{
  "abstract": "We give a short proof of the density-one assertion posed by Gao for consecutive Collatz starts, independent of the first public proof, due to Said Duran (September 2026). Let C be the ordinary map and let H(n, m) count halving branches among the first m ordinary steps. Write Gk(n) when n and n+1 have, by time k, a common endpoint, equal halving counts, and have stayed above one up to that endpoint. If d̄k = 2−k·#{1 ≤ n < 2k − 1 : Gk(n)}, then d̄k → 1. A uniform absorption estimate for the 2-adic pair process gives the quantitative estimate 1 − d̄k = O(k−1/2), and the exact leading constant is determined in a companion paper. The argument concerns synchronization of two paths; it proves neither termination of every path nor the Collatz conjecture. Companion papers: Exact rates in Collatz synchronization; Quantitative synchronization of Collatz trajectories; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "David Leen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5q58b6gjv0rr7.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.23268761",
  "fulltext_url": null,
  "id": "5q58b6gjv0rr7",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23268761",
  "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",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221079292",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/5q58b6gjv0rr7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7221079292",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23268761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Mathematical Dynamics and Fractals",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Gao's dyadic density-one coalescence conjecture",
  "title_text": "Gao's dyadic density-one coalescence conjecture",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/5q58b6gjv0rr7",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}