Large gaps and BTZ entropy in modular spectra with positive integer degeneracies

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney–Witten–Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For $c_L=c_R=c$ and $a=(c-1)\mathrm{/}12$, the construction realizes primary dimension gaps ${\mathrm{\Delta }}_1=(1+\kappa )a$ for sufficiently small fixed $\kappa >0$, and ${\mathrm{\Delta }}_1=a+\delta$ for any fixed $\delta \ge 0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar{h}\ge (c-1)\mathrm{/}24$. In the fixed-$\delta$ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in $1\mathrm{/}c$. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein–Hawking entropy and its corrections at every fixed positive $E\mathrm{/}c$, where $E=\mathrm{\Delta }-c\mathrm{/}12$. This includes $0<E<c\mathrm{/}12$, where thermal AdS dominates the canonical ensemble.