{
  "abstract": "We study cyclotomic cosine evaluations beyond the fully symmetric setting by combining Galois-orbit structure with finite-frequency expansions. First, for truncation-compatible poly?nomial families over Q, we distinguish invariance of the evaluated value from invariance of the polynomial formula. Summability by a rational function of the level forces eventual rationality of the evaluated values. Conversely, every rational-valued sequence can be realized by a compatible family whose total degree is uniformly bounded by two, showing that value-level Galois invariance alone carries essentially no stable-range rigidity. We then introduce explicit orbit-structured pattern sums indexed by multiplicative configurations. For every fixed pattern, at each sufficiently large admissible level n, its cyclotomic cosine evaluation collapses to an affine function n times S0(r,h) minus 2 to the abs(h) power, where S0 is the zero-frequency coefficient of the associated Laurent polynomial. Consequently, every family uniformly generated from finitely many such patterns by a single polynomial recipe has eventually polynomial evaluation. Finally, decomposing the punctured index set into gcd-orbits yields an exact orbitwise formula in terms of Ramanujan sums. The semiprime and prime-power cases become transparent specializations of this general identity. These results identify both the obstruction to extending symmetric rigidity from value-level Galois invariance and a concrete non-symmetric class exhibiting uniform stable-range collapse",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27084",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan D. Vélez"
    },
    {
      "name": "Carlos A. Cadavid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c417aejht7j5b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27084",
  "fulltext_url": null,
  "id": "c417aejht7j5b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27084",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.27084",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c417aejht7j5b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27084",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27084"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas",
  "title_text": "Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/c417aejht7j5b",
  "venue": "arXiv",
  "versions": []
}