{
  "abstract": "The polynomial algebra generated by the spectral moments of a family of circulant graphs consists of all eventually polynomial sequences in the graph order; as an unfiltered algebra, it therefore carries little information about the jumps. We show that its natural power-sum weight filtration retains substantially finer arithmetic data. For a finite jump set $S$ with $\\gcd(S)=1$ and $M=\\max S$, the relevant invariant is the numerical semigroup $Σ_S=\\langle M,M-s,M+s:s\\in S\\rangle$, which arises from the lengths of closed walks winding around the cycle. We prove that the eventual global filtered dimension is $MD+\\lfloor D/2\\rfloor+1-M(M+1)-g(Σ_S)$. Restricting graph orders to $n\\equiv a\\pmod q$, for $q\\mid M$, replaces the genus term by the number of gaps of $Σ_S$ in the class $-a\\pmod q$; the class $a=0$ gives the genus of the quotient semigroup $Σ_S/q$. These formulas yield an inverse reconstruction theorem: the unrestricted dimensions determine $M$, and the dimensions in all classes modulo $M$ at one common stable degree determine the Apéry set and hence $Σ_S$. We give effective thresholds, exact examples, and a finite-rank algorithm valid in every degree. Reconstruction of the semigroup need not identify the original jump set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.14316",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Diego Velez Caicedo"
    },
    {
      "name": "Carlos A. Cadavid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0t9jy52ygbnb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.14316",
  "fulltext_url": null,
  "id": "f0t9jy52ygbnb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.14316",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.14316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0t9jy52ygbnb.ris",
  "sources": [
    {
      "first_seen": "2026-11-11T06:21:55Z",
      "native_id": "2611.14316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.14316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Reconstructing Numerical Semigroups from Circulant Spectral Dimensions",
  "title_text": "Reconstructing Numerical Semigroups from Circulant Spectral Dimensions",
  "type": "preprint",
  "updated_at": "2026-11-13T06:20:11Z",
  "url": "https://research.draconitic.com/paper/f0t9jy52ygbnb",
  "venue": "arXiv",
  "versions": []
}