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 $\Delta_1=(1+\kappa)a$ for sufficiently small fixed $\kappa>0$, and $\Delta_1=a+\delta$ for any fixed $\delta\ge0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar h\ge(c-1)/24$. 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 $1/c$. 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 $E=\Delta-c/12$. This includes $0