{
  "abstract": "For a positive integer $n$, let $S(n)=\\{σ(d): d\\mid n\\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\\mapsto σ(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with $ω(n)=r$, every width from $1$ to $\\binom{r}{\\lfloor r/2\\rfloor}$ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset $D_m=(\\{1,\\ldots,m\\},\\mid)$, the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit $V(D_m)\\to 1.0367934201598\\ldots$, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02264",
  "author_total": 1,
  "authors": [
    {
      "name": "Felix Huber"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dspenpx89b52d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02264",
  "fulltext_url": null,
  "id": "dspenpx89b52d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02264",
  "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"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220405833",
  "pdf_url": "https://arxiv.org/pdf/2610.02264",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dspenpx89b52d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02264",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02264"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Antichains among Divisor Sums of Divisors",
  "title_text": "Antichains among Divisor Sums of Divisors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/dspenpx89b52d",
  "venue": "arXiv",
  "versions": []
}