Maximal Torus Topological Entanglement Entropy in WZW Theories at Bounded Ground-State Degeneracy

For an untwisted Wess–Zumino–Witten modular tensor category $\text{cC}(\mathfrak{g},k)$, let $r(\mathfrak{g},k)$ be the number of simple objects and $\text{cD}(\mathfrak{g},k)$ its total quantum dimension. The vacuum-flux state on a torus divided into two cylinders has positive universal entropy contribution ${\mathrm{\Gamma }}_{T^2}=2\log \text{cD}$. We maximize this quantity over all simple Lie algebras and positive integral levels subject to $r(\mathfrak{g},k)\le R$. If $q_R={\log }_{2}R$, then ${\mathrm{\Gamma }}_{\max }(R)\sim [7\zeta (3)\mathrm{/}(4{\pi }^2)]q_R^2$ as $R\to \mathrm{\infty }$. The sequence $Sp(2n{)}_n$ attains the asymptotic coefficient. At equal rank and level, the Fourier expansion of the type-$C$ root product loses its even modes, leaving $2{\pi }^{-2}{\sum }_{m\mathrm{o}\mathrm{d}\mathrm{d}}{m}^{-3}=7\zeta (3)\mathrm{/}(4{\pi }^2)$. An entropy–spectral inequality proves optimality among the classical families, while rank–level duality and fixed-rank estimates control unbalanced and exceptional sequences.