{
  "abstract": "Let C be the Collatz map, C(n) = n/2 for even n and C(n) = 3n+1 for odd n. Following Gao (1993), two consecutive integers n and n+1 coalesce if C^k(n) = C^k(n+1) for some k, with the same number of halvings on both trajectories and without either trajectory passing through 1. Gao computed the normalized count d̄_k of the n < 2^k − 1 for which this happens within k steps, proved that d̄_k is nondecreasing, and conjectured that d̄_k → 1. In the form recorded by Lagarias, the conjecture states that the set of n with C^k(n) = C^k(n+1) for some k ≤ log_2 n has natural density one. We prove both statements, and also that the proportion d(x) of n < x for which n and n+1 coalesce tends to 1; Gao did not know whether this limit exists. The proof follows a pair of trajectories through a Markov chain on states {v, 3^k v + c} driven by fair coin flips, which describes the residue classes modulo 2^t exactly, and shows that the chain is absorbed almost surely. The ingredients are a contraction of the square root of |c|/3^k, a harmonic function for the level k, a clock controlling long runs of equal parity, a window in which the offset returns to a bounded range, a universal absorbing word of length O(log^2(2+|c|)), and a counting argument for repeated trials that never conditions on the future. A short lemma shows that a meeting within log_2 n steps is already a coalescence in Gao's sense. Every step is formalized in Lean 4 with Mathlib, and the final theorems depend only on the axioms propext, Classical.choice and Quot.sound. The results concern coalescence only: they do not show that any integer reaches 1, and the convergence is extremely slow.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Omar Javier Said Duran",
      "orcid": "0009-0009-1418-2558"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egsztbgy5thxt.bib",
  "cited_by": 9,
  "doi": "10.5281/zenodo.23003523",
  "fulltext_url": null,
  "id": "egsztbgy5thxt",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.23003523",
  "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": "W7214676410",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/egsztbgy5thxt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7214676410",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23003523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Mathematical Dynamics and Fractals",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Consecutive integers coalesce with density one under the Collatz map: a formally verified proof of a conjecture of Gao",
  "title_text": "Consecutive integers coalesce with density one under the Collatz map: a formally verified proof of a conjecture of Gao",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:21Z",
  "url": "https://research.draconitic.com/paper/egsztbgy5thxt",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}