{
  "abstract": "Let $φ=(1+\\sqrt5)/2$. The four-point set $\\{0,φ,φ^2,φ^3\\}$ is classical: up to a dilation it consists of the two endpoints and the two crossing points on a side of the regular pentagram (Euclid, Elements XIII.8), and, by the identity $φ^2=φ+1$, its six pairwise differences are $1,φ,φ^2,φ^3$, with multiplicities $1,2,2,1$. We write this configuration as the golden quadruple $S=(F, F-a, F-b, F-a-b)$, where $u>0$, $a=φu$, $b=φ^2u$ and $F>a+b$, and study its extremal and rigidity properties. We prove that it is maximal and unique: no five distinct real numbers have all their positive differences in $\\{u,φu,φ^2u,φ^3u\\}$ (indeed, not even in the whole set $\\{uφ^n:n\\in\\mathbb{Z}\\}$), and every four-element set with this property is a translate of $S$. We prove that the golden ratio is forced: $φ$ is the only ratio $x>1$ for which some four-point set of reals has the four-term progression $\\{c,cx,cx^2,cx^3\\}$ as its set of distinct differences. For the analogous quadruple with base $x>1$, $x\\neqφ$, the failure is measured by two closure residues, $(x^2-x-1)u$ and $x(x^2-x-1)u$, which along Fibonacci quotients equal $\\pm u/F_n^2$ and $\\pm F_{n+1}u/F_n^3$. Finally, the cosine sum $C_S(t)=\\sum\\cos(2πd t)$ over the six differences equals $8\\cos^2(πa t)\\cos^2(πb t)-2$; it is not periodic, but it is almost periodic in the sense of Bohr, with explicit almost periods $F_n/u$. For an arbitrary scale $K>0$ the golden ladder $ν_n=Kφ^{-n}$ realises the construction with $u=ν_{203}$, so that the differences are $ν_{200},\\dots,ν_{203}$; we evaluate it for $K=\\sqrt{c^5/(Gh)}$. We state explicitly what is classical and what is new, in relation to difference sets, Golomb rulers, Sidon sets, Fibonacci numbers, quasicrystals and almost periodic functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00973",
  "author_total": 1,
  "authors": [
    {
      "name": "Michał Przybylski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ge21axzy4sv8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00973",
  "fulltext_url": null,
  "id": "6ge21axzy4sv8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00973",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ge21axzy4sv8.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum",
  "title_text": "The Golden Quadruple: Four Reals Whose Differences Form a Geometric Progression, Rigidity of the Golden Ratio, and a Bohr Almost Periodic Cosine Sum",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/6ge21axzy4sv8",
  "venue": "arXiv",
  "versions": []
}