The problem of the long established thermal stratification discrepancy between numerical and experimental results is investigated in three companion articles. The Part I article establishes reference solutions by means of three-dimensional (3D) spectral direct numerical simulations of a buoyancy-driven flow (RaH = 1.5 × 109). Two configurations of differentially heated air-filled cavity are considered: an idealized cavity (perfectly adiabatic cavity, PAC) and an Intermediate Realistic Cavity (IRC) making use of experimentally measured temperature distributions ( Salat, 2004) on its top and bottom walls. The IRC flow structure as well as its associated rms fluctuations correspond to the experimentally observed flow dynamics. However both configurations keep resulting in a core thermal stratification value equal to 1.0 whereas experiments lead to a stratification of about 0.5. It is proved that this stratification paradox is neither related to three-dimensional effects nor to the experimental thermal distributions applied on the horizontal walls. Resolving this stratification discrepancy is the subject of the parts II and III articles ( Sergent et al., 2013 and Xin et al., 2012).