Excess charge in the Engel-Dreizler model of atoms and molecules

We prove an explicit excess-charge bound for the exchange-free Engel--Dreizler functional of atoms and molecules. If αλ≤1/1233 and α³Zₖ²≤1/137 for each nucleus, every finite-mass electronic minimizer whose total energy, including nuclear repulsion, is nonpositive satisfies N-∑ₖZₖ<(5913/1000)λ3/2K. For the gradient coefficient $λ=1/9$, this gives N-∑ₖZₖ<0.219K. The energy hypothesis holds automatically for atoms and for local minima of the Born--Oppenheimer energy at which the electronic infimum is attained. The proof uses no symmetry of the density, restriction on the total nuclear charge, or minimum nuclear separation. We combine molecular Sommerfeld estimates with a convex nonlinear comparison in the variable $F(p)$. The failure of the latter comparison at very high density is represented by a nonnegative Newton potential. A quartic coercivity estimate, obtained from nuclear repulsion and a signed localized Hardy inequality, controls the total mass of this source. The chemical potential is retained throughout. Exact covariance yields the invariant parameter domain, and all finite numerical inequalities are reduced to rational arithmetic.