{
  "abstract": "For a positive integer n let m(n) be the least value of its Collatz orbit and τ*(n) the first time the orbit reaches it; both are finite for every n, and nothing is assumed about the Collatz conjecture. We prove that the descent of a typical orbit to its minimum is a Brownian motion, independent of where the orbit ends. For n uniform in [2, X], the time to fall to or below n1−s (or to reach the minimum, if sooner), centered at μ s log n and scaled by σ√(log n), converges as a process in s ∈ [0, 1] to standard Brownian motion, jointly with and independently of m(n); here μ = 2/log(4/3) and σ² = 2(log 3)²/(log(4/3))³. The same holds on every arithmetic progression, in logarithmic density, and jointly with the residue that produces the teeth of the stopping-time histogram. Consequently half of all positive integers take at least μ log n steps to reach their minimum, and the fractions of the descent spent above its chord and behind its average schedule follow the uniform and arcsine laws. For fixed 0 < s < 1 the clock has Gaussian moderate deviations with relative error up to K√(log n log log n) for every K, and for s > 1/1001 also given a fate of positive density. Every set invariant under the map above some point has a natural density. The argument uses no numerical computation. This paper extends A Gaussian law for Collatz total stopping times (Zenodo 23046896). Companion paper: Gaussian but not locally Gaussian.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "David Leen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gdjgsm1jwbg5.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.23252394",
  "fulltext_url": null,
  "id": "9gdjgsm1jwbg5",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23252394",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221196302",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/9gdjgsm1jwbg5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7221196302",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23252394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Brownian motion in Collatz orbits",
  "title_text": "Brownian motion in Collatz orbits",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/9gdjgsm1jwbg5",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}