{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39930",
  "author_total": 3,
  "authors": [
    {
      "name": "Chi-Ming Chang"
    },
    {
      "name": "Reiko Liu"
    },
    {
      "name": "Wen-Jie Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5gwr4cfr2mhp8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39930",
  "fulltext_url": null,
  "id": "5gwr4cfr2mhp8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39930",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7215127215",
  "pdf_url": "https://arxiv.org/pdf/2609.39930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5gwr4cfr2mhp8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2609.39930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39930"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Large gaps and BTZ entropy in modular spectra with positive integer degeneracies",
  "title_text": "Large gaps and BTZ entropy in modular spectra with positive integer degeneracies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/5gwr4cfr2mhp8",
  "venue": "arXiv",
  "versions": []
}