{
  "abstract": "We study the circle-relation subspace $O_n^\\circ(V)$ in the defining relations of the level-$n$ Zhu algebra of a vertex operator algebra $V$. We assemble the residue relations generating $O_n^\\circ(V)$ into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least $-2n-1$, modulo $O_n^\\circ(V)$. This recursion is encoded in a left truncated field $Y_L$. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for $Y_L$ needed for finite-length calculations. For a finite homogeneous strong generating set $S$, we define the finite-length circle subspaces $O_n^\\circ(N,S,V)$ and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the $k=0$ strip and to last vacuum mode $-1$, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for $n=1$ and $n=2$. The $Y_L$ identities are the input for the companion paper (Part II).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19501",
  "author_total": 1,
  "authors": [
    {
      "name": "Nathan Vander Werf"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eq9yx0femr8s6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19501",
  "fulltext_url": null,
  "id": "eq9yx0femr8s6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19501",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7222174051",
  "pdf_url": "https://arxiv.org/pdf/2610.19501",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eq9yx0femr8s6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19501",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19501"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On generators and relations for $O_n^\\circ(V)$ in higher-level Zhu algebras, Part I",
  "title_text": "On generators and relations for O_n^∘(V) in higher-level Zhu algebras, Part I",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/eq9yx0femr8s6",
  "venue": "arXiv",
  "versions": []
}