{
  "abstract": "Let $T(n)=n/2$ if $n$ is even and $T(n)=(3n+1)/2$ if $n$ is odd. We prove that for each $m\\ge1$, exactly $F(m+1)$ odd integers $n$ in $\\{1,\\ldots,2^m\\}$ have the property that none of the iterates $T(n),T^2(n),\\ldots,T^{m-1}(n)$ lies in the residue class $4\\pmod6$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(φ/2)^m$, $φ=(1+\\sqrt{5})/2$. Equivalently, these are the odd $n\\le2^m$ for which no two consecutive terms of $n,T(n),\\ldots,T^{m-1}(n)$ are even. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\\{1,2,4,5\\}]$. Removing vertex $4$ from $G'$ yields a subgraph of spectral radius $φ$, against $ρ(G')=2$; the Fibonacci count follows from this spectral gap. We construct an explicit bijection $Ψ_m:\\{1,\\ldots,6\\cdot2^m\\}\\to\\mathcal{P}_m(G)$ onto the directed paths of length $m$ in $G$. We further show that no vertex of $G'$ is dispensable: removing any single vertex reduces the spectral radius strictly below $2$, with hierarchy $1<\\sqrt{2}<φ<2$. In particular, every positive cycle of $T$ must visit residue class $2\\pmod6$, and a flow conservation identity forces this class to account for more than $18\\%$ of the steps in any such cycle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.02621",
  "author_total": 1,
  "authors": [
    {
      "name": "Manuel-Alejandro Reyes Jiménez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1k3ggp7nczvex.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.02621",
  "fulltext_url": null,
  "id": "1k3ggp7nczvex",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.02621",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7163401101",
  "pdf_url": "https://arxiv.org/pdf/2606.02621",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1k3ggp7nczvex.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2606.02621",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.02621"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Fibonacci theorem for Collatz trajectories via modular graph structure",
  "title_text": "A Fibonacci theorem for Collatz trajectories via modular graph structure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1k3ggp7nczvex",
  "venue": "arXiv",
  "versions": []
}