. We propose a simple
semi-analytical criterion to determine whether layer formation is expected or
not, and at what rate it proceeds, as a function of the background
stratification and of the diffusion parameters (viscosity, thermal diffusivity
and compositional diffusivity) only. The theoretical criterion matches the
outcome of our numerical simulations very adequately in the numerically
accessible "planetary" parameter regime, and can easily be extrapolated to the
stellar parameter regime.
  Subsequent papers will address more specifically the question of quantifying
transport in the layered case and in the non-layered case.
