Gaia’s catalogue completeness S(G | M₁₀) as a function of G magnitude for a given completeness parameter M₁₀. Top left: empirically determined Gaia G completeness derived from the comparison with DECaPS photometry in thirteen DECaPS patches that are each colour-coded by their M₁₀. Top right: model completeness, from Eq. (1), for the corresponding M₁₀ values. Bottom left: empirically determined completeness derived from the comparison with DECaPS photometry in 105 DECaPS patches with M₁₀=19.4, 19.8, 20.2, 20.6, and 21 (within 0.01 mag). The dispersion at a given M₁₀ corresponds to the scatter seen in Fig. 1 (right panel) and the residuals in Fig. 5. Bottom right: model completeness for these five values of M₁₀.