Large gaps and BTZ entropy in modular spectra with positive integer degeneracies
Chi-Ming Chang, Reiko Liu, Wen-Jie Ma
hep-th
Sep 30, 2026 · v1
TL;DR
The two primary-gap families and the all-order smoothed BTZ entropy expansion are stated to have been formalized in Lean 4.
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 $κ>0$, 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.
Problem
The Maloney–Witten–Keller modular completion of the Virasoro vacuum is modular invariant. Its nonvacuum spectrum, however, is continuous and has negative densities, so it cannot be the partition function of a compact unitary CFT. The goal is a modular-invariant torus partition function with a unique vacuum, discrete levels, positive integer degeneracies and large primary gaps, whose density of states matches BTZ entropy.
Approach
The MWK completion is repaired recursively. Exact local modular repairs, built from Poincaré-type seeds with a threshold anchor, prescribe the spectrum in a finite energy band. Finite moment matching replaces successive continuous spectral bands with integer-multiplicity atoms while preserving earlier levels. Uniform estimates control the evolving continuum and give convergence of the modular functions to a discrete integer spectrum.
Results
At every sufficiently large real c, the construction gives primary gaps Δ₁=(1+κ)a for small fixed κ>0 and Δ₁=a+δ for any fixed δ≥0, with all nonvacuum primaries satisfying h, h̄ ≥ (c−1)/24. In the fixed-δ family, the smoothed density of states matches a single perturbative BTZ saddle to every finite order in 1/c at every fixed positive E/c. The gap theorems and the entropy expansion are reported as formalized in Lean 4.
