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

Abstract: 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 $a=(c-1)/12$, the construction realizes primary dimension gaps $Δ_1=(1+κ)a$ for sufficiently small fixed $κ&gt;0$, and $Δ_1=a+δ$ for any fixed $δ\ge0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar h\ge(c-1)/24$. In the fixed-$δ$ 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 $1/c$. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein--Hawking entropy and its corrections at every fixed positive $E/c$, where $E=Δ-c/12$. This includes $0<E<c/12$, where thermal AdS dominates the canonical ensemble.