{
  "abstract": "For integers $h,k\\geq 1$, let $hA$ be the $h$-fold sumset of $A$ and put $\\mathcal{R}(h,k)=\\{|hA|:A\\subset\\mathbb{Z}, |A|=k\\}$. Previously, the fixed-cardinality exponent law was known only for $k\\leq 3$; every fixed $k\\geq 4$ remained open. We settle the problem in full by determining the complete fixed-cardinality exponent law: $|\\mathcal{R}(h,k)|=\\begin{cases}1,&k\\leq 2,\\\\ h,&k=3,\\\\ h^{k-1+o_k(1)},&k\\geq 4\\end{cases}$. Here $o_k(1)\\to 0$ as $h\\to\\infty$ with $k$ fixed. More sharply, for fixed $k\\geq 4$, an interval of length $Θ_k(h^{k-1})$ contains at least $h^{k-1-o_k(1)}$ attainable values. At $k=4$ we prove $|\\mathcal{R}(h,4)|=Θ(h^3)$ with positive lower density in its ambient interval, disproving Nathanson's proposed $o(h^3)$ and $O(h^2)$ bounds. One bounded addition-table geometry drives these results, coupling Hilbert-energy amplification to optimal finite-observation compression. Every ordered real $k$-set ($k\\geq 2$) has an integer model in $[0,O_k(h^{k-2})]$ preserving every sum equality and strict comparison through degree $h$; the exponent $k-2$ is sharp. The universal label-realization length is therefore $Θ_k(h^{k-2})$, one power sharper than Nathanson's $O_k(h^{k-1})$ bound. For $h\\geq 2$ and $k\\geq 3$, minimum active rank equals realization-frequency codimension, exponent-shape codimension, and sampling-rarity exponent; a full-exponent family has maximal-rank witnesses with Cohen-Macaulay toric coordinate rings. At rank zero, for $h\\geq 2$, it proves the conjectural OEIS A227589 formula $\\binom{h+2}{2}+\\mathbf{1}_{\\{2\\nmid h\\}}$ for the least normalized diameter of a four-point $B_h$-set. It also gives exact fixed-$(h,k)$ popularity laws for $k$-subsets of $\\{1,\\ldots,q\\}$ as $q\\to\\infty$, resolving Nathanson's Problems 9 and 10.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01690",
  "author_total": 1,
  "authors": [
    {
      "name": "Henry Shin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fk9em44c6wn99.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01690",
  "fulltext_url": null,
  "id": "fk9em44c6wn99",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01690",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7207531302",
  "pdf_url": "https://arxiv.org/pdf/2609.01690",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fk9em44c6wn99.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01690",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01690"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Iterated-sumset spectra: The complete exponent law and its rank geometry",
  "title_text": "Iterated-sumset spectra: The complete exponent law and its rank geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/fk9em44c6wn99",
  "venue": "arXiv",
  "versions": []
}