Fixed-particle-number optimizers for the Lieb--Oxford inequality

Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $Λ_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb--Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration--compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $Λ_N>Λ_k$ for $k<N$ exclude limits with a fixed lower particle number. Once attainment at particle number $N$ is known, the compact-support theorem of Di Marino and Lelotte arXiv:2607.11440, valid for all $0<\mathsf{s}<\mathsf{d}$, permits a non-product one-particle extension and yields $Λ_{N+1}(\mathsf{s},\mathsf{d})>Λ_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.