We reexamine the problem of inverting global transfer functions to constrain the internal structure of the Earth. We go beyond the conventional approach of inverting electromagnetic induction data by inverting directly for chemical composition and thermal state, using the model system CaO-FeO-MgO-Al2O3-SiO2, rather than subsurface electrical conductivity structure, which is only an indirect means of estimating the former parameters. Using minimisation of Gibbs free energy, we calculate the stable mineral phases, their modes and densities. The mineral modes are combined with recent laboratory measurements to estimate the bulk lunar electrical conductivity structure from which transfer functions are calculated. To further constrain the radial density profile in the inversion we added constraints imposed by mass and moment of inertia. Our results indicate that electromagnetic sounding and experimental mineral electrical conductivity data are consistent with a silicate earth, with a composition close to the pyrolite model and additionally seem to require a low temperature mantle geotherm.